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Density corrections to transport coefficients from time correlation functions Alavi, Saman 1999

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DENSITY CORRECTIONS TO TRANSPORT COEFFICIENTS FROM TIME CORRELATION FUNCTIONS by Saman Alavi B . S c , Shiraz University, 1988 M . S c , Isfahan University of Technology, 1992 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Chemistry) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A June 1999 © Saman A l a v i , 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of CZk&r\i$-lYy The University of British Columbia Vancouver, Canada Date ZSiAW DE-6 (2/88) A b s t r a c t A new method for deriving first order density corrections to transport coefficients using projection operators in the time correlation function formalism is developed. Low and moderately dense gas transport coefficients are standardly calculated from a form of the generalized Boltzmann equation. This equation being solved to first order density corrections for repulsive potentials at the binary collision level by Snider and Curtiss and later extended to include the effects associated with the static presence of a third particle on a binary collision by Hoffman and Curtiss. Rainwater and Friend added extra contributions for the presence of bound pairs when the molecules have an attractive potential. They utilized the Stogryn - Hirschfelder theory for the bound pairs and performed detailed numerical calculations of the resultant formulas. While the numerical calculations give good agreement with experiment, questions remain as to the nature and rigor of the assumptions made in obtaining the final formulas, especially the ad hoc addition of bound pair contributions to the density corrections of systems with repulsive potentials, and the extent that these approximations affect the final numerical results. To study these questions, the time correlation function formulas for the transport coeffi-cients were chosen as an alternative route to determine first order density corrections. The time correlation formulas are formally exact and so the density corrections can be usefully compared to those of the generalized Boltzmann equation. Kawasaki and Oppenheim had previously derived formal expressions for first order density corrections to the shear viscosity for a gas of molecules with a repulsive potential, but their results had not been reduced to a form that could be directly compared to those of Snider and Curtiss. As a first step in the study of the time correlation function formalism, the density corrections of Kawasaki and Oppenheim are shown to be equivalent to those of Snider and Curtiss along with an additional correction for three-body collisions. The projection operator method developed in this thesis does not have the infinite series resummation procedure used by Kawasaki and Oppenheim and is an alternative route to obtaining density corrections from the time correlation functions. At low pressures, pro-jection operators are defined which only consider kinetic contributions to the flux function and expressions for the lowest order transport coefficients along with their higher moment corrections are derived. These expressions are consistent with the solution of the Boltzmann equation. ii The first order density correction from bound pairs on the transport coefficients are approximated by treating the system as a binary gas mixture consisting of free molecules and bound pairs. The results of viewing the system from the molecular picture and the atomic picture with appropriate projection operators are shown to be consistent with one another and also with the Boltzmann equation for binary mixtures. Density corrections in moderately dense gases also arise from potential contributions to the flux. Projection operators which account for both the kinetic and potential flux contributions are required in order to derive explicit expressions for the first order density corrections to the viscosity and thermal conductivity. It is observed that these corrections are consistent with those of Snider and Curtiss with the added Hoffman and Curtiss correction and a term which takes explicit account of three-particle collisions. In the treatment of mixtures and potential interaction effects, the calculation of a trans-port coefficient is reduced to an equivalent matrix inversion problem. The binary collision expansion of the respective resolvent in the matrix elements in these formulas allows the transport coefficient to be expressed in terms of integrals over functions of the intermolecu-lar potential. The projection operator for each system is determined in a straightforward manner with reference to the particular flux tensor in the time correlation formula. Reduction of the gen-eral formula to relations suitable for numerical calculation involves the resolvent expansion onto the appropriate projected subspace, and the subsequent binary collision expansion to reduce the iV-particle resolvent to a tractable form. 111 Contents Abstract ii Table of Contents iv List of Figures viii Acknowledgements ix 1 Introduction 1 2 Overview of the Kinetic Theory of Transport Coefficients 5 2.1 Introduction 5 2.2 Shear Viscosity: A General Survey 6 2.3 Generalized Boltzmann Equation for Moderately Dense Gases 14 2.3.1 Moderately Dense Gas Transport Coefficients from the Generalized Boltzmann Equation 19 2.4 Time Correlation Functions for Transport Coefficients 24 2.4.1 Moderately Dense Gas Transport Coefficients from Time Correlation Functions 27 2.5 Binary Collisions 29 2.5.1 Collision Parameters 29 2.5.2 Collision Dynamics 35 2.5.3 Collision Operators 37 3 Moderately Dense Gas Shear Viscosity from the Kawasaki - Oppenheim Approach 39 3.1 Formulation for the Shear Viscosity 40 3.2 Contributions to 77(e) 41 3.3 Equivalence of the Two Approaches 47 i v 3.3.1 Evaluation of b 0 ; 48 3.3.2 Evaluation of 49 3.3.3 Collisional Transfer Contribution 50 3.3.4 Triple Collision Contribution 51 3.3.5 Shear Viscosity 51 3.4 Inclusion of Bound Pairs 52 3.4.1 Viscosity of a Mixture of Free Molecules and Bound Pairs 53 3.4.2 Solving for the B^s 54 3.5 Kawasaki and Oppenheim's Modifications Due to Attractive Forces 58 4 Complete Binary Collision Approximation for Dilute Gas Transport Co-efficients via the Time Correlation Formulation 60 4.1 Rotational Invariance of the Resolvent Matrix Elements 62 4.2 Projection Operators 65 4.3 Projection Operator - Resolvent Relations 66 4.4 One Particle Properties at Low Density 69 4.5 The Complete Sonine Expansion 75 4.6 Transport Coefficients of a Binary Mixture 77 4.7 Discussion 80 5 Bound-Free Gas Transport Coefficients via the Time Correlation Formu-lation based on an Atomic Picture 82 5.1 Introduction 82 5.2 Distribution Functions for an A^-Particle System with Unbound Particles and Bound Pairs 84 5.3 Transport Coefficients for a System With Bound Pairs 86 5.3.1 Free Atom Contributions 89 5.3.2 Bound Pair Contributions 93 5.3.3 Solving for By and B 6 95 5.4 The Kawasaki - Oppenheim Treatment of Bound States 97 5.5 Discussion 99 6 Moderately Dense Gas Shear Viscosity Coefficient via the Time Correla-tion Formulation 101 6.1 Introduction 101 6.2 Distribution Function for the Af-Particle System and Shear Viscosity . . . . 102 v 6.3 Projection Operators for Shear Viscosity 103 6.4 Reduction to Projected Liouville Superoperators 107 6.5 Binary Collision Expansions for the Matrix Elements 110 6.5.1 Expansion of K^v I l l 6.5.2 Expansion of KKV 114 6.5.3 Expansion of KKK 115 6.5.4 Evaluation of Wvv 122 6.5.5 Expansion of VVK 122 6.5.6 Expansion of VyK 123 6.6 First Order Density Corrections Within the Two Moment Approximation . . 124 6.7 Discussion 125 7 Moderately Dense Gas Thermal Conductivity Coefficient via the Time Correlation Formulation 129 7.1 Introduction 129 7.2 Thermal Conductivity 129 7.3 Projection Operators for Thermal Conductivity 130 7.4 Reduction To Projected Liouville Superoperators 134 7.5 Binary Collision Expansions for the Matrix Elements 135 7.5.1 Expansion of K^v 136 7.5.2 Expansion of K^v 139 7.5.3 Expansion of KKK 140 7.5.4 Evaluation of wvv 144 7.5.5 Expansion of VVK 144 7.5.6 Expansion of V^K 145 7.6 First Order Density Corrections Within the Two Moment Approximation . . 146 8 Discussion and Conclusions 148 Bibliography 150 A Identification of Fv 156 B Identification of Nv 159 C Binary Mixture Viscosity Cross Sections 161 D Determination of the Fourth-Rank Identity Tensor 162 vi E Evaluation of C 164 F Determination of Hn + Rv 165 G Evaluation of Fx 168 H Evaluation of Nx 170 I The Second Virial Coefficient and its Temperature Derivatives 172 J Determination of H\ + R\ — I\ 173 List of Symbols 176 vii L i s t of F i g u r e s Figure 1. The force F on the surface A establishes a velocity gradient perpendicular to the direction of the flow 7 F igure 2. Random motion of molecules in different layers acts to dissipate the momentum gradient among them. There is no net transfer of mass between the layers 9 F igure 3. Collisional transfer of momentum from the fast layer to the slow layer, without an exchange of particles 11 F igure 4. Plots of the different contributions to the reduced second viscosity virial coeffi-cient for argon 13 F igure 5. In the Boltzmann equation, the collision is assumed to occur at a single point r. 15 F igure 6. Position coordinates during a binary collision relevant to the generalized Boltz-mann equation. The radius of the large circle is the cutoff distance of the potential 17 F igure 7. The coordinate system for a binary collision. A reference point of the trajectory with position r and momentum p, and also a general point on the trajectory with position £ and momentum are indicated on the diagram. Various angles defined in reference to these points are also shown 31 F igure 8. Binary collision parameters. R is the cutoff distance of the potential and r m is the closest distance of approach. Various angles are defined with reference to these vectors 33 F igure 9. Primed position variables and their related collision parameters, b is the impact parameter 34 vii i A c k n o w l e d g e m e n t s I owe a depth of gratitude to many individuals without whose direct or indirect assistance, completion of this work would not have been possible. First and foremost, I wish to express sincere thanks to my research supervisor, Professor R. F . Snider. I clearly recall meeting Prof. Snider on the first day of coming to U B C and since then, through his rigorous standards, hard work and exceptional command on the one hand, and his unwaivering patience and enthusiasm on the other, has taken me through the often difficult, but always fascinating field of kinetic theory. The climb has at times been demanding, but under his guidance I have occasionally reached openings in the path and glimpsed at the landscape and marveled at the scenery below. It has indeed been a pleasure and a privilege to work under Prof. Snider's supervision. Professors at U B C with whom I have had the opportunity of working with also deserve my gratitude. I have taken courses with Profs. J . A . R. Coope, M . C . Gerry, L . G . Harrison, M . McMil lan , A . J . Merer, B. Shizgal, R. F . Snider, J . Trotter and I. Waller, all of which contributed to my scientific knowledge and outlook. It has been a pleasure interacting with Prof. M . Thachuk on a daily basis in matters scientific and otherwise. I would like to thank Profs. G . N . Patey, B. Shizgal and M . Thachuk for organizing the excellent Friday seminars of the Physical/Theoretical groups at the Department of Chemistry at U B C . I would also like to acknowledge Drs. B . Clifford and Y . Koga for the many practical points learned while working as a teaching assistant in their labs. The financial support of the University of British Columbia and the Department of Chemistry at U B C is very gratefully acknowledged. This support made it possible for me to come to U B C and through a university graduate fellowship lightened my teaching load to a significant extent. From the days prior to my coming to U B C , work with Profs. A . Boushehri, M . H . Dehghani and N . Maleki of Shiraz University and Profs. B . Najafi and G . A . Parsafar (my M.Sc. supervisor) of the Isfahan University of Technology has left a lasting impression on me and has my gratitude. I would also like to thank Profs. Najafi and Parsafar for the numerous occasions in which they have also offered guidance and assistance in non academic matters. Teachers from earlier stages to whom I owe gratitude are too numerous to mention! O f the visitors to our research group, discussions with Drs. J . G . Muga, B . Najafi, J . C . Rainwater and R. E . Turner are acknowledged and some work with them is still producing interesting results. I've enjoyed many stimulating discussions on science and many other ix topics with our former research group member Prof. G . W . Wei. It was through discussions with Guowei, that I took the first steps towards contributing to kinetic theory. M y present lab mate, Dr. R. Baranowski deserves my gratitude for among other things the numerous times he has offered help with my computer problems. I thank Dr. J . F . Kenney and M r . M . Gettner for their friendship and for introducing me to into fields I would not have otherwise explored. Knowing B. C . Cha, A . and K . Ghandi, A . and C . King, M . Kono, B . Kuipers, R. and T . Murdey, F . Pacheleh, and M . and Y . Terazono has been one of the most enjoyable aspects of my stay in Vancouver. The friendship of M . Lang and E . Watanabe is very valuable to me. Eriko and Michael have greatly enriched my life. To my family, my father Mohammad, mother Zari, sisters and brother Sepideh, A l i and Niloufar, I owe infinite gratitude for their support and encouragement throughout my studies. Their letters, calls and e-mails have always been a source of energy. Finally, I dedicate this thesis to the memory of my cousin, Mehdi Alavi . Chapter 1 Introduction The rigorous theory of density corrections to transport coefficients for moderately dense gases has been studied since the late 1950's. Distinct formalisms based on generalizations of the Boltzmann equation and time correlation functions have been used to approach this problem. These developments are summarized briefly in this introduction with relevant references and further details presented in the following chapters. Historically, studies in this direction began with the generalizations of the Boltzmann equation due to N . N . Bogoliubov who provided a systematic method of extending the Boltzmann equation to all orders of density and with the work of H. S. Green who in-dependently developed a physical method to include first order density corrections in the Boltzmann equation. The Bogoliubov approach involves proposing a generalization of Boltz-(s) mann's Stosszahlansatz to express the s-particle reduced distribution functions p\...s, in terms of products of the one-particle distribution functions pj\ j = 1, • • • s at times in the distant past and also assuming that at all other times, higher order distribution functions can be expressed as functionals of the p^ 's . Choh and Uhlenbeck implemented this method to obtain a set of kinetic equations which includes up to ternary collisions among particles. Cohen reformulated the Bogoliubov theory in terms of cluster expansions of the B B G K Y hierarchy so as to remove a number of objections with regard to the functional assumption. The solution of the generalized Boltzmann equation within the Chapman - Enskog frame-work has been achieved subject to different approximations. To make the general problem tractable, both the method of generalizing the Boltzmann equation and the finding of the solution of the resultant equations require simplifying assumptions. In particular Snider and Curtiss limited the intermolecular interactions to repulsive potentials and binary collisions, and Hoffman and Curtiss included some three-particle effects in the derivation. Necessarily, it is also not clear to what extent these approximations have affected the form and magnitude of the final results. 1 At approximately the same time, formalisms for determining transport coefficients from time correlation functions appeared. The initial formulas for the time correlation function approach are general and valid to all orders of density, but in order to solve these general expressions, simplifying approximations have to be made. Historically, the equality of the transport coefficients as calculated via the Boltzmann equation and the time correlation functions has been investigated by many people, especially within the lowest order approximation. But the question also arises as to the relation between the expressions for the density corrections from the generalized Boltzmann equation and from the time correlation function formalism. Advances to answering this question were made by starting from the time correlation functions and relating them to a set of kinetic equations which are then shown to be equivalent to the generalized Boltzmann equation of corresponding order in density. But the method of treating the kinetic equations in the time correlation function formalism is the same as that used in deriving the generalized Boltzmann equations so this comparison does not appear to be completely independent. In this manner the equivalence of the time correlation function formalism with the kinetic equation approach has been demonstrated. Once the comparison of the equations is done, there is still the question of solving the kinetic equations. Presumably, the Chapman -Enskog solution can be used to complete the derivation of the transport coefficient. While this helps to clarify some of the assumptions and conditions under which the generalized kinetic equations are derived, it still leaves unanswered questions regarding the common approximations made in both formalisms. The generalized Boltzmann equations and the equivalent time correlation function ex-pressions contain collision integrals over operators which for example are convergent when the potential is repulsive. If the potential is attractive, a number of these integrals diverge when calculated for bound states and different approximations have been introduced to account for these problems. These approximations have all been introduced within the framework of the generalized Boltzmann equation. Despite giving rise to reasonable numerical estimates of the first order density corrections, the rigor of the assumptions that have gone into the derivation of the final expressions for the transport coefficients in the presence of attractive forces is under question. To our knowledge, a problem-free study of similar scope has not been performed using the time correlation function approach. It was with these questions in mind that the following thesis was proposed. It was decided to start with the time correlation function formulas and try to develop a method for their evaluation which is independent of the Chapman - Enskog solution, but in a final form that is suitable for numerical calculation. Simplifying assumptions would undoubtedly 2 be necessary, bu t a contras t of the m e t h o d (and results) w o u l d he lp to c la r i fy the ex i s t i ng ques t ions r ega rd ing the r igor of the approaches a n d the range of v a l i d i t y o f the results . T h e s t u d y involves a number of steps w h i c h are presented i n th is thesis a n d w h i c h r o u g h l y fo l low our ch rono log ica l development of the work . A f t e r a chapter of p r e l i m i n a r i e s i n w h i c h the m a t e r i a l of th i s i n t r o d u c t i o n is expanded a n d relevant references are g iven , the m a i n s t u d y begins. T h i s thesis owes a lo t to the w o r k of K a w a s a k i a n d O p p e n h e i m w h o used the t i m e c o r r e l a t i o n func t i on app roach to o b t a i n expressions for the first order dens i ty cor rec t ions to the v iscos i ty . T h e y es tabl i shed the equivalence o f the i r f ina l equat ions to those of the C h o h a n d U h l e n b e c k a p p l i c a t i o n of B o g o l i u b o v ' s approach , but they d i d not reduce the i r expressions fur ther . In C h a p t e r 3, i t is demons t ra t ed tha t the f ina l equat ions of K a w a s a k i a n d O p p e n h e i m (and thus C h o h a n d Uhlenbeck) indeed reduce to resul ts s i m i l a r to the C h a p m a n - E n s k o g so lu t i on o f the genera l ized B o l t z m a n n equa t ion for repuls ive po ten t i a l s as o b t a i n e d by Sn ide r a n d C u r t i s s w i t h an added three-par t ic le c o l l i s i o n c o n t r i b u t i o n . In t h i s chapter , a p p r o x i m a t i o n s w h i c h can be used to ex tend the app roach to i nc lude a t t r ac t i ve po ten t i a l s are proposed . P r o b l e m s we feel exist i n the K a w a s a k i a n d O p p e n h e i m theory for the r igorous i n c l u s i o n of a t t r ac t ive po ten t ia l s a n d b o u n d states are also discussed. T h e m a i n b o d y o f th i s w o r k is presented f r o m C h a p t e r 4 onwards . I n C h a p t e r 4, a pro-j e c t i o n opera tor f o r m a l i s m is i n t r o d u c e d to o b t a i n expressions for the ze ro th order dens i ty t r an spo r t coefficients a n d the i r Son ine p o l y n o m i a l correct ions . T h i s verifies the w e l l k n o w n fact tha t the ze ro th order t r anspor t coefficients of the B o l t z m a n n equa t i on are i d e n t i c a l to those of the t i m e co r r e l a t i on func t ion expressions. T h e expressions for t r a n s p o r t coefficients of b i n a r y m i x t u r e s are also o b t a i n e d w i t h the p ro j ec t i on opera to r f o r m a l i s m . T h e b i n a r y m i x t u r e v i e w p o i n t is one of the ways of a p p r o x i m a t i n g the effects of a t t r ac t i ve forces. A t -t r ac t i ve forces can l ead to the f o r m a t i o n of b o u n d pa i rs w h i c h change the de t a i l ed na tu re of the fluxes i n the sys tem. In the s imples t case, the number o f b o u n d pa i rs are a p p r o x i m a t e d by an e q u i l i b r i u m s t a t i s t i ca l m e c h a n i c a l c a l cu l a t i on , a n d the sys t em is cons idered to be a b i n a r y m i x t u r e of an appropr i a t e number of u n b o u n d par t ic les a n d b o u n d pa i r s . In C h a p t e r 5 the i n c l u s i o n of b o u n d pa i rs i n the sys t em f rom a n a t o m i c v i e w p o i n t is presented. T h i s c a n be considered to be a p r e l i m i n a r y step to the r igorous t r ea tment of a t t r ac t i ve po ten t ia l s . T h e results are the same as v i e w i n g the sys t em as a b i n a r y m i x t u r e , bu t the b o u n d pa i rs are cons idered to be composed of the c o n s t i t u t i n g par t i c les of the sys tem, a n d not jus t a general second species as is done i n the b i n a r y m i x t u r e p o i n t of v i e w . In C h a p t e r s 6 a n d 7, the in t e rmolecu la r p o t e n t i a l is i n c o r p o r a t e d in to the p ro j ec t i on ope ra to r f o r m a l i s m a n d expressions for the first order dens i ty cor rec t ions to the t r anspo r t 3 coefficients are de te rmined . C h a p t e r 6 deals w i t h general aspects a n d the i r a p p l i c a t i o n to the shear v i scos i ty w h i l e C h a p t e r 7 deals w i t h the a p p l i c a t i o n to t h e r m a l c o n d u c t i v i t y . T h e a s sumpt ions m a d e a n d te rms neglected are enumera ted a n d ways of i m p r o v i n g o n t h e m seem c lear i n t h i s f o r m a l i s m . It r emains to pu t these aspects together to derive a f o r m a l i s m w h i c h inc ludes b o t h the effects of the i n t e rmo lecu l a r in te rac t ions a n d the b o u n d pai rs i n a u n i f o r m manner . W e feel t ha t t h i s thesis lays the g r o u n d w o r k for th is . 4 Chapter 2 Overview of the Kinetic Theory of Transport Coefficients 2.1 Introduction The state of a fluid at equilibrium is completely specified on a macroscopic scale by the density of the components and the temperature, which are uniform throughout the system [1]. Wi th the equation of state, and the use of the three laws of thermodynamics, the hydrostatic pressure and other thermodynamic properties of the fluid are determined. From a microscopic point of view, equilibrium is associated with a Maxwellian distribution of velocities for each of the species with constant temperatures, and a positional dependence of the particles determined by the Boltzmann distribution associated with the intermolecular potentials between the particles. In a non-equilibrium fluid, the densities of the components, temperature, and/or pressure can be different in different parts of the system. The equation of state no longer necessarily relates these quantities. Moreover, if the fluid is flowing, the pressure can have directional properties and must be described by a pressure tensor. A single scalar hydrostatic pressure is no longer sufficient. As a consequence of being out of equilibrium, density, temperature and/or velocity gra-dients arise in the fluid which cause macroscopic flows. Density and concentration gradients lead to mass and species flow, temperature gradients to heat flow, and velocity gradients to momentum flow. The flows may have conductive (or diffusive) and convective compo-nents. The rates of the conductive transport for these flows are determined by the diffusion, thermal conductivity and viscosity coefficients respectively, which are the simplest transport coefficients for an electrically neutral, non-reacting fluid. Coupling between flows can occur, leading to other transport coefficients. If the gradients are not sustained by external sources, 5 they dissipate and the system relaxes to a uniform equilibrium state. The microscopic de-scription of these phenomena for gases is the subject of gas kinetic theory. If the fluid is not too far from equilibrium, macroscopically small, but microscopically large elements of fluid can still be assigned a set of densities for the components, a tempera-ture, a pressure and a flow velocity. Such volume elements are said to be in local equilibrium. Within this approximation a qualitative description of the microscopic origin of the shear viscosity is given in the next section. In this chapter, and indeed in most of this thesis, any molecular internal states are assumed unaffected by the flow and ignored in the discussion. 2.2 Shear Viscosity: A General Survey A n insight into the physical aspects of the theory developed in this thesis can be gained by first describing the phenomena under study. Of the various transports, shear flow is chosen for this purpose. Viscosity (Latin viscum: birdlime) is the phenomenon of the dissipation of a momentum gradient in a fluid system. Just as friction produces a resistive force to the motion of rigid bodies which counters their motion, viscosity opposes non-uniform flow in a fluid. Due to so called "no slip" boundary conditions [2], viscosity is always present when a fluid is flowing adjacent to a stationary object (which may be the walls of its container). Viscosity con-verts the ordered, directional macroscopic flow of the fluid to random microscopic molecular motions which are manifested as an increase in temperature of the fluid. The viscosity coefficient is defined for fluids that are microscopically not too far from equilibrium. For a fluid flowing in a container, no slip boundary conditions cause the fluid adjacent to the boundary to be stationary while further away from the wall, the velocity is non-zero. This situation is represented in Fig. 1 by a lengthening of the momentum vectors upon moving away from the wall. The actual momentum profile in a fluid depends on the nature of the flow process. Newton's Law of Viscosity, it = <"> defines the viscosity coefficient 77 in the simple geometric arrangement shown in Fig. 1, as the proportionality constant between the directional tangential force per unit area Fxz/A, required to maintain a velocity gradient, and the magnitude of that velocity gradient dvx/dz perpendicular to the direction of the flow [3]. The viscosity coefficient of a pure fluid depends on the nature of the substance, the temperature, and at high pressures (more than about 1.5 MPa) the density. Since the 6 Figure 1. The force F on the surface A establishes a velocity gradient perpendicular to the direction of flow. 7 fluid in a flow process is not in equilibrium, the definitions of its temperature and density require some care. The temperature and density are assumed to be local properties of macroscopically small but microscopically large volume elements in the fluid. In general, these properties are not equal in different volume elements in the fluid, although in simple flow experiments they can be made constant throughout the fluid. Thus, for example, when viscosity is measured experimentally in the laboratory [4], the temperature and density of the fluid are kept uniform. From a macroscopic point of view, analogous to the virial expansion of equilibrium thermodynamic properties, the temperature and density dependence of the viscosity of a gas can be expressed as a virial expansion, which at least to the first order in density is well defined, V = VQ(T)[l + nBv(T)+ ...]. (2.2) T]Q is the low density, ideal gas limit to the viscosity, Bn is called the second viscosity virial coefficient and n is the number density of the gas. Inclusion of terms up to the second viscosity virial coefficient suffices for representing the viscosity for pressures up to about 10 MPa . For comparison, terms up to the third pressure virial coefficient are adequate for the description of equilibrium properties of a gas in this range [ 5 ] . A qualitative description of the microscopic nature of the phenomena contributing to r/o(T) and BV(T) follows. Consider as in Fig. 2, two adjacent layers of the fluid in a laminar flow, one having a larger momentum in the direction of flow than the other. If the width of the layers is of the order of the mean free path, the random nature of molecular motions guarantees that a portion of the molecules from each layer wander into the adjacent layer. Molecules from the fast layer entering into the slow layer tend to increase the average momentum of the slow layer. On the other hand, a similar number of molecules from the slow layer enter the fast layer, decreasing its average momentum. The net effect of this transfer of molecules is a decrease in momentum for the fast layer and an increase in momentum for the slow layer. This mechanism for momentum transfer between layers is purely kinetic in nature. The initial velocity gradient is dissipated through the operation of this momentum transfer and in the absence of a tangential external force, the velocity gradient dissipates and the fluid "decays" to an equilibrium state with zero net momentum flow. The average distance a molecule moves before colliding is determined by the mean free path which is inversely related to a collision cross section. A larger collision cross section implies a smaller distance over which the transverse component of the momentum flow is transferred between layers, effectively a smaller transfer of momentum and thus a decrease in the viscosity. At higher temperatures, the particles have a larger momentum and thus transfer more momentum between layers and the viscosity of a gas becomes larger. It is emphasized that in this 8 Figure 2. Random motion of molecules in different layers acts to dissipate the momentum gradient among them. There is no net transfer of mass between layers. 9 process there is a net transfer of momentum between layers without a net transfer of mass. At higher densities, mechanisms involving the intermolecular potential also contribute to the transverse momentum flux in a fluid. The simplest effect is due to spatial correlations among the particles and is related to the non-ideality in an equilibrium gas. For a given pressure, the density of an ideal gas is different from that of a real gas. The mean free path in a gas is inversely proportional to the density and a correct value for the mean free path will not be obtained if the ideal gas estimate for the density is used. A correction for gas non-ideality must be introduced at higher densities and to first order in density, this leads to a correction to the viscosity that is proportional to the second pressure virial coefficient. A second mechanism which affects momentum transfer at higher densities is referred to as collisional transfer. As illustrated for hard spheres in Fig. 3, collisions can occur in which the centers of mass of both molecules stay within their respective layers. In a binary collision, a molecule with greater momentum loses some of its momentum to the molecule with less momentum. Thus momentum is transferred from the fast layer to the slow layer without an exchange of molecules. As the gas becomes more dense and the mean free path decreases, contributions from these collisions become more important in transferring momentum between layers. The same mechanism for momentum transfer applies to molecules with smooth long range forces. A molecule in the fast layer may exert a force over a distance on a molecule in the slow layer (and vice versa). This force leads to a transfer of momentum between the layers without exchange of molecules. Collisional transfer becomes predominant at high densities. In liquids, where motions are much more restricted, collisional transfer may be up to two orders of magnitude more effective in transferring momentum than the kinetic mechanism of molecular exchange [6]. If interactions between molecules in different layers are attractive, collisional transfer will hamper the relative motion of the layers and give rise to an effective "activation barrier" to the flow. This picture has been used by Eyring as the basis for a theory of viscosity of liquids [7]. At higher temperatures, the average kinetic energy increases, and the molecules can overcome this activation barrier to motion more readily, and collisional transfer contributions to viscosity become smaller. This is consistent with the decrease of viscosity observed in liquids as the temperature is raised. Collisional transfer contributes to first and higher order density corrections since it depends on the interaction of two particles. Another aspect of this momentum transfer mechanism is described in terms of the non-locality effect. Even at the distance of closest approach, the centers of mass of two colliding molecules are not at the same point. This allows interconversion of kinetic momentum flow into "virial" (potential; force related) momentum flow and affects the momentum transfer between the layers. The occurrence of at least binary 10 Figure 3. Collisional transfer of momentum from the fast layer to the slow layer, without an exchange of particles. 11 collisions is required for this mechanism so it contributes to first order or higher density corrections. The non-ideality and collisional transfer effects require at least two particle interactions and occur even for repulsive intermolecular potentials. At higher densities, the possibility arises for three molecules to interact simultaneously. At a static level, a third particle perturbs the spatial distribution of pairs of colliding molecules and adds a density dependent weight factor to the binary collision cross section. Depending on the range of temperature, this can increase or decrease the viscosity. True three-body collisions also affect momentum transfer in a fluid. The probability of a three-body collision occurring is one order of density higher than a binary collision and so leads to an extra first order density correction to the viscosity. Finally, in systems with attractive intermolecular potentials, the formation of bound pairs becomes possible. This introduces a new species in the system with its specific collisional properties. The formation of bound pairs also decreases the density of the free species avail-able for momentum transfer. At low abundance, the number of bound pairs is proportional to the square of the density of the fluid. The kinetic transfer of bound pairs between layers contributes to the momentum transfer in the fluid, giving rise to a first order density cor-rection to the viscosity. Calculation of the binary cross sections between free molecules and bound pairs (treating the bound pairs as point particles) [8, 9] indicates that the presence of bound pairs increases the viscosity. This is probably attributed to the fact that bound pairs carry more momentum due to their larger mass and so their transport between layers is more efficient in dissipating the momentum gradient. At higher temperatures where the bound pairs dissociate more readily, their contribution to the viscosity decreases. Calculations of Najafi et al. [9] for accurate intermolecular potentials, see Fig. 4, show the relative magnitude of the various contributions to the reduced second viscosity virial coefficient at different temperatures. The corrections, designated by (2), are related to two particle potential effects, those indicated by (3), take into account the static effect of a third particle on a binary collision cross section, and finally the (M — D) correction is related to the binary collision cross section between free molecules (monomers) and bound pairs (dimers). To our knowledge, numerical calculations of the contribution of true three-body collisions to the first order density corrections have not been performed. A theory which sets out to describe the viscosity of a moderately dense gas should account for the above mentioned mechanisms. A l l density corrections should in principle arise from a unified approach, but in practice, due to conceptual and mathematical difficulties, the mechanisms are usually treated separately. 12 Figure 4. Plots of the different contributions to the reduced second viscosity virial coefficient for argon. 13 Two general approaches exist for determining the viscosity of a moderately dense gas. The first approach is based on solving the generalized Boltzmann equation, whose method and results are summarized in Section 2.3. The other major approach utilizes the time correlation function formalism. While this is the main topic of this thesis, a short summary of past work is given in Sec. 2.4. 2.3 Generalized Boltzmann Equation for Moderately Dense Gases Systematic generalizations of the Boltzmann equation which lead to the first order density corrections to the transport coefficients were first developed by N . N . Bogoliubov [10] and H . S. Green [11] and later worked out by Choh and Uhlenbeck [12] to the level of including three-body collisions. M . S. Green [13] and E. G. D. Cohen [14] proposed a derivation for the generalized Boltzmann equation based on cluster expansions which removed a number of objections in the original calculation. Many excellent expositions describing these gener-alizations [15, 16, 17, 18] are available. Formal derivations are not presented here whereas the emphasis will be on the physical aspects of the theory and establishing connections with the qualitative discussion of the previous section. The Boltzmann equation describes the time evolution of the state of a non-equilibrium gas at low density. The distribution function f^(v±, r, t) for a particle being within the range dr about a position r and the range dvi about the velocity v i at time t, changes with time if the system is not in equilibrium. Changes in / ^ ( v ^ r , t) occur due to streaming-motion and to collisions which remove or introduce particles from the neighborhood of this phase space point. This leads to the Boltzmann equation which in the absence of external forces, has the form, M ^ M + Vl.M i^w) = / / / { / « ( v ; , r , t ) / « W , M ) - A ( 1 ) (v x, r, t ) / 2 ( 1 ) (v 2, r, t)} g12bdbdedv2. (2.3) In obtaining this equation, the distribution functions of the colliding molecules are both considered to be at the position of the first molecule r. The collision is also assumed to occur instantaneously at time t. Studied from a macroscopic viewpoint and on hydrodynamic length and time scales, the Boltzmann equation takes the collisions to occur instantaneously, and at a single point in space. In this context, the collision term is said to be local in space and time. The assumptions of collision locality are suitable for dilute gases. 14 1 v, Figure 5. In the Boltzmann equation, the collision is assumed to occur at a single point r. 15 The initial and final (incoming and outgoing) velocities appearing in the Boltzmann equa-tion are shown in Fig. 5. In deriving the Boltzmann equation, the well known molecular chaos assumption or Stosszahlansatz is evoked which states that there are no correlations among one-particle distribution functions prior to collision. The Boltzmann equation de-scribes the time change of the one-particle reduced distribution function, f[l\ at a point Vx and r in phase space at time t. The terms on the left hand side represent the change in fl1'1 as a result of streaming motion in the fluid, while the right hand side approximates the changes of the distribution function due to collisions. A particle with a velocity vx that collides with a second particle of velocity v2, is transferred outside the vicinity of that par-ticular phase space point. This event decreases the probability of having particles at Vi and thus decreases values of the distribution function at r, Vi. On the other hand, a particle with a specific velocity v[ at r, when colliding with another particle at a specific velocity v2, produce particles with velocity vx and v2, thus increasing the probability at the phase space point r, vx. The velocities v[ and v 2 involve the inverse collision of particles with velocities Vx and v2. This inverse collision is represented by the first term in the integral on the right of Eq. (2.3). The integral is over the velocity of the second particle, dv2, and also relative volume per unit time, gnbdbde, where gu is the relative speed of approach of the pair of colliding molecules, b is the impact parameter of the collision and e is the azimuthal angle in the cylindrical coordinate system defined by the relative velocity of approach in the collision, cf. Ref. [7]. The values of v[ and v2 are determined from vi and v 2 with a knowledge of the intermolecular potential by applying binary collision dynamics. The Chapman - Enskog solution of the Boltzmann equation gives the lowest order in density expressions for the transport coefficients. To obtain higher density corrections, the details of the collision must be studied within smaller length and time scales than that of the Boltzmann equation. The collision can no longer be considered to occur at a single spatial point and a finite collision time must be allowed. Inclusion of these details into the kinetic equation leads to the first generalization of the Boltzmann equation to higher densities. The quantities involved in a binary collision, when studied on a collisional length and time scale are indicated in Fig. 6. To study the collision, it is convenient to define some cutoff distance R, beyond which the intermolecular force is zero. This distance is later taken to be infinite, but it serves as a useful computational tool. This description gives an indication of the approximations made in the Boltzmann equation. First, in the Boltzmann equation, the sizes of the molecules are neglected, and it is assumed that for particle 2 undergoing a collision in the small interaction region, /21')(v2, r2, t) = ,/'21^(v2, ri, t), i.e., the position of particle 2 can be displaced to coincide with the position of particle 1. In essence, this 16 Figure 6. Position coordinates during a binary collision relevant to the generalized Boltzmann equation. The radius of the large circle is the cutoff distance of the potential. 17 approximation implies the distribution function has no explicit positional dependence inside the interaction region. Second, the particles are taken back from time t, to a time t 0 where their separation is R and they are about to enter the interaction region. Since the interaction region is considered small in the Boltzmann equation, the difference between the times to and t is neglected, the particles are not put at their actual positions of ri(to) and r2(to), but at the position r of particle 1 at time t. H . S. Green [11] started with the first B B G K Y equation to obtain a more rigorous expression for the collision term. In the spirit of the Stosszahlansatz, it is still assumed that the distribution functions of particles 1 and 2 are not correlated at the instant they enter the interaction region of radius R. Considering the details of a binary collision, the time evolution of the one-particle distribution function is written as, zV}^+ ? f P = L [[I d A ( 1 ) [vi (to), ri (t0), tolfP [v2 (tp), r 2 (t 0), t0] \ ^ ( du \ dt dri mJJ | dv\ J \<9ri J (2.4) in which m is the mass of the particles and u the intermolecular potential. H . S. Green demonstrates that the "exact" integral for the effect of collisions on the distribution function as given in E q . (2.4) differs from the approximate integral of E q . (2.3) by a correction factor a/i(1)(vi) ^ d/i(1)(vi) -^r~ + V i — d ^ ~ A ( 1 ) ( v i ) / 2 ( 1 ) ( v 2 ) - f[1)(v1)fP(v2)}gv2bdbdedv2 + J(vu n ) , (2.5) //{• where the correction term, J(vi,ri) is defined as, J<™> = UII (l-JI) = J L K * ) + A K r J . (2.6) r = ri — r2 is the relative position of particle 1 with respect to particle 2. The decomposition into Ji and J 2 arises from splitting up of the correction factor A into two parts, o A = Ai + A 2 Ai = / ^ i M . r i C i o ) , ^ ] / ^ A 2 = / J ^ i M . M / i ' W ^ . M ( 2- 7) R is the center of mass of the pair at time t, i.e. R = | (ri + r2). The first correction Ai arises from using the center of mass position of the pair in the distribution functions instead of their actual positions at time t 0 , and also replacing the explicit time t 0 with t. 18 The second correction A 2 is related to using r i , the position of particle 1, at time t, instead of the center of mass position of the pair. These terms account for the explicit dependence of the distribution function on position in the collision region and thus the effect of the intermolecular potential. The factor A i introduces a correction involving both a time and a spatial displacement, while A 2 only involves spatial displacements. By expanding the A, 's as a Taylor series with respect to the position of particle 1, and retaining only first order gradients, the difference of the terms in E q . (2.7) are further simplified. Expanding the terms in A i about r i and time t gives, A i = ^ .{/fWM.r,,*]^^ (2.8) The center of mass position in A 2 is expressed in terms of r i and r. Using a Taylor expansion about r i gives, 2 2 | 3r i j r' is defined in Section 2.5.1. Snider and Curtiss [19] use these forms in E q . (2.6) to derive first order density corrections to the viscosity coefficient and thermal conductivity. The first correction reflects both time and position aspects of the collision through r', while the second correction only has a positional aspect and is simpler in structure. 2.3.1 Moderately Dense Gas Transport Coefficients from the Gen-eralized Boltzmann Equation Snider and Curtiss use the Chapman - Enskog method for solving the generalized Boltzmann equation for repulsive potentials, including only binary collisions. This involves solving the Boltzmann equation in the presence of the corrections of E q . (2.6) The first step in the solution is linearizing Eq . (2.5), o (•[()] /)ft°l dT + V l '^r7 = / / / l [ 0 l / ' ° ] ^ l ( V ' l } + " ^ l ( V l ) ~ ^ 2 ( V 2 ) ] 9nbdbdedv2 + Jx + J 2 (2.10) where / { ° ' ( v i , r i , t) and / 2 ° ^ ( v 2 , r 2 , t ) are local equilibrium Maxwellian distribution functions in which the densities, mean velocity and temperature are position and time dependent. The (piS represent perturbations of the distribution function from Maxwellian. B y recognizing the tensorial properties of a number of the integrals which arise in the Chapman - Enskog solution of E q . (2.10), Snider and Curtiss were able to simplify the cal-culation of the transport coefficients. The result was the explicit expression of the first order 19 density corrections to the transport coefficients for soft repulsive potentials as integrals over functions of the intermolecular potential and the collision parameters. Snider and McCourt [20] later simplified the results considerably by recognizing that some of these integrals can be expressed in terms of the second virial coefficient and its temperature derivatives. These latter forms are stated below. The viscosity coefficient to first order in density is, 5kT V 8ft(2>2)(T) and the thermal conductivity given as, 15k2T 1 + ^ ( T ) - ^ ( T ) - ^ ^ ( T ) (2.11) A = 32mfi('2'2)(T) 2 0 n c r 3 ^ / _ , Ann1'2 a3 _ /rTnX 5 + 5nB(T) - ^ T ^ A C O NX(T) 7 ^dB(T) 5 ^2d2B(T) —nT 1 ; - -nT2 ir}0 ' 3 dT 3 dT2 (2.12) Q(2>2) (j1) i s the omega integral [7] which is proportional to the kinetic cross section, and a is a scaling parameter, which for attractive potentials corresponds to the separation at which the intermolecular potential is zero. The physical origins of the first order density corrections to the viscosity were discussed in Section 2.2. The thermal conductivity corrections have similar origins. In Eqs. (2.11) and (2.12), the second virial coefficient, which accounts for the non-ideality of the gas in the equations of change, originates from the non-dynamic (time-independent) J 2 correction. Fa (a = ry, A) corrections are dynamic in nature and originate from J x . The Nv term and the sum of the N\ and temperature derivatives of the second virial coefficient arise from contribution of the perturbative r/>, terms to the potential contributions to the pressure tensor and heat flux vector respectively. The latter are independent of J\ and J 2 . The Fa and Na terms are integrals over functions of the relative momentum and position, drj d p F ( r . p ) = to2 J r2dr J p2dp sin ed6F(r,p, 8) (2.13) The reduction of the six-fold integrals to three-fold integrals occurs since the functions F ( r , p) depend only on the magnitudes of the relative separation r, relative momentum p and the angle 9 between the r and p vectors of the colliding particles. The integrand F(r,p, 0) may also include one or more of the possible binary collision parameters defined in Section 2.5.1. These collision parameters themselves involve an additional integration, and so evaluation of the density corrections require the numerical calculation of a set of nested four-fold integrals, a far from simple task [8]. 20 Equations (2.11) and (2.12) need to be modified in two ways. First, Snider and Curtiss limited their calculations to systems with repulsive intermolecular potentials which do not support bound states, and second, the occurrence of three-body collisions was not considered in their work. Both these effects contribute extra first order density corrections to transport coefficients. Bound pairs in moderately dense gas systems were first studied in a modern setting by Stogryn and Hirschfelder. Based on the work of Hil l [21], they developed a specific division of the integral of the second virial coefficient into regions of phase space where a pair of particles are free (corresponding to an open collision trajectory), regions where they are bound and regions where they are metastably bound. Making use of equilibrium statistical mechanics, they obtained an expression for the equilibrium constant of the formation of bound pairs and then showed that if a limited amount of association occurs, the mole fraction Xb of bound molecules is proportional to the number density n of the particles in the fluid Xb = nKeq. (2.14) In the above relation Keq is the equilibrium constant for the "dimerization" reaction and the number of double molecules is small relative to the number of free molecules so that n « rif. The presence of bound pairs implies a new mode of momentum transfer in the system. Stogryn and Hirschfelder treat the system as a binary mixture of free and bound molecules. The viscosity for such a mixture with any general composition, not just small bound mole fractions Xb is, XbHff + xjHbb - XfXb{Hfb + Hbf) V = HJT Wn ' (2'15) The are related to binary collision cross sections of the species £ with C. Considering E q . (2.14) for low densities of bound molecules to first order in density only free-free and bound-free collisions contribute. The result is expressed as, X = \M[l + nB{MD)] (2.16) where a^ M ' is the transport coefficient for a system composed of pure monomers (unbound molecules), equivalent to Eqs. (2.11) and (2.12), and B^MD^ is the first order correction from the monomer - dimer (free molecule - bound pair) collisions. This purely kinetic contribution to momentum transfer assumes the flow process does not change the equilibrium population of the bound species. Considering the bound pair to be a point molecule and using estimations for the parameters in the free - bound intermolecular potential, Stogryn 21 and Hirschfelder obtained numerical values for B^MD\ However, they treated non-locality and collisional transfer effects crudely by using adaptations of the Enskog theory, not the rigorous corrections of Eqs. (2.11) and (2.12). Formal relations for the effect of three-body collisions on the transport coefficients were first developed by Choh and Uhlenbeck [12]. Their expressions however, are difficult to implement in a numerical calculation. Hoffman and Curtiss [22] proposed a different formal-ism with a similar scope. Starting with the B B G K Y equations, they made a factorization assumption for higher order reduced distribution functions similar to the superposition ap-proximation of equilibrium statistical mechanics, and after detailed calculations obtained the Snider - Curtiss density corrections, collectively denoted as B^M\ along with an extra first order density correction B^M\ which approximates the effects associated with three-body dynamics, 77 = r/o [ l + n X = X0[l + n(B^-rB^)]. (2.17) The 3 M superscript emphasizes this correction accounts for three monomer collisions. The explicit form of this correction is expressed by Hoffman and Curtiss [22] as, " 87T 2ft( 2> 2)* 1 > where R(T) is an integral which has the general structure of E q . (2.13) and is defined.in E q . (6.91). The detailed form of the three monomer correction shows that the presence of the third particle in this correction is actually static in nature. It does not involve the dynamics necessary to obtain the contribution of true three-particle collisions to the transport coefficient. In this correction, the third particle affects the spatial distribution of a colliding pair and thus gives an extra weight factor to the binary collision cross section. For potentials with an attractive well, Curtiss and coworkers [22, 23] calculate the correc-tions of E q . (2.17), but use the full second virial coefficient, and not only its free part when it occurs in B^M^ of Eqs. (2.11) and (2.12). This includes contributions from the bound pairs, which it should not, since B^M^ arises from open collision trajectories. At this stage of the development, Kuznetsov [24] proposed that only the free part of the second virial coefficient should be used whenever it occurs in B^2M\ the density corrections of Curtiss and coworkers, E q . (2.17). The separate contribution of Stogryn and Hirschfelder for the bound pairs should then be added on to this to account for the presence of bound pairs. Thus 7/ = r/o[l + nBr, + 0(n% 22 A = A 0 [ l + nBx + 0(n2)}, (2.19) with the second transport virial expressed as, Bx = B ^ + B[MD) + E | 3 M ) (2.20) The BfM^ and B^M"> terms only involve the free part of the second pressure virial coefficient according to the division of Stogryn and Hirschfelder. The argument is that if bound pairs are accounted for by including the Stogryn - Hirschfelder correction, then using the full second virial coefficient in the Snider - Curtiss corrections overcounts the effects of the pairs. Kuznetsov's scheme leads to numerical results in good agreement with experiment for the viscosity coefficient, but in poor agreement for the thermal conductivity. The separate adding of bound pair contributions to the transport coefficients, as proposed by Kuznetsov appears physically reasonable but is not rigorous. Rainwater [8] accounted for the discrepancies with thermal conductivity data by realizing that for attractive intermolecular potentials, there is an ambiguity in the way which the second pressure virial coefficient can be partitioned into free and bound regions. He studied cases and stated the importance of choosing the appropriate division for the second pressure virial coefficient and its temperature derivatives in the first order density corrections. The first calculation of first order density contributions with correct partitioning for Lennard-Jones fluids appeared in 1984 with the work of Rainwater and Friend [8] which they later extended to lower temperatures in 1987. The calculations of Rainwater and Friend and their later extension to accurate inter-molecular potentials show good agreement with experiment [9]. But from a theoretical point of view the addition of the contributions in E q . (2.20) is not rigorous. The corrections have not been derived from a uniform starting point which still leaves open the possibility of some overcounting or neglection of contributions. O n the other hand, each of the individual evalu-ations are rather complicated and involve both formalistic and numerical approximations. It is difficult to assess the effect of these approximations on the final expression of the density corrections. In a recent thesis by Wei [25] a uniform approach was used to derive density corrections in the presence of bound pairs and pair correlations. The method utilized a kinetic equation approach in which the TV-particle distribution function accounts for free molecules, bound pairs, and correlated pairs. A system of coupled kinetic equations was derived for these species which was closed and made convergent by introducing a decay time for the corre-lated pairs. The coupled equations were solved simultaneously with the Chapman - Enskog 23 method. The solution leads to first order density corrections which include contributions from the free molecules, bound pairs and correlated pairs. Due to the introduction of the particular structure in the TV-particle distribution function, the final results are expressed in a manner different from previous theories. The density corrections discussed above may in principle be derived using the Choh and Uhlenbeck theory [12]. They implement the Bogoliubov method [10] to higher order density corrections, but express their results in a set of integral equations which must still be solved in order to derive explicit density corrections. It is expected that if this program is explicitly carried out, the density corrections of Snider and Curtiss will be obtained, although to our knowledge this has not been demonstrated. It was with the goal of clarifying the relations among kinetic equation approaches that a study of alternate derivations of the density corrections based on time correlation functions was initiated. This is also desirable from another viewpoint. Numerical calculations of transport coefficients are invariably done with some approximate intermolecular potential. The degree of disagreement with experiment may either be related to the approximate nature of the theoretical expressions for the transport coefficients, or the approximate nature of the potential [26]. A n independent verification of the theoretical expressions could help establish the validity of the density correction expressions and so relate any residual disagreement with experiment to the approximate nature of the intermolecular potential. 2.4 Time Correlation Functions for Transport Coeffi-The relation between a transport coefficient and the time correlation of a phase function was introduced by Einstein [27] in 1905 in the study of Brownian motion. The relation states that the diffusion coefficient, D , is related to the integral over all times of the ensemble average of the velocity auto-correlation function, The (( )) notation implies a thermal average over an equilibrium canonical ensemble of an A^-particle system of volume V and temperature T (other ensembles can also be used), and v(0) and v (r) are the velocities of a particle at time 0 and r. Over all members of the ensemble, the initial velocity of a chosen particle v(0), has a Maxwellian distribution. For times equivalent to a small number of mean free times, the initial magnitude and direction of the velocity of a particle have not been randomized and v(0) and v(r) show some correlation. cients (2.21) 24 As time progresses, more collisions randomize the velocities and the initial correlation is lost. The ensemble average of the velocity auto-correlation decays from its classical equipartition value of 3kT/m at r = 0 to very small values over the time scale of several collision times. The nature of this decay can be determined from the study of the detailed dynamics of the system or from numerical simulations. The former approach is pursued in this thesis. The Einstein relation is an example of a general principle named the fluctuation - dissi-pation theorem which was anticipated by Onsager in 1931 [28], and given a general proof by Callen and Welton in 1951 [29]. This theorem states that the rate of decay to equilibrium of a perturbed non-equilibrium state not very far from equilibrium is related to the time correlation of spontaneous fluctuations at equilibrium [30]. In the case above, the diffusion coefficient which describes the rate of dissipation of a density gradient in a non-equilibrium system, is proportional to the integral over time of the velocity auto-correlation function at equilibrium. This theorem allows the use of equilibrium properties of a system to obtain information about its non-equilibrium properties. Extensive study of transport coefficients utilizing time correlation function methods be-gan with the pioneering work of M . S. Green [31] and R. Kubo [32] in the mid 1950's. Many excellent introductions [33, 34, 35, 36, 37, 38] and review articles [39, 40] regarding the general principles of time correlation function methods are available. A multitude of approaches are available for deriving the time correlation function formulas for transport coefficients [39]. In particular, the method due to Helfand [33, 41] obtains the time correlation expression for the transport coefficient from the corresponding equation of change by utilizing an appropriate form for the distribution function of the system. As a result, the time correlation formula for the viscosity coefficient is J p is the phase space function for the transverse momentum flux tensor. For a system with only pairwise additive forces, the momentum flux tensor has a kinetic contribution which is a function of the momenta pj, of particles and a potential contribution which depends on the relative separation Vji and force Fji between pairs of particles j and £, The bracket notation [ p) indicates a symmetric traceless second rank tensor. From here onwards, when a dyadic tensor is constituted of two identical vectors, the vector is repeated only once within the bracket, i.e. (2.22) (2.23) [AA]<2> = [ A p . (2.24) 25 For the thermal conductivity, A = 3 ^ ^~dr<<Je(0).Je(r)>)> (2.25) where J e , the phase space function for the heat flux vector is, l v ~ ^ - ™ + U j 7 U ) - ^ J . (2.26) Uj£ is the intermolecular potential between particles j and i, h the average enthalpy per parti-cle, and Uss the second rank identity tensor. The enthalpy term eliminates any contribution from the convective energy flux. Clearly the heat flux vector also has kinetic and potential contributions. Explicit expressions for other transport coefficients have been derived but will not be stated here. For an TV-particle system, a time correlation function for a transport coefficient has the general form, C(t) = 2 ^ V l « J ( 0 ) © J (* )» = ^ V l « J ( ° ) © e " l t £ j ( ° ) » > (2-27) where J is a tensor valued phase space function representing the flux of the property of interest. Moreover, J is a tensor of rank a belonging to an irreducible representation of dimension 2a + 1 of the three-dimensional rotation group 50(3), © represents an a-fold dot (tensorial) contraction, and C is the TV-particle Liouville superoperator. Wi th this notation the general expression for transport coefficients in terms of time correlation functions has the following form, a = lim lim — / dte _ r f((J(0) 0 e _ , t £ J ( 0 ) » . (2.28) JV/V=const. v ' The thermodynamic limit is needed to eliminate edge effects. The B factor introduces a scale factor for each transport coefficient, e is related to a frequency and the convergence factor e~et implies the adiabatic approximation, namely the slow switching on of a perturbation from equilibrium in the distant past [42, 43]. Equations (2.21), (2.22) and (2.25) are consistent with this general form. The flux J is an extensive property and may be a scalar (bulk viscosity), a vector (diffusion and thermal conductivity) or a symmetric traceless second-rank tensor (shear viscosity). Introduction of the volume factor V recognizes the extensivity of a flux and the intensivity of a transport coefficient. For diffusion and thermal conductivity, a is 1, so the tensorial contraction in the time correlation function reduces to a simple dot product, while for the shear viscosity, a is 2 and the tensorial contraction in the time correlation function is a double dot product. 26 A transport coefficient a is the particular limiting form of the Fourier transform of the time correlation function of the flux. Calculation of the time integral specifically leads to, a = lim lim -—^ - r - 7 ( ( J © — ^ J » . (2.29) /V/V=const. v ' In most of the literature it is standard to use the tensorial components of the transverse momentum flux for the shear viscosity, 77, r , = }\f+ * f & » ^ W > - (2-30) N / V=const. Since SPtXy is one component of a second rank symmetric traceless tensor, it belongs to a five-dimensional irreducible representation of the rotation group. By the rotational invariance of C and of the thermal average, the quantity of E q . (2.30) is l / 5 t h the sum over all traceless symmetric components, which appears in E q . (2.29). A factor of 1/2 is also required to symmetrize the components and obtain E q . (2.22). The advantage of expressing the transport coefficient in a rotationally invariant form is that the rotational properties of the different operators can be used to simplify its evaluation. 2.4.1 M o d e r a t e l y D e n s e G a s T r a n s p o r t C o e f f i c i e n t s f r o m T i m e C o r r e l a t i o n F u n c t i o n s Before discussing any analytical approach for the evaluation of a transport coefficient using the time correlation function formalism, it is useful to mention simulations on equilibrium systems which utilize E q . (2.28). The earliest molecular dynamics simulation, and indeed one of the pioneering works in the history of computer simulations is that of Rahman [44] who studied the velocity auto-correlation and the diffusion coefficient through E q . (2.21). In the liquid range, the work of Hoheisel and coworkers [6] on viscosity and thermal conductivity is notable. Simulations explicitly show the nature of the time decay of the correlation function (an aspect bypassed in our theory) and the magnitude of the kinetic and potential contributions of the flux function. Much work has been done to show that the results of E q . (2.28) are equivalent to the generalized Boltzmann equation. Most treatments begin with expressions like E q . (2.28) and proceed to derive a set of kinetic equations . Resibois [45] uses this approach and with a generalized master equation demonstrates that the time correlation function equations are equivalent to the generalized Boltzmann equation to all orders of density. But the explicit evaluation of a transport coefficient is not attempted in his work. Dufty and Lindenfeld 27 [46] take this a step farther and show the equivalence between the non-linear Boltzmann equations and the time correlation function expressions. Other approaches explicitly use the time correlation function formula to obtain expres-sions for the transport coefficients. Ernst, Dorfman and Cohen's [47] method is in the spirit of Cohen's cluster expansion for generalizing Boltzmann's equation. They introduce a series of reduced time correlation functions in a fashion reminiscent of the definition of reduced distribution functions and proceed to derive a set of kinetic equations for the reduced time correlation functions which are analogous to the B B G K Y hierarchy. By utilizing cluster ex-pansions for the reduced time correlation functions, and assuming an appropriate functional dependence of the higher order reduced time correlation functions on the first order reduced time correlation function, this hierarchy is closed and Boltzmann type equations are derived. The nature of the assumptions limits the scope of the work to repulsive potentials. The solution of these equations leads to density corrections for the transport coefficients. To the extent that the resulting equations are similar to the generalized Boltzmann equation, their solution involves the same sort of approximations and limitations mentioned in Section 2.3.1. A solution of the equations to obtain explicit expressions for the transport coefficients is not presented in this work. Later Ernst [48] extended this approach to general orders of density and showed that its results are equivalent to those of the generalized Boltzmann equation to general order in density. Ono and Shizume [49] calculate the time correlation functions for transport coefficients at moderate densities, but their formulas are not reduced to a form that is easily comparable to other work in this field. They derive an equation which describes the time evolution of the flux function J(t) in terms of an unperturbed Liouville operator and a perturbation. After developing what is equivalent to an interaction representation for the time evolution, they utilize a Mayer cluster expansion technique quite similar to that used in deriving virial coef-ficients for equilibrium properties. The time evolution is expressed as a series of consecutive binary collisions. The formal development is complete, but no comparison with previous work is given. Kapral and Desai [50] evaluate transport coefficients for a binary mixture and obtain interesting results for the diffusion and thermal diffusion coefficients. They express trans-formations of the time correlations in terms of equivalent Boltzmann type kinetic equations. These kinetic equations are solved within certain (two Sonine) approximations and explicit time-decays of the time correlation functions associated with the transport coefficients are obtained. The work of Kawasaki and Oppenheim [51] is studied in greater detail in the next chap-28 ter. This method is an extension of the resolvent expansion of Zwanzig [52] to a flux with both kinetic and potential contributions. Zwanzig introduced an expansion of the resolvent in powers of 1/e, which was then resummed to retain a finite expression for the transport co-efficient. Zwanzig's treatment was applicable to purely momentum dependent flux functions which are sums of one particle properties, irrespective of their particular functional nature. The Kawasaki and Oppenheim method was initially extended to include potential contribu-tions to the flux function, on condition that the intermolecular potential be repulsive. The final results are a set of integral equations for the various contributions to the viscosity that are similar to the collision integrals of the Boltzmann equation. Unfortunately they do not reduce the final equations to a form suitable for comparison with the results of Snider and Curtiss. However, Kawasaki and Oppenheim do show detailed equivalence of their equations with those of Choh and Uhlenbeck who utilize Bogoliubov's generalized Boltzmann equation. It is recalled that this was also the starting point of the calculations of Snider and Curtiss. The Choh and Uhlenbeck equations themselves have not been independently evaluated to our knowledge. Subsequently, Kawasaki and Oppenheim extended the method to attractive intermolecular potentials [53]. Thus in principle, the theory should account for bound pairs, though explicit expressions for contributions to the viscosity in terms of a functional of the intermolecular potential were not given. In the next chapter, an explicit derivation of density corrections to the viscosity from the Kawasaki and Oppenheim approach is presented. It is demonstrated that their equations reduce to the Snider and Curtiss formulas within certain well defined approximations. This is encouraging as it provides independent verification of the forms of the density corrections. In our opinion, their treatment of bound states has a number of problems which are also discussed. 2.5 Binary Collisions 2.5.1 C o l l i s i o n P a r a m e t e r s Detailed knowledge of binary collision dynamics is required in order to calculate the density corrections to the viscosity. Multiple integrals over the relative position and momentum of the type given in E q . (2.13) arise. The collision parameters in the integrals must be determined as functions of the relative position and momentum so that the phase space integrals can be evaluated. The kinematics and dynamics of a binary collision are summarized in this section and the notation used in later chapters is introduced. The first step in the kinematic study of binary collisions is the conversion of the positions 29 and momenta of the pair of particles to the center of mass and relative motion variables (with the position and momentum of particle 2 taken as the origin). The transformations are well known, r = r 1 2 = n - r 2 R = i ( n + r2), (2.31) and P = ^ ( P i - P 2 ) P = P l + p2, (2.32) P i = ^ P + P p2 = i p - p . (2.33) The Jacobian for this transform is unity, namely d r i d r 2 d p i d p 2 = drdRdpdP. (2.34) With in classical mechanics, for a known intermolecular potential, the trajectory of a collision is uniquely determined by the position and momentum of a single point on the trajectory. This reference point is given by the relative position vector r and relative mo-mentum vector p. The z axis is chosen to be in the —p direction. The polar angle 9 of the position vector determines whether it is on the incoming or outgoing part of the trajectory, with 9 < 7r/2 for the incoming part of the trajectory and 9 > n/2 for the outgoing part. The angle between r and p is TT — 9. These relations are shown in Fig. 7. For computational purposes, the potential is considered to have a cutoff distance R, beyond which it is zero. This distance is eventually taken to infinity. The internal states of the colliding molecules are assumed not to change during the collision. If there is no external torque, the collision trajectory is planar and remains in the plane of r and p at all times. A general point on the trajectory is characterized by a position vector £ and a relative momentum vector p^ . The unit vector along the £ direction is which makes an angle (3^ with respect to the z-axis. The unit vector along the direction of p^  is K.^ which makes an angle of 7^  with respect to the z-axis. The magnitude of the angular momentum L of the initial reference point, L = |r x p| = rpsin(7r — 9) = rpsin9 (2.35) and its direction is the same for all points along the trajectory. The magnitude of the angular momentum for a general point is, £ = | £ x p f | = & f s i n ( » f e - & ) . . (2.36) At the cutoff distance R where p^  = p# = po (the constant momentum in the absence of interaction), the relation between the angles rjR and fiR is, V r = pR + TT - a r c s i n [ L / . R ( r a £ ) 1 / 2 ] . (2.37) 30 z 11 K e n-P Figure 7. The coordinate system for a binary collision. A reference point of the trajectory, with position r and momentum p, and also a general point on the trajectory with position £ and momentum p4 are indicated on the diagram. Various angles defined in reference to these points are also shown. 31 If the distance R —>• oo, the above angles approach their asymptotic values on the trajectory, in which case they are no longer denoted by the subscript R, i.e., lim /3R = p lim nR = n (2.38) i?.—>-oo R—s-oo The asymptote of the incoming branch of the trajectory makes an angle j3 with respect to the z-axis. The vector r m denotes the point of closest approach of the two particles. The angle between r m and the asymptotic direction of approach is —to and the angle between r m and the reference position r is denoted by £. These relations are illustrated in Fig. 8 for r on the incoming part of the trajectory. From this figure, it is obvious that the following relations hold among these angles, (3R — 9 = toR + ( incoming (2.39) Pn-9 = LOR-C outgoing (2.40) The angles between the position and momentum vectors of the reference point r and p with those of the position and momentum vectors at the cutoff distance R are obtained from the following set of dot products, P ' P o K f l = PPO C0S(7T - 77ft) •= -ppo COS T]R (2.41) r-Po '^ft = rp0 cos(r?ft - 9) (2.42) P-RKR = pR cos (n — /3ft) = —pR cos (3R (2.43) T>RKR = rRcos(pR-9) (2.44) p«r = pr cos(7r — 9) = — pr cos 9 (2.45) PQK!R-RK,R = p0Rcos(7]R - PR) (2.46) Another important collision parameter is the time So it takes to go from RKR on the incoming part of the trajectory to r. During this time the momentum changes from p 0 to p. A somewhat related collision parameter is r' = RKR + 2p0so/m. The magnitude and direction of this vector are obtained by starting at the vector r and tracing backwards a time —s0 along the trajectory to reach R. The intermolecular potential is then "turned off' and the relative position allowed to evolve freely forward in time for a duration of s0. The value of r under this transformation is r'. The relative momentum p, under the same set of transformations is po- These quantities are illustrated in Fig. 9. It is convenient to define two additional collisional parameters, X and Y with reference to Fig. 9. The distance go^ o — R along the asymptotic direction is denoted by b(\X\ + \Y\), where go is the relative speed of approach of the particles and b is the impact parameter. 32 Figure 8. Binary collision parameters. R is the cutoff distance of the potential and rm is the closest distance of approach. Various angles are defined with reference to these vectors. 33 Figure 9. Primed position variables and their related collision parameters, b is the impact parameter. 34 From the figure, this sum is seen to be the projection of r' onto the asymptotic direction. If r m undergoes the same operations mentioned in deriving r', the result is a new vector r'm. The projection of this vector along the asymptotic direction of the incoming part of the trajectory is b\X\. The parameter Y is defined by the difference of these two projections. 2.5.2 Collision Dynamics The dynamics of a binary collision allows the determination of the expressions for the collision parameters as integrals over functions of the intermolecular potential. Angular momentum conservation constrains the relative motion to a single plane, taken to be the zrr-plane. As stated previously, the z-axis is chosen in the direction of —p. Thus in a Cartesian system, the time behaviour of the x and z components of the relative position coordinate and the conjugate momentum describe the dynamics of the collision. The total relative energy of this system, as given by the Hamiltonian function, is H=±(p22+p2x) + u(0 (2.47) where it is assumed that the intermolecular potential is spherically symmetric so depends only on the separation between the particles, £ = \J x2 + z2. Due to planar symmetry, it is more convenient to convert to polar coordinates, based on the relative separation £, and the angle /?£. With reference to Fig. 8, z = £ c o s i = £ c o s s i n ( 2 . 4 8 ) x = £sin/% x = £sin/?f + £cos/? f/5 f (2.49) and so, UPI+PD = ^ M 2 + (^ )2] = £ (e2+e&). (2.50) 2 / j ^ 2 rx' 2 fx 2 In this polar coordinate system the Hamiltonian is, H = f (£ 2 + i2k2) + «(0 = ^ (P? + p P f t J + (2-51) where p^ — dH/d$, = /i£ is the radial component of the relative momentum, and pp = dH/d/3^ — /^£2/3f the angular momentum. From the conservation of angular momentum, the magnitude of the angular momentum before the collision, bp0 = bfig0 equals that at any other instant during the collision, i.e. Vh = tfk = A%o =^^f=bf- (2-52) 35 This equation shows the variation of the angular coordinate (5^ with time. The conservation of energy shows that the energy before the collision, E = p,g\j2 equals that of any other instant during the collision. On substituting Eq. (2.52) into Eq. (2.51), ^a2 2 9 0 ^ 2 + ^ o 2 9l 9l A 2 1/2 (2.53) This equation shows how the relative separation £ changes with time. The negative sign corresponds to the incoming part of the trajectory where the relative distance £ decreases with time and the positive sign is for the outgoing segment of the trajectory where £ increases with time. Equations (2.52) and (2.53) are parametric equations in time t. Time can be eliminated from these equations to give the variation of the angle coordinate dfe directly as a function of the variation of the relative position coordinate e [i £ MO]1/2' (2.54) Integrating the right hand side of this equation from the position of the reference point r, assumed to be on the incoming part of the trajectory, to the distance R is equivalent to integrating the left hand side from the angle 6 to PR, £ _ " ( Q l 1 / 2 ' e E J (2.55) From the definition of the collision parameters, LOR and ( in Eq. (2.39), and with reference to Fig. 8, for r on the incoming part of the trajectory, the integral of the right of Eq. (2.55) is decomposed to give, bdi UR - -r-J r m £2 Jr-n i _ £ _ MQ. bd£ 1/2 1 _ 6 2 _ 1 e E 1/2' (2.56) (2.57) A particular case of the first of these equations is lim 2UJR: — 7T — x R-±oo (2.58) with x being the angle of deflection. 36 With reference to the definition of so, it is easily seen from Eq. (2.53) that for the incoming part of the trajectory, So f Jr dt; 00 [ l E (2.59) It is possible to decompose the above integral to obtain contributions corresponding to the collision parameters XR and Y. Thus with reference to the definitions of XR and Y, g0s0 - R = + rB. Jr 1 _ £ _ ^ £2 £ 1-1/2 - J R = 6 ( ^ + y ) , (2.60) so that, XR = Y = t*(fl £7 -1/2 -1/2 6 V m (2.61) Analogous definitions can be stated for reference points r and p on the outgoing part of the trajectory. These relations are stated by Snider and Curtiss [19] and also by Rain-water [54]. In principle all collision parameters can be determined for a particular collision trajectory from a knowledge of the intermolecular potential. The parameters cuR and XR are properties of the trajectory as a whole, while £ and Y are parameters different for each particular reference point, r, p of the trajectory. 2.5.3 Collision Operators A number of processes which occur on a pair of molecules during a binary collision are described by operators. For quantum mechanical calculations, the superoperator equivalents to these operators are used and depending on the context, the operator and/or superoperator forms are used interchangeably throughout this thesis. These arise naturally while studying the dynamics of the system, in particular as a result of the binary collision expansion of the Af-particle resolvent as described in future chapters. For completeness of the treatment of binary collisions, two of the relevant superoperators are discussed in this section. The quantum two-particle transition superoperator [55] is encountered in the study of binary collisions, Tl2 = V i , + V-'12 z-C v 1 2 = V-12 12 z-C 12 •(z - JC12). (2.62) 37 As applied in this thesis, z is a small positive imaginary number which is taken to be zero at the end of the calculation. The classical analog of this, the transition operator, involves the potential part V i 2 , and the kinetic part /C 1 2 of the Liouville operator £ 1 2 , r ir , v , M .P i 5 ,p2 3 .duu ( d d \ C12 = £1 + JC2 + V12 = -i—^ * — + * A "a ( 2 - 6 3 ) m dr i m dr 2 ovi2 \ c 7 p i c>p2 / When the transition operator is in an integral of the structure Tl f n1 = — d x 2 i T i 2 ( l + P i 2)^W2 (2.64) kpi J and operates on a function of the momentum -F(pi), the result is related to the linearized (Boltzmann) collision operator H\, VN ftiF(pi) = / dr2dp2 [F(p'2) + F(pi) - F ( P l ) - F(p2)] ^ (2.65) where the p^ - are the momenta before the collision and Pj are those after the collision. The (fj are normalized Maxwellian momentum distributions, 1 e-pj/2mjkT ( 2 6 g ^ ^ (2TtmjkTyf2 The M0ller superoperator Q i 2 for the pair of particles 1 and 2 [56, 57] is also encountered on numerous occasions, flia = lim e - i t £ l 2 e i t / c . (2.67) t - J - O O The classical M0ller operator takes an interacting pair of particles back an infinite amount of time to the past (in practice to a time where particles are no longer considered to interact), and then brings them forward in time by the same amount, without the interaction. The M0ller operator (superoperator) converts the relative position r to r', f ] 1 2 r = r'. (2.68) and is related, when acting on functions of the momentum, to the two particle resolvent by, lim ^ V i 2 = Q' 1 2 - 1. (2.69) The prime is used to indicate that this equality applies only when the resolvents are operating on functions of momentum (functions with zero eigenvalues with respect to the operator /C). A correction term is needed when the resolvent form of the M0ller operator operates on functions of the position. 38 Chapter 3 Moderately Dense Gas Shear Viscosity from the Kawasaki -Oppenheim Approach In this chapter, the Kawasaki and Oppenheim method of evaluating time correlation func-tions for first order density corrections to the shear viscosity for gases with repulsive po-tentials is outlined [51]. Hamer and Oppenheim have carried out similar calculations for thermal conductivity and bulk viscosity [58, 59]. The kinetic and potential contributions to the shear viscosity and the binary collision expansion of the resolvent are introduced in Section 3.1. Of the various terms, the kinetic - kinetic contribution is treated in detail in Section 3.2 and the results for the other contributions are quoted. The Fourier transforms of operators introduced by Kawasaki and Oppenheim are not used as these do not appear to be crucial to the development and their deletion simplifies the notation considerably. Our treatment utilizes the symmetric traceless form of the momentum flux tensor, making the relations rotationally invariant and independent of the choice of coordinate system. The new work of this thesis begins with Section 3.3 where it is demonstrated that the results of Section 3.2 are equivalent to the Snider - Curtiss first order density corrections to the shear viscosity. A n extra term which accounts for triple collisions is also obtained in the Kawasaki and Oppenheim approach. A binary mixture calculation based on the method of Section 3.2 follows in Section 3.4. This accounts for bound pairs in the system at the level of the Stogryn and Hirschfelder approximation. By including bound pairs in this manner, it is demonstrated that the approach can produce results equivalent to the Rainwater theory. Section 3.5 outlines the theory originally proposed by Kawasaki and Oppenheim to account for systems with attractive potentials. A discussion on what we believe to be problems in this treatment of attractive potentials is presented. 39 3.1 Formulation for the Shear Viscosity From Eq. (2.22), the time correlation function expression for the shear viscosity of a pure fluid of volume V at temperature T, composed of particles of mass m is, r? = lim lim 77(e) JV/V=const. 1 r°° where a convergence factor e _ e t with e > 0 is introduced. The flux at time —t is expressed in terms of the flux at t = 0 with the help of the time evolution operator. The time integration of this expression can be equivalently expressed in terms of an TV-particle resolvent operator, 1 r°° ^ = I b W o * e - « J ( 0 ) : e ^ * J ( 0 ) » , 1 ; « J (0 ) : - ^ - , J (0 ) ) ) (3.2) lOVkT" z — C Viewed in this manner, the transport coefficient is proportional to the Laplace transform of the time correlation function, «J(0):J(—t))), with a vanishing real variable e (or vanishing complex variable z = ie). In Eq. (2.23) the momentum flux tensor was seen to have both kinetic and potential parts, j = J K + J y (3.3) where JK = £j[p.?PV rn a n d Jy = T,j<e[*jtFje}^• Substituting this sum into Eq. (3.2), the viscosity breaks up into four contributions, lOVkT = VKK(c) + VKv(e) + VvK(e) + Vvv(e)- (3-4) The evaluation of TJKK^) is presented in detail. Other contributions are obtained by similar arguments but only the final results of these evaluations are given. Before evaluating the ^ ^ ( e ) , a number of preliminaries need to be discussed. Kawasaki and Oppenheim introduce the notation, G(e) = -^—r (3.5) ie — L for the resolvent where C is the iV-particle Liouville operator. Determining the effect of the TV-particle resolvent operator on SK requires a detailed knowledge of the dynamics of the TV-particle system. At low densities where the present 40 calculations are valid, interactions among particles are predominantly pairwise additive. It is physically reasonable to expect that in this situation, the dynamics of the system are expressed as combinations of binary collisions. Recalling the definition of the transition operator Ta from Eq. (2.62), (Greek subscripts will designate pairs of particles from this point onwards) and defining Go as the resolvent for a system of non-interacting particles and G2(a) as the resolvent of a system with only the pair a interacting, it is seen that, Ta = Va - iVaGoTa = Va - iTaG0Va = VaG2(a)Gol = Go1G2(a)Va. (3.6) With these relations it is straightforward to derive the binary collision expansion of the iV-particle resolvent developed originally by Watson [60] and stated in the present form by Siegert and Teramoto [61], G = Go — i GoTaGo — "^2 GoTaGoTpGo + i ^2 GoTaGoTgGoTjGo + . • • (3.7) The iV-particle resolvent in Eq. (3.7) is decomposed into a free streaming term, a sum of single binary collisions, a sum of two consecutive binary collisions (the prime on the summation indicates that the pairs a and (3 do not involve the same pair of particles), and so on. The theory of Section 3.2 assumes the potential is repulsive and so none of the pairs in the above expansion are bound. The operators are thus well defined and convergent. Before proceeding, the normalization convention for reduced distribution functions is defined, NJ. (N^s) Consistent with this normalization, the expressions for and p^} are, pis) = j drs+1dps+1 • • • drNdpNpW. (3.8) p? = p g = n V i ^ e " - ^ 1 2 . (3.9) <Pi is the normalized Maxwellian momentum distribution and n is the number density. 3.2 Contributions to 77(e) To evaluate the shear viscosity, Kawasaki and Oppenheim write the integral of r]Ki<{e) i n explicit form, VKK(e) = fj dvNdVNSK:G{z)]Kp^\ (3.10) 41 where is the ./V-particle distribution function. Using the equivalence of the N particles in the system and substituting the binary collision expansion of Eq. (3.7) for the resolvent operator gives, xG0\l-iY,TaG0-J2 TaGoTpGo + ... 1 JKp{N)- (3.11) Since the function to the left of the first G0 resolvent only depends on the momentum, the equation becomes, riKi<(e) = ^^jjdvNdvN{Vlp-}-U^ = VKK (e) + VKK (e) + VKK (e) + VKK (<0 • The numerical superscript on each term represents the number of transition operators T , or physically the number of binary collisions involved in its structure. The r superscript represents the contributions from all remaining interactions with at least three collisions. rfKK(e) only involves the free streaming of the particles and is simplified to, = d o ^ / * ' ^ * ^ ( 3 - 1 3 ) Kawasaki and Oppenheim demonstrate that the only non-zero contributions from ] K in VKK(6) a r e those in which the index j of [pjp ' is one of the particles in the pair a in the collision operator Ta- Furthermore, one of the particles of the pair a must be particle 1. Therefore, *«W = ^ / / r f r " * > ' l ( 2 , : ! ? r " G ° J ' ^ ) = €l0^2kTfj <Wpi r fp 2 [ P l ] ( 2 ' : iT i2Go( l + Pi2)lPif}p{3 (3.14) P12 is the permutation (exchange) operator for the 12 pair, and Eq. (3.8) has been used. With the introduction of the Mayer /-function F 0 ( r i 2 ) = eT®Uvi - 1 , Eq. (3.14) is split up into a sum of a term where the pair 12 do not have any spatial correlation prior to collision and a term where 1 and 2 are spatially correlated prior to colliding, VKM = e2^2kTJS ^ i 2 e f r i d p 2 [ p i j ( 2 ) : i T i 2 ( l + P12)[PlpW2 2 n el0m,2kT fj driadpidpalpiJ^tiriaGoCl + P ^ ) ^ ! ] ^5 ^ ^ ) ^ ! ^ . (3-15) 42 The operand to the right of Go in the first integral in Eq. (3.15) is only momentum dependent and so this operator produces a 1/e factor. The linearized collision operator £ x for an uncorrelated pair of particles is defined as, £i(pi ) = J dr12dp2iTi2(l + Pn)<P2- (3.16) Kawasaki and Oppenheim use the script £ rather than the fraktur £ for this quantity. The £ is related to the linearized collision operator Hi of Eq. (2.65) by, Hi = —£i(pi)v>i. (3.17) In the limit of z —¥ 0 when operating on a function of the momentum F(p i ) , £i gives, Zi(Pi)F(pi)<Pi = - / dp2 j** di> / ° ° M 6 | P l ~ P 2 1 J Jo Jo m x [F(p[) + F(p' 2) - F ( P l ) - F ( p 2 ) ] < ^ 2 (3.18) Momenta with the prime superscript in F{p'-) have values prior to the collision, while mo-menta without superscripts in F(pj) have values after the collision has occured. Kawasaki and Oppenheim also define a collision operator for a correlated pair, t i (pi) = J dvl2dp2iTl2G0Fo(xl2){\ + Pl2)<p2. (3.19) In terms of these new operators, Eq. (3.15) is written as, VKM = J <WPI] (2)= { - n e - 2 £ i - ne^h] [P lpVi- (3-20) Evaluation of rj^^e) follows in a similar manner. Nonzero contributions are, = € l Q ^ 2 V k T J *3dp3[Pi](a):ri2Go(r13 + r 2 3 )G 0 [i + p12 + Pi 3][Pi] ( 2Vi 3 2l (3.21) At the level of approximation required for first order density corrections, the position depen-dent part of p[% is replaced by 1 + F0(ri3) + -Po(r 2 3). The result is T&M « J ^ P i [ P i ] ( 2 ) : [n2e-H21 + n V 2 t 2 2 ] [ p ^ V - (3.22) The operator £ 2 i(pi) involves two consecutive binary collisions among three particles, with no prior spatial correlation among the particles, *2i(Pi) = - j dr13dT23dp2dp3Ti2(Ti3 + T2S)(l + P12 + F 1 3 )^ 2 v? 3 , (3.23) 43 while the operator £22(pi) has a similar interpretation but includes spatial correlation among the initial colliding pair, t 2 2 (p i ) = - j d r 1 3 dr 2 3dp 2 o!p3Ti2Go[Ti3Goi^ (3.24) Finally, the expression for rfKK(e) includes all the contributions from more than two consecutive collisions, elOmkT 1(2). SKP{N). (3.25) Approximations are introduced at this stage to simplify this expression. At the lowest order in density, only three or more collisions among the three particles 1, 2 and 3 are retained. The sums of operators that remain in the bracket in Eq. (3.25) is then denoted by r(123) = Y, iTaGoiTpGoiTyGo + (3.26) To first order in density, spatial correlations among the three particles are neglected. Within these approximations, the contributions from three or more consecutive collisions are, ,3 e220m2VkT n J d r 3 <i P 3 [ P l p : r (123) ( l + P 1 2 + P 1 3 ) [ P i ] ( 2 ) ./dPl[Plp) [ - n W ( P l ) ] [p 10m2kT where the operator t r (pi) is defined as, * r (Pi) ~\j dp2dp3dri3dr2 3r(123)(l + p 1 2 + p13)<p2<p3. rjKK(e) is obtained by adding Eqs. (3.13), (3.20), (3.22) and (3.27), n with V K K { e ) = m^kr I ^ P i [ p i ] ( 2 ) ^ ^ [ p i ] ( 2 V i + o(n 2) 1 / ( 0(e) = , £1 , h\ . 2 {hi . hi t e V e2 e J V e3 e2 e (3.27) (3.28) (3.29) (3.30) This is a powers series in the number density whose inverse 1 Q-l(e) = e + n£i + n2 which, in the limit of e —> 0 + becomes - t 2 2 + (£ x£i + t i£ x ) + f + - (£f - t 2 1 ) + e(nti + n2t{) + ... (3.31) G+l = lim G~l{e) = n[£i+ + n(ti+£l+ + t\ + Rx + R2)}. (3.32) 44 The operators R\ and R2 are denned as, Ri = - t2i+) R2 = £i+ti+ - t22+ (3.33) The crucial assumption required is that a function BKK(PI)[PI\^ifi, as yet undeter-mined, exists which satisfies n - ^ K K f o i H P i p V i .= 0 + [ P i ] ( V - (3-34) If such a function exists, the resummation of the divergent series of Eq. (3.30) can be accomplished. A n equivalent way of stating Eq. (3.34) is that the operator Q'1 is well defined and n - 1 ^ 1 5 ^ ( p 1 ) [ p 1 ] ( 2 ) = [ p 1 ] ( 2 ) ^ 1 . (3.35) The notation B K K is chosen to indicate the similarity of this function to the B function in the Chapman - Enskog [7, 62] solution of the Boltzmann equation of the shear viscosity. The tensorial properties of the unknown function must be equivalent to [pip- 1 since this is the only symmetric traceless second rank tensor that can be constructed from the momentum. Since Q~X is in the form of a density expansion, it is convenient to also expand BKK(p{) in powers of density, BKK(pi) = £ ( 0 ) ( P i ) + nB^M + 0(n 2 ) . (3.36) Substituting the expansion into Eq. (3.35) and equating powers of density gives rise to a pair of equations, £ + ( p i ) 5 ( 0 ) ( P i ) [ p i ] ( 2 V i = [ P i ] ( 2 V i , (3-37) and £ + ( P i ) ^ ( P i ) [ P i l ( 2 V i = - i i + [Pi] ( 2 Vi - * r + B ( 0 ) ( P i ) [ P i ] ( 2 V i - (3-38) The contributions of the operators Ri and R2 in Eq. (3.38) ultimately cancel each other [51]. B^(pi) and B^K(pi) are unknown functions which must be determined by solving Eqs. (3.37) and (3.38). Equation (3.34) is then used to substitute these functions into Eq. (3.29) to give an expression for the low density viscosity and its first order density correction from kinetic - kinetic contributions, VKK = YOnl^kf I ^P i [P i ] ( 2 ) = [ P i ] ( 2 ) ^ ( 0 ) ( P i ) ^ i + j d P i t P x l ^ r l P i l ^ s g ^ ) ^ = r/°> + nr$K. (3.39) 45 -qi0) gives the purely kinetic momentum transfer, while rf^K is the kinetic momentum transfer between layers of the flowing gas when the particles moving among the layers are spatially correlated and/or suffering triple collisions. The other contributions of Eq. (3.4) are obtained in a similar manner. The kinetic -potential term gives a first order density correction. This contribution involves the transfer of potential momentum flux of correlated particles to kinetic momentum flux. Determining this contribution involves solving the equation, £ I + ( P I ) B K V ( P I ) [ P I ] ( V = - / d p 2 d r T i 2 G o [ r ] ( 2 ) ^ e - t t / A ! T ¥ ) l V ? 2 , (3.40) and substituting BKV(pi) into the following integral, VKV = Y^kf I ^ P i [ P i ] ( 2 ) : ^ v ( P i ) [ P i ] ( 2 V i - (3-41) The potential - kinetic contribution corresponds to the collisional transfer contribution in the Boltzmann equation, VVK = Vet = ~Yo^kf II ^ P ^ P ^ [ r ] ( 2 ^ : G 0 T 1 2 J B ( ° H p i ) [ P i ] ( 2 V i . (3-42) Finally, it can be demonstrated that rjVv does not contribute a first order density cor-rection to the viscosity. It is convenient to define = B$K + BKV such that the addition of Eqs. (3.38) and (3.40) gives, - / d p a d r T i a G o e - ^ W W — ^ ^ . (3.43) Kawasaki and Oppenheim show that the right hand side is equivalent to corrections which appear in the Choh - Uhlenbeck approach (designated with a tilde), £ i + ( p i ) B ( 1 ) ( p i ) [ p i ] ( V = - - M ( p i ) - tT(Pi)5 ( 0 ) (Pi)[Pi] ( 2 Vi- (3-44) where M(Pl) = mj dr2dp2 • ^ n12[prpV lV22 - B ( T ) [ P l ] < 2 V i , (3-45) and £+(pi) is defined in Section 3.3.4. M. is a combination of the kinetic - kinetic term for correlated particles and the kinetic - potential term. This expression includes the M0ller operator f2i 2, and the second pressure virial coefficient B(T). With the definition of B^l\ the first order density correction has a slightly more compact form, V{1) = j d p i [ P i ] ( 2 ) : £ ( 1 ) ( P i ) [ P i ] ( V + Vet- (3.46) 46 3.3 Equivalence of the Two Approaches Kawasaki and Oppenheim's derivation ends at the stage of the previous section. This cal-culation is limited to repulsive potentials, since for a bound pair, the linearized collision operator £ x is divergent. They show their relations to be equivalent to those of Choh and Uhlenbeck [12], but do not make comparisons with the density corrections of Snider and Curtiss [19]. In this section, the equations of Section 3.2 are solved within certain approxi-mations and the equivalence, or otherwise, of the approaches is studied. To our knowledge, this has not been done previously. The final form of the first order density corrections of Snider and McCourt [20] is given by their Eqs. (20), (21), and (27). These are the equations to which the Kawasaki and Oppenheim results are compared. In the previous section it was seen 77 = r / 0 ) + nr? ( 1 ), (3.47) where Eq. (3.39) gives the shear viscosity to the zeroth order in density and Eq. (3.46) the first order density correction. To obtain an explicit expression for the viscosity, B^ and B^ must be determined by solving Eqs. (3.37) and (3.44). This is accomplished by expanding B^(pi) in terms of Sonine polynomials in of index 5/2, £ ( 0 ) ( P i ) [ P i ] ( 2 ) = £ b r S $ ( ^ i ) [ P i ] ( 2 ) (3-48) j where W j = Pj/(2mkT)1/2 is the dimensionless momentum of particle j. The orthonor-mality conditions of these polynomials [16] makes them a suitable choice for the expansion. p ( i)[p 1 ](2) i s similarly expanded in terms of Sonine polynomials, £ ( 1 ) bi)[Pi] ( 2 ) = E ^ ) ^ / 2 ( ^ i 2 ) [ P i ] ( 2 ) - (3-49) 3 Keeping the lowest order in the Sonine polynomial expansion, which is the order of approx-imation of Snider and Curtiss, only the constant first Sonine term Sf^2{W2) = 1 in these expansions are retained. Therefore 5(°Hpi)[Pi] ( 2 )^b 0 0 )[P l]( 2), (3.50) and similarly B ( 1 ) (Pi)[Pi] ( 2 ) *b<Vl ( 2 )- (3-51) 47 Consistent with the one-moment approximations for and B^l\ Eqs. (3.37) and (3.44) are dot contracted on the left into 5 (^14 )^ [pip/10 = [pi](2)/10 and integrated with respect to P l , giving 1 J dp 1 [p 1 ] ( 2 ) :£ 1 + (p 1 )bS° ) [p 1 ] ( 2 )^ 1 = ± J dp i [p i ]« : [p i ]< 2 Vi (3-52) and ^ / ^ P i [ p i ] ( 2 ) : £ i + ( p i ) b ^ [ p i ] ( 2 V i = ^1 d p i t P i l ^ r t T f P i J b ^ I p i l ^ V i - ^ / r f p i [ P i ] ( 2 ) : - M ( P i ) . (3.53) The factor of 1/10 leads to a simplification of the numerical factors in later stages. These two equations are solved in the following sections. Higher moment approximations to the viscosity can be obtained by including, more terms of the expansions in Eqs. (3.48) and (3.49). Equations (3.37) and (3.44) will then have to be contracted into a function which includes more terms of the corresponding series ^ 5 / 2 ( ^ / 2 ) i P i F ' ) and then integrated with respect to P i - The result will be a set of equations for the unknown bf^ expansion coefficients. 3.3.1 Evaluation of 4°^  The right hand side of Eq. (3.52) is easy to evaluate ^ / ^Pi[Pi] ( 2 ) : [Pi] ( 2 Vi = / d W ^ F M W i p e ^ 2 = (mkT)2. (3.54) Then using the definition of the operator £ i + ( p i ) , Eq. (3.52) becomes, (mkT)2 = d p 1 d p 2 / o 0 0 6d6b<° )p[p 1]W:{[p' 1]W + [P'2P - [Plp - [ p 2 ] ( 2 ) } ) W 2 m ° (3.55) It is convenient to transform the momentum tensors into relative and center of mass coordi-nates, [Pi] ( 2 ) = ^ [P] ( 2 ) + [p] ( 2 ) + [Pp] ( 2 ) , (3.56) [Pi] ( 2 ) + [P 2 ] ( 2 ) = ^ [P] ( 2 ) + 2[P](2)- (3.57) Making use of the invariance of the center of mass momentum during a collision, P' = P, Eq. (3.55) becomes, (mkT)2 = J dp jf MM P p) :{[p ' ] ( 2 ) - [p]^Mp). (3.58) 48 The integration with respect to P has been carried out. Introducing the dimensionless relative momentum 7 = (p2/mkT)1!2 and contracting the tensors gives, , _ . 2 16b 00)m2(kT)5^ / T T X 1 / 2 /•«> / • » 2 7 2 (mkTY = — - — — / / d'jbdbe 7 7 (cos x — 1) (3.59) 5 \mj Jo Jo where x is t n e angle of deflection in a binary collision. The conservation of energy shows the collision does not change the magnitude of the relative momentum and p' = p is used in deriving Eq. (3.59). The 0^ , , s) collisional integrals [7], defined as, fi«.*)(r) = ^ ^^y/2°^° j r ° ° d 7 6 d 6 e - 7 2 7 2 5 + 3 ( 1 _ c o s * x ) ( 3 6 0 ) are used to express Eq. (3.59) in compact form, (mkTf - 8 f e * Z 2 ! n M (3.61) 0 p is the reduced mass of the colliding pair. The value of the expansion coefficient b00^ is obtained from this equation, C = ^ry (3.62) 3.3.2 Evaluation of FromEq. (3.45), ^ / d P l [ P l ] ( 2 ) : 7 W ( P l ) = ^ / d P l [ P l ] ( 2 ) : [ P l ] ( V / / dr2dp2dPl[Plp): {^~) ^ [ p r p W a - (3.63) The first term on the right hand side is easily evaluated and Eq. (3.63) written as, ^ / d P l [ P i ] ( 2 ) : . M ( p i ) = B(T)(mkT)2 UJ drfadtoM®: • ^  n 1 2 [pr]( 2 )^^ 2 . (3.64) The M0ller operator does not affect the center of mass momentum, it converts the relative position to the vector r' defined in Section 2.5.1 and the relative momentum to its precollision value, po- Integrating over the center of mass momentum then gives, - 1 10 / dp i [p i ] ( 2 ) -A*(P i ) = B(T)(mkT) r.[nl(2)- I dr dp IJ drdp[pp>: [ p o r f ' V M e - * (3-65) 49 The integral on the right is integrated by parts and converted into dimensionless form, ^// drdp~([rp]<2> + [prp>):[p„r']<2V(j,)e - 4 ( T 3 ( 2 ) e - 7 2 e - 0 « — (mkTYF,. (3.66) The calculations required to identify Fv are given in Appendix A. The left hand side of Eq. (3.53) is treated in a manner similar to the left hand side of Eq. (3.52) and in the one moment approximation, only the constant term b ^ is retained. Collecting the terms of Eq. (3.53) gives, l(mkT)h^n^ = B(T)(mkT)2 - -^-(mkT)2Fv. 5 1DTT1'Z (3.67) The triple collision contribution in Eq. (3.53) has not been included in this expression. It is discussed briefly in Section 3.3.4. The expansion coefficient b0^ is, 8fi(2>2) (3.68) In solving the generalized Boltzmann equation, the two terms on the right of this equation arise from the non-locality correction terms A 2 and A x in the collision operator respectively. Here they are the result of a combination of the correlated kinetic - kinetic and kinetic -virial time correlation functions. By comparing the sum of Eqs. (3.62) and (3.68) with the result of Eqs. (20) and (21) of Snider and McCourt, 1 h 5 1 + " B ( T ) - 1 5 ^ (3.69) the combination b ^ + nb0^ can be identified as nbQ/2. It is to be noted that the perturbative amplitude b0 is density dependent. 3.3.3 C o l l i s i o n a l T r a n s f e r C o n t r i b u t i o n FromEq. (3.42), / dr fj d P ldp 2^i[r]( 2):n 1 2b( 0 ){[Pi] ( 2 ) + [p2](2)M</?2- (3.70) -n Vet lOmkT 50 The momenta are converted to dimensionless center of mass and relative coordinates and then the M0ller operator is applied, The tensors are contracted and integration over dimensionless center of mass momentum, G, carried through, The calculation of Appendix B shows that rjc t = - n m i 0 ) ^ J V , . (3.73) The corresponding term in Eq. (27) of Snider and McCourt is, 1 -47T 1 /V 3 -nkTbo N„. (3.74) This term originates from the effect of perturbations on the distribution function in the potential part of the pressure tensor. The two corrections agree if b ^ is identified as the lowest order density term of n6 0/2, which is consistent with the relation from Eq. (3.69). 3.3.4 T r i p l e C o l l i s i o n C o n t r i b u t i o n Kawasaki and Oppenheim obtain a contribution to the first order density corrections from three-body collisions which they demonstrate to be equivalent to a similar contribution from Choh and Uhlenbeck. The explicit term in Eq. (3.44), involves the operator ^ ( P i ) = - / / d*2dx3e12 J™dtS$(12) X {(013 + 023^123 - ^12^13^13 + ^ 3 ^ 2 3 ] } ( l + Pl2 + A 3 ) W 3 - (3.75) Hoffman and Curtiss [22] have studied this correction extensively and shown the relation between their correction and that of Choh and Uhlenbeck. This correction is not discussed further and it is only mentioned that the Hoffman-Curtiss correction, r / 3 M ) can be added to the shear viscosity to approximate this correction. 3.3.5 S h e a r V i s c o s i t y Returning to Eqs. (3.47) for the shear viscosity, " = rn^f I * I N ( 2 M P I ] , 2 ) » ? V + loJw / <WP>] ( 2 MPI] ( 2 , ^V . + »*. (3.76) 51 The first two terms of the right hand side are kTb00^ + nkTb0l\ Substituting the values obtained for b0°\ b^ and r?ct from the previous sections gives, 5kT f 4 n a 3 „ 4n7r1/2<r3 . ^ s o ^ r ^ - i ^ ^ — 1 5 Comparison with Eq. (27) of Snider and McCourt shows that the two expressions agree exactly. (3.77) 3.4 Inclusion of Bound Pairs In this section the first density contribution to the viscosity from the presence of bound pairs is determined within the scope of the Rainwater - Friend theory. In a system with an attractive intermolecular potential formation of bound pairs becomes possible. Accounting for bound pairs by considering the gas to be a nonreactive binary mixture gives rise to density corrections to the shear viscosity which are straightforward to evaluate. The general time correlation function expression for the viscosity of a binary mixture was derived by Mori [63]. This section illustrates that by extending the method which Kawasaki and Oppenheim used to determine the viscosity of a pure gas to a binary mixture of free molecules and bound pairs, an expression identical to the viscosity of binary mixture from the Boltzmann equation approach can be derived. Only contributions from the kinetic part of the flux of the nonreactive mixture are considered. Adding this result to the relations of Section 3.3.5 takes the time correlation formalism for shear viscosity to the level of the Rainwater theory. The viscosity of a mixture of Nf free (/) and Nb bound (b) molecules is, V = Yol7/cT /f drNdpNJK:G(e)JKpW (3.78) with the kinetic contribution to the momentum flux ] K , i K ^ £ W » + £ W P = i K t + i K t . (3.79) The distribution function for an ideal gas mixture consisting of Nf free and Nb bound molecules is Pw = n ^ n ^ M (3.80) j=x v e=i v A binary collision expansion for the resolvent of the binary mixture is similar to Eq. (3.7), G = Go — i £ G0TaGo — £ GnTaGnTpGo + i £ GoTaGoTpGnT-fGn + ... (3.81) a a,0 a>/8,7 52 except now the indices of the summations include all possible collision combinations between free and bound species. 3.4.1 Viscosity of a Mixture of Free Molecules and Bound Pairs A n approach similar to Section 3.2 allows the calculation of the viscosity of a binary mixture. The expression obtained is compared to the standard expression for the viscosity of a binary mixture from the Boltzmann equation [7]. Utilizing the binary collision expansion, the analog of Eq. (3.29) for the viscosity of a binary mixture is, W ^ £ ^ / * £ I f W ) ! f W ) (3.82) where £ and £ denote the molecular species present in the mixture. The sum over species is written explicitly as, with the Q operators defined by, C7//(e) = I_n I£//_n6l£/" + . . . ) (3.84) gf(e) = - n b ^ b + ..., (3.85) gb\e) = - - n b - 2 H h b - n f ^ b f + ..., (3.86) gbf(e) = -UfjZbf + .... (3.87) The form of the operators is obtained by retaining only terms involving up to a single binary collision among uncorrelated particles in the binary collision expansion, Eq. (3.81). The operators have meanings similar to the one component case, with the superscripts indicating the colliding species in the 7^ operator. Specifically, = J dr f 2 dpf2«7i i{ 2 ( l + ^if2)v(Pfi) and ^ = J drcdpci%c<p{pc)- (3-88) Equation (3.83) is thus written, (3-89) mb nb r ir>J(2) / dPb^-: fi - nb\zbb - r v V ' a + Pfb) J mb le ez el lOkTJ r u mb le 53 To simplify the notation, species permutation operators are defined which interchange the species labels on the flux functions (but not the Maxwellians) which follow them, for example, !p£ = (Ee ( 3 9 0 ) m f mb The species permutation operators are contrasted with the permutation operators used pre-viously which interchange the labels of molecules of the same species. Each of the brackets in Eq. (3.89) is a series in powers of 1/e and appears to be divergent. The series are inverted and terms of the lowest order in e and number density n retained as in Eq. (3.30) of Section 3.2. The resumed series are assumed to be finite and give the following set of equations, [ n ) ^ + n ^ f t l + n , ) ] ^ / ) ^ (3.91) [ng£» + wf£»(l + Pfb)]Bb(pb)[2!!^ (3.92) TTlb TYlb The functions B^(p^) are to be determined by solving these equations. 3.4.2 S o l v i n g fo r t h e B^s The Bf(pf) and Bb(pb) functions are expanded in terms of Sonine polynomials S^\(W^) and in a one moment approximation only the constant terms, and b^ are retained. Eq. (3.91) is then right contracted into ^ ^ j - and integrated over py, J rrif nif 10  f rrif ' +  10 J rrif mf 1 10 J rrif mb = ( a) + (6) + (c) (3.93) The first term (a) represents the loss of polarization (non-equilibrium flux) of a free molecule associated with free - free binary collisions. From the definition of the free - free linearized Boltzmann collision operator , njbf , [p,p „{p,p 54 o rrif rrif P/iP ^ [P/aP [ P / i ] ( 2 ) [P/2P + zZIil— - _ i £ ^ f i _ v l ( p (3.94) I m/ m , m,f mf J The two characteristic free particles of this collsion are labeled by the subscripts fl and /2 and pff is the relative momentum of a pair of free particles. Converting to center of mass and relative coordinates of the free - free pair [P / i ] ( 2 ) + [P/2](2) = [P / / ] ( 2 ) + [P/ / ] ( 2 ) , (3-95) and intergrating over the constant center of mass momentum leads to (°) = ^ / b / ° } / ^ / / ^ / / [ P / / ] ( 2 ) : ( [ P / / ] ( 2 ) - [P / / ] ( 2 ) }^ / = n l b f - — ( 3 . 9 6 ) 5 The middle term, (6) of Eq. (3.93), represents the loss of polarization from a free molecule as a result of a binary collision with a bound pair. This term can be handled in a manner similar to that of (a) namely, w 10 J f rrif + rrif r / - ^ f d P f d p t r U b m ^ M . M l \ v m (3.97) 5 J Jo afb rrif rrif rrif r roc \ p f b \ [ p f p : \ [ p ' f p [pfp Hfb rrif '{ rrif mf The center of mass and relative momentum coordinates of the colliding free and bound pair are, P/6 = Pf + Pb P/ft = ; P / f Ph rrif + mb rrif + mb pf = !^Pfb + pfb p 6 = M p p (3.98) mb rrif \ifb is the reduced mass of the free - bound collision. Converting to these new coordinates, gives, 5(ifb J Jo ymfiml rrif rrifmb rrif rrif rrifmb 55 h(o) / r M M p / 6 P / 6 [ p / 6 ] ( 2 ) : { [ p y ( 2 ) - [P / 6 ] ( 2 ) }V(P /^ -nn bm'jufb (2). 4.7rnfnbbf rr°° x j [ P / ^ ] ( 2 ) - [P / 6 P / h ] ( 2 ) }cp (p / f c ) ^ (P / h ) . (3.99) After contracting and integrating over the center of mass momentum the structure of the omega integrals is recognized, (b) = 5m2,1 7//J/6 J Jo 3(m f.f bdbdpfbp7fb(cos2 Xfb - l)<p(Pfb) 7T UfUbb) r r°° —• ^ / / bdbdpfbpfb(cosXfb -if + mb)2nfbJ Jo J J J = nfnbbf0) (kT)2 16 %60(2,2) 32 / i / 6 5 m 2 f» + 3 mf + mb fb (3.100) Finally, the third term, (c), of Eq. (3.93) gives the transfer of polarization from the bound molecule to the free particle as a result of a free - bound collisions, (c) (0) n/nbbb 10 , (o) -Trnfnbbb / d p , [p/] ( V M.2) m/ mb / d p , d p 6 / ^ M i P £ ; ( i P i P _ M ! l ( 3 1 0 1 ) 0 /J/6 771/ [TTlft 7776 J Calculations similar to those used in the evaluation of the term (b) show this integral can be simplified to, c = -nnfnbbb0) f r u ^ m _ \ J J ^ ( 2 ) + [ p / 6 ] ( 2 ) M 2j±fb_[P „ n „ 1 (2) / / bdbdpfbdPfbpfb 5afb J 7 o {mfm2L~ J"' ' rrif 'mfmb^PfbPf^ ^ - ^ - ( | P " P ' " P - P ^ ) } ^ ™ 16M/60(2,2) 32 /^ ,/6 n ( l i l ) 3 777/ + 777,6 7776 = nfnbbb(kT) 5 m 2 / b (3.102) Terms from (a), (6), and (c) are put together to get a simplified form for Eq. (3.93), namely , 2l n (2 ; 2) , Pfb r 3 777/ + 7776 ^ 6  16/i/60(2,2) 32 /i/6 (^1,1) 5 77T 1 + nfnbbb (o) 16Ai/6Q(2,2) 32 /i/fc o ( 1 ) 1 ) o S ' f l 0 Ji7fc 3 777/ + 7776 5 7772 ^ 6 (3.103) 56 A n analogous form is derived from Eq. (3.92), R O ( 2 , 2 ) n 2 = n 2 b ( 0 ) 8 ^ + nbnfbf 1 6 / 4 f (^2,2) _ 32 fj,bf 5 mI bf 3 rrif + mb bf + nbnfbl (0) 16M6/ 0 (2,2) 32 nbf n ( 1 ; 1 ) 5 m 2 b f ' 3 rrif + mb' (3.104) Equations (3.103) and (3.104) can be written in terms of mole fractions. Using M — rrif + mb for the total mass of the colliding pair, 0 OI 4 f f X2f-4^ + XfXb 16 m 2 ( 2 ) 2) 3 2 m / 6 ( l i l ) . h(o) - J0* — Of. 16 771/7716 (2,2) y y 32 r^ff l j ( l j l ) + h ( 0 ) +b6 V28iAfc A i —-5 M 2 " 6 / 3 M 2 " 6 / ' ° " 4. y v 1 6 m ' o<2'2) 4- X X 3 2 M / M & 0(1,1) 3 M 2 (3.105) From the identities in Appendix C, the cross sections for the binary mixture shear vis-cosity H (Greek eta) are identified and the system of equations becomes, bfHff + bb0)Hfb = X}/kT b{f0)Hbf + b{b0)Hhb = Xl/kT These are solved for b f and b^ to give, (3.106) (3.107) kTb (o) _ XjHbb - X^Hfh kTb{0) = X b H f f ~ XfHbf * HffHbb — HfbHbf1 0 HffHbb — HfbHbf These factors are substituted in Eq. (3.89) to obtain the viscosity of a mixture (3.108) V kT [bf + bi0 )] P/ ] ( 2 ) . , (o ) [P/] (2) •7 -<Pf [P»]{2) .AO) [P6] (2 ) rrif J rrif 10kT J mb mb XbHff + X'fHbb — XfXb(Hfb — Hbf) HffHbb — HfbHbf (3.109) which is identical to the standard result for the viscosity of a binary mixture. Assumptions similar to those of Stogryn and Hirschfelder can be made to get a first order density correction to the viscosity due to bound pairs. 57 3.5 Kawasaki and Oppenheim's Modifications Due to Attractive Forces Attractive forces and the possibility of formation of bound pairs were dealt with by Kawasaki and Oppenheim in a separate article [53] which does not use the binary mixture calculation of the previous section. The treatment formally considers the possibility of reactive collisions which lead to the formation, decomposition or exchange in bound pairs. The potential contribution to the momentum flux allows for these possibilities. Kawasaki and Oppenheim did not attempt to compare their results with previous work on transport coefficients in the presence of bound pairs, in particular the work of Stogryn and Hirschfelder. Within our limited understanding of their work a number of problems exist which are discussed in this section. The results of their calculations is that the viscosity is expressed as a density expansion 77 = 7 7 ( 0 ) + nr/ 1 } . (3.110) The density independent term is the same as in the absence of bound pairs, ^ = WrTrWkf I ^ P ^ t P ^ P V i (3-111) The first order in density correction is composed of terms analogous to the repulsive potential case, V{1] = VKKI + VKKT + VKI + V{V]K + Jvl (3.112) where VK\<I 7 7 ( 1 ) 'iKKr 1 f d p i f e i £ : £ ; i ( p 1 ) t l + ( p 1 ) f e l £ v , 1 > (3.113) J 771 777, bf I ^ ^ < ( P ^ V P I ) £ ; V ) ^ ^ (3.H4) lOVkT WVkT V{K]V = 41 = YoTTfcT / ^ P i ^ 2 ^ : [ r - 1 ( 1 2 ) e G 2 ( 1 2 ) ] + [ r 1 2 F 1 2 ] ( 2 ) e - ^ c l > ( p 2 ) ) ( 3 . 1 1 5 ) rf& = i ^ / d P l d x 2 [ r 1 ^ (3.116) In these equations, n _ 1 £ + 1 ( P i ) is the binary collision operator or the modified propagator for a free particle which can be considered a measure of the mean free time between collisions for particle 1. The n _ 1 r - 1 ( 1 2 ) similarly represents the modified propagator for a pair of interacting particles and can be considered a measure of the mean free time between collisions of the pair 12 with other particles. Explicit expressions for each of the inverse operators are given in Ref. [53]. 58 Eq. (3.114) represents the effects of collisions involving three particles. These particles may all be free, in which case the expression represents a triple collision (this is illustrated in Fig. 6(b) of Ref. [53]), or if two of the particles are bound, it represents a free - bound collision. The contributions from bound - bound collisions are not explicitly discussed. When analyzed, these terms may lead to corrections of the type given in Section 3.4. At lowest gas densities, bound pairs should produce a density correction reflecting the cross section of free - bound collisions which arise solely from kinetic effects. It is this form that is expected to arise from (3.114) when considering the system to be a binary mixture of free and bound species. Corrections related to potential contributions to the momentum flux for repulsive po-tentials have been discussed in detail in Section 3.2. For an attractive potential, other than non-locality and collisional transfer corrections, contributions to the viscosity from the formation and decomposition of bound pairs and particle exchange are considered in Eqs. (3.115) and (3.116). Within our understanding of their work, some problems appear to ex-ist with regard to these expressions. In the derivation of the viscosity for repulsive forces, the potential contributions to the momentum flux [r^F^p) involve forces among separate particles j and £. If the particles j and £ are bound together by the force Fje, a term of the structure [r^Fj^p) no longer contributes to the momentum flux. Therefore, forces in the potential contributions to the momentum flux must only be between separate particles and not internal to a bound pair. In deriving Eqs. (3.115) and (3.116), it was assumed that the pair 12 may constitute a bound pair. In this case it would be incorrect to have the flux function [r^F^P^ as the potential contribution to the momentum flux since it is assumed that the momentum of the bound pair is carried by the motion of its center of mass and the relative motion does not contribute to the flux of momentum in the system. The rigorous evaluation of the kinetic - virial cross terms in the presence of bound pairs requires a detailed knowledge of the three-particle collisions. The cross sections of various outcomes like bound pair formation and decomposition would be needed in order to evaluate the rifty term. 59 Chapter 4 Complete Binary Collision Approximation for Dilute Gas Transport Coefficients via the Time Correlation Formulation As stated in Chapter 2, the original motivation for examining the time correlation function formulation of the transport coefficients was to understand the density corrections to the dilute gas transport coefficients, in particular to establish any relation to the viable compu-tational method of Rainwater and Friend [8]. The latter theory is a composite of a number of developments, whereas a derivation from a unified formalism would be desirable. This would also require understanding the role that bound pairs play in gas transport. As a closed equation for the transport coefficient, Eq. (2.29) should be capable of evaluation without recourse to the Boltzmann equation, and in doing so, provide an independent approach to justify or contradict the density corrections obtained by the Boltzmann equation and its generalizations. It appears that most published comparisons between the time correlation function formulation and the results of the Boltzmann equation [34, 45, 47, 50, 64] evalu-ate the time correlation function using the Boltzmann equation or some generalized kinetic equation. In contrast, Zwanzig [52] uses a binary collision expansion to deduce that the eval-uation of Eq. (2.29) requires the solution of a series of integral equations, the lowest order of which is recognized as the linearized Boltzmann equation. But in order to carry out this procedure, Zwanzig had to resum an infinite series in powers of 1/e whose individual terms formally diverge as e —> 0. As seen in Chapter 3 Kawasaki and Oppenheim [51] extended Zwanzig's method to obtain density corrections to the viscosity coefficient and make contact with the theory of Choh and Uhlenbeck [12]. The method presented here avoids the 1/e expansion by using another technique, also introduced by Zwanzig [65], namely the use of 60 projection operators. While examining the density corrections to the dilute gas transport coefficients, it was noticed that the standard treatments that make connection between these disparate formula-tions (time correlation versus Boltzmann equation) appear to focus on only the first moment approximation to the Boltzmann equation and do not consider the derivation of corrections associated with higher moments (Sonine polynomials) for the accurate calculation of dilute gas transport coefficients. It is the purpose of this chapter to show how these higher moments arise solely from the time correlation formulation. Thus the applications presented in this chapter are limited to gases at low density to deduce expressions for the transport coeffi-cients in terms of kinetic cross sections, while retaining all binary collision effects. The same method of approach is adapted to the inclusion of the necessary multiple moments required for the calculation of the transport coefficients of mixtures. The presentation for mixtures is limited to keeping only one expansion term per species, but it is clear this could be extended to all binary contributions. Many of the techniques shown here are used when expressions for the density corrections to the transport coefficients are derived in later chapters. In Sec. 4.1 the properties related to the rotational invariance of the resolvent operator and flux functions are discussed. It is shown how these lead to simplifications in the evaluation of the ensemble averages which appear in the time correlation function expressions for the transport coefficients. Section 4.2 gives a brief introduction to projection operators used in this work and demonstrates their idempotency using the results of Sec. 4.1. In Section 4.3 properties of the projection operators are used to carry out a direct resolvent expansion of the time correlation function for a transport coefficient. A result similar to that of Zwanzig [65] is derived. For dilute gases it is clear that a binary collision expansion [60, 61] is needed at some point in this relation. In Section 4.4, the flux function J of Eq. (2.29) is assumed to be a sum of 1-particle functions, and with the use of a binary collision expansion, an analytic solution is obtained for the low density part of the transport coefficient. In Section 4.5, this solution is interpreted as including the complete set of Sonine polynomials. Section 4.6 reinterprets the formalism of Section 4.4 to derive an expression for the transport coefficient of a system which is a mixture of two species. In that case, for simplicity, only a single moment for each species is considered. The chapter ends with a discussion. Most of the material in this chapter has appeared in a previous publication [66]. 61 4.1 Rotational Invariance of the Resolvent Matrix El-ements From Eq. (2.29) it is recalled that matrix elements of the resolvent with respect to the flux vectors must be determined in order to evaluate the transport coefficients, a = lim lim 7 7 - ^ 7 7 <(J © — ^ ; J » . (4-1) JV/V=const. v ' In the absence of external fields, the resolvent operator is rotationally invariant and its form does not change when the coordinate system is rotated. This rotational invariance leads to the simplification of subsequent calculations. To make use of these simplifications, a number of concepts and some notation are introduced in this section. The continuous group of all rotations in three-dimensional space, SO(3), has an infinite number of irreducible representations, corresponding to the infinite number of elements and classes which it possesses. These irreducible representations have weights 0,1, 2, 3, • • •, a, • • • corresponding to their tensorial rank, and dimension 1 , 3 , 5 , . . . , 2 a + ].,•••, corresponding to the number of independent elements in each representation. Thus for example, the infinite set of vectors (a = 1) which have 2a + 1 = 3 independent components, form an irreducible representation of the three-dimensional rotation group. The dimension of the irreducible representation should be distinguished from the three spatial dimensions x, y and z which are common to all irreducible representations. If the operation of rotation in the three-dimensional space is represented by a matrix D, the rotational invariance of the resolvent operator implies, D — r D - 1 = — r - (4-2) z — L z — L The rotational invariance of the resolvent allows the use of a generalized version of the Wigner - Eckart theorem [67, 68]: When acting on J , a symmetric traceless tensor of rank £, the rotationally invariant operator 1 / ( 2 - / 2 ) produces a tensor T, i h 3 = T ( 4 - 3 ) of the same rank as J , which belongs to the same irreducible representation. That is, if a component of J transforms under a rotation R according to J , = X>(*)fe,J<„ (4.4) where the S)(R)$ are matrices for the irreducible representation of weight a of the rotation R, then the components of T also satisfy Tp = J2®(R){pajTa. (4.5) 62 It follows that if J is a three-dimensional Cartesian vector, then so is T and if J is a symmetric traceless second rank tensor, then so is T. The canonical distribution function is rotationally invariant in the absence of external fields and as a result, the inner product ((3(z — £) _ 1 J)) = ((J|T)) is an isotropic tensor of rank 2a. This isotropic, rotationally invariant tensor must be proportional to E^a\ the rotationally invariant tensor of rank 2a which acts as the Cartesian identity in the 2a + 1 irreducible representation of the three-dimensional rotation group of J, «J|T» = aE< a\ (4.6) with some scalar a, which is determined by magnitude and directional properties of J and T. The rotationally invariant 2a rank tensor acts like an identity for the irreducible representation. As such, it is useful to impose upon it the condition of idempotency, E ( « ) § E ( a ) = E (a)_ ( 4 ? ) The idempotency property is required when constructing projection operators in the next section. It is now appropriate to consider in greater detail the two cases that are encountered in this thesis. The flux functions studied in this thesis are the heat flux vector which belongs to the irreducible representation of dimension 3 of the 50(3) group and the symmetric traceless second rank transverse momentum flux tensor which belongs to the irreducible representation of dimension 5. The detailed form of the corresponding identity tensors and E ^ are studied below. For a vectorial flux function, a = 1 and the second rank rotationally invariant tensor E ^ must be a multiple of U, the unit rotationally invariant second rank tensor [69], U = £ ij Sij = xx + yy + zz. (4.8) For a three-dimensional space, the i and j indices, and unit vectors i and j are summed over the x, y, and z directions. A schematic geometric representation of the summation is suggested by the shape of the symbol U [70]. The two "branches" of the symbol U represent the number of indices or in other words, the rank of the tensor. The connectedness of the branches at the bottom shows that the indices must be identical, and in this manner indicates the presence of the Kronecker delta function. A summation over three spatial dimensions is implied. For vector flux functions J, the inner product ((J|T)) is a scalar multiple of the second rank rotationally invariant identity tensor, «J|T» = oU. (4.9) 63 To obtain the scalar multiple a, both sides are double contracted onto U, aU:U = U:((J|T» (4.10) which gives, « = ™ . (4.11) This follows from U:U = 3 and U : A B = A - B . Using the schematic representation described above, the branch of the U closest to the contraction is dotted into the vector closest to it (A), and the farthest branch of U is contracted onto the next vector (B). Since the indices on the two branches are identical, the result is the dot product of the two original vectors. For a symmetric traceless second rank tensorial flux function J like the transverse mo-mentum flux in shear viscosity, the identity is a fourth rank isotropic Cartesian identity tensor. This tensor must be a linear combination of the three rotationally invariant fourth rank tensors [69]. These isotropic tensors are given in terms of delta functions, U U = £ S^1 SijSkt = (xx + y y + zz) (xx + y y + zz) , (4.12) ijkt y = £ ijicl 6ik6jt, (4.13) ijkt (UJ = £ i j k i ^ f c = x ( J x + y | J y - t - z | J z . (4.14) ijkt The schematic representation introduced previously is also useful in this case. The four branches indicate that the tensors have four indices and are thus of fourth rank, and the connectivity at the bottom of the branches indicates the specific combination of indices which are identical. In three-dimensional space, the indices are summed over the three spatial directions x, y, and z. E^ 2 ' must be a linear combination of these three tensors and can be expressed as E ( 2 ) = aUU+fr(UJJ + y ) - (4.15) In Appendix D the coefficients chosen to make E ^ idempotent are determined to be E ( 2 ) = |(UJJ + y) - |UU- (4-16) For the mometum flux tensor J , ((J|T)) is a multiple of E^ 2 ' , « J | T » = aE(2>. (4.17) Contraction of both sides of Eq. (4.17) into E ^ allows the identification of the scalar multiple a, a E ( 2 ) £ E (2 ) = E ( 2 ) £ ( ( J | T ) ) ^ a = (JJ^l^ (418) 64 4.2 Projection Operators In the context of irreversible statistical mechanics, projection operators were first introduced by Nakajima [71] and Zwanzig [72]. Starting with the Liouville equation Zwanzig derived a generalized master equation using projection operators [65, 73]. More recent applications of projection operators have been reviewed by Hynes and Deutsch [74] and Grabert [75]. The nature of the calculation involved in determining the matrix element of the resolvent operator suggests the use of a projection operator. A complete set of orthonormal iV-particle basis functions { ^ ( r ^ , p^)} can be constructed of which J is a member, J = * 0 ( r i V , p i V ) (4.19) The tyj basis functions have tensorial properties identical to J . In general, they are not sums of individual particle functions and involve correlations among particles in a complicated manner. The resolvent produces a linear combination of these basis vectors in iV-particle space, 7 ^ J = ^ * 0 = £ a ^ ' ( 4 2 0 ) i but this linear combination is projected back onto \&o when calculating the matrix element, Z L. j It is seen that a direct calculation of the result of the operation of resolvent on the flux function J is computationally quite wasteful, since only that part of the complete evolution which remains in the J subspace has a nonvanishing contribution in the final result. By projecting the resolvent onto the subspace of the flux function J , the extra calculations can be avoided. This projection and its consequences are introduced in the next section. Any projection operator must possess the condition of idempotency, i.e., V2 = V. (4.22) For a flux function J of tensorial rank a, the projection operator is V = A\3)) Q ((3\. (4.23) The condition of idempotency requires that, V2 = A2\3)) © « J | J » 0 <(J| = V = A\3)) © «J | . (4.24) 65 Determining the constant A requires the evaluation of ((J|J)). If there are no external fields in the system, ((J| J)) is a rotational invariant, and in particular, a scalar multiple of E^a\ the rotationally invariant tensor of rank 2a which acts as the Cartesian identity tensor in the 2a+l-dimensional representation of the rotational group in three spatial dimensions. Proper normalization factors can always be chosen for the flux function J such that {(J j J)) = T V E ^ . For a vectorial flux function (tensor of first rank), if ((J| J)) is to be rotationally invariant, it must be a scalar multiple of the only possible second rank rotationally invariant tensor U [69], « J | J » = /VU. (4.25) Thus in Eq. (4.24), A 2 |J)).((J|J)).((J| = A 27V|J)).U.((J| = 4J».«J| , (4.26) and so A = 1 /N. For a tensorial flux function J which is a symmetric traceless tensor of second rank, verification of the idempotency property of Eq. (4.24), requires recognizing that ((J|J)) = J V E ^ . The calculations are otherwise analogous to that given for a vector flux function. 4.3 Projection Operator - Resolvent Relations The thermal average bracket (( | )) involves the "bra" and "ket" defined either in quantum or classical mechanics as « X | = J dxNX = T W X | Y » = Y / W = i [ Y , p W ] + « X | Y » = J dxNXYfW = ±Tvl...Nx[Y,pW}+, (4.27) where, if only translational degrees of freedom are involved, a phase space element is dx.N = dpNdrN, while in quantum mechanics, the noncommutation of an observable Y and the equilibrium density operator p^ requires, for example, the introduction of the anticommu-tator to preserve hermiticity. Obvious generalizations are required if there are internal states and/or the molecular system is a mixture. It is the classical terminology and language that are used here, but the classical formalism can be immediately transcribed into a quantum formalism. The bracket acts as a scalar product of two phase space functions so that a vector space language can, and is, used in the rest of the chapter. The "bra" and "ket" tensors are seen to be asymmetric with respect to the iV-particle distribution function f(N\ Different positionings of the distribution function are possible, but the above inclusion of in the 66 "ket" is the most convenient. Due to the asymmetric distribution of the distribution func-tion, even though the bracket combination is finite, the individual "bra" and "ket" tensors are not components of a Hilbert space. While J(t) changes as it evolves with time, only its component along the phase space J direction is of importance for the calculation of cr. This is equivalent to determining the matrix element of the resolvent that is diagonal in J . The generalization of Zwanzig's [65] projection operator is the projection operator V onto the 2a+l dimensional subspace spanned by the components of J V = A | J)) 0 ((J | . (4.28) The normalization constant A is introduced to make the projection operator idempotent V2 = A2 | J)) 0 ((J | J)) © ((J |= A | J)) © ((J |= V. (4.29) On the assumption that the equilibrium density operator (distribution function) is rotation-ally invariant, and that the flux operator J has no preference for any particular direction in space, it follows that ((J | J)) is a 2a rank tensor which must be a rotational invariant. ((J|J)) is thus a scalar multiple of E^a\ the rotationally invariant tensor of rank 2a that acts as the Cartesian tensor identity for the 2a + 1 dimensional irreducible representation of the three dimensional rotation group [36, 76]. Thus ((J | J)) = a E ( a ) (4.30) with a proportionality constant a determined by taking the contraction of both sides of Eq. (4.30) with E ( a ) to show that and by substituting Eq. (4.30) into Eq. (4.29), A = + (4.32) «J © |J» The general expression for a transport coefficient is given in Eq. (2.29). The right flux function begins in the J subspace. The resolvent changes both the magnitude and direction of this "vector" in the function space, but the dot product projects the resultant vector back onto the J subspace. Since V projects onto the subspace of functions determined by the components of J , it follows that the transport coefficient can be written a - l i m lim _ i £ _ ( ( J © \ V ^ — z PJ)), (4.33) e^o N,V^OO (2a+l)Vxx -ie-C " JV/V=const. v ; 67 involving the projection of the resolvent operator. This can be expanded in terms of the resolvent of the projected Liouville operator according to (for convenience and possible gen-eralization, this is expressed in terms of z — ie) P J ^ P = vz _ lvcv [z-c + (i- v)c(i -v) + veil - v) + (i - v)cv] -^—^v ~ 1 :V z-VCP The first term describes the evolution within the J subspace while the second term describes the influence of the rest of the function space on the subspace determined by J (the subspace complimentary to J) . Solving for the projected resolvent gives -p I j> = I (A 35) Z _ C z - V C P - V C z _ ( 1 J ) c { l _ v ) (1 - V)CV V ' ; Since the J's for the transport coefficients belong to irreducible representations of the rotation group with time independent normalization a, the projected Liouville operator VCP = 0. The proof [65] states that from the Liouville equation, ((J | L | J)) can be written as ((J | J)), which is proportional to d{{J\J))/dt. Since the equilibrium ensemble is constant with time, so is ((J|J)), its time derivative vanishes, TCP = 0 and Eq. (4.35) is simplified to ^ T 2 ^ = Z _ V C \ cp- (4-36) Wi th this expression for the projected resolvent, the transport coefficient expression, Eq. (4.33), becomes . . . . %B a 1 a = lim lim — TT—UJ Q - — i — — J)) = lim lim — — — ((J © |X)) (4.37) JV/V=const. v ' where |X)) is defined by l X » = Z . V C . £P-|J»- (4-38) 4 r ^ z-{l-V)C{l-V)^r Inverting this defining relation and using the e —> 0 limit, it is seen that |J)> = ~ V C ; ^ 7 (4.39) z - (1 - V)C(1 - V) 1 " x ' 68 The phase space inner product of this equation with ((J| implies that E<"> = - ^ ( ( J £ 2 _ ( l _ p 1 ) £ ( l _ p ) £ l J ) ) 0 A((J\X)), (4.40) where Eq. (4.30) has been used. Since both brackets on the right in this expression are 2a rank rotationally invariant tensors, they are each a scalar multiple of E^a\ Manipulations similar to those stated in Eqs. (4.30) and (4.31) give which results in an expression for the transport coefficient in the form 1 A2V a i - = lim lim — — ((J © £ — T - — £ I J)). (4.42) ° ^ % I J v l l Z , ( 2 a + 1 ) B z-(l-V)C(l-V) 1 / 7 1 1 It can be noted that the above manipulations have led to an operator combination that has the structure of an JV-particle transition operator. The Liouville operator in this expression is projected onto the subspace complimentary to J . The e —> 0 limit of a transition operator has well understood convergence properties. Physically, the study of general time evolution in the system, Eq. (4.33) has been reduced to the study of collision events. Zwanzig [65] simplified this to the equivalent form 1 A2V a i - = lim lim — ((J © £ — - £ | J)). (4.43) a e_x) N . V ^ O O (2cx+l)Bxx z-(l-V)C  1 "  y J N/V=const. V 1 / V ) His derivation involved the generalized Master equation with the consequent identification of the transport coefficient as the inverse of the Laplace transform of the memory kernel of that equation. 4.4 One Particle Properties at Low Density For a low density gas, the measured physical properties are all associated with individual molecular attributes. In particular, transport is due solely to the kinetic motion of the individual molecules, as distinct from transport associated with the intermolecular potential, which becomes significant at higher densities, see Chapters 6 and 7. Collisions disrupt these free kinetic motions, giving rise to the transfer of flux from one molecule to another. This interrupts the transport and gives rise to the resistance to flow of the fluid (viscosity), finite transfer of energy (thermal conductivity), etc. In such cases: 1) the phase function J is a sum of one particle properties; 2) the equilibrium averaging involves a distribution function 69 f(N) which is a product of one particle Boltzmann factors; 3) the intermolecular potential for the system is pairwise additive so that the Liouville operator has kinetic and potential contributions £ = £ ^ + £ ^ = - * £ ^ f + ^Y^~^ (4-44) fe fe fe m dvi fe d ™ d p k e and 4) termolecular collisions are to be ignored. It is the combination of these four restrictions that are used to derive otherwise exact expressions for the transport coefficients involving only binary collisions. In later sections the general formalism developed here is specialized, first to identify how all higher Sonine polynomial terms of the Chapman-Cowling solution of the Boltzmann equation arise from the time correlation formalism. This shows how the binary collision expressions derived here are the same as those obtained by exactly solving the Boltzmann equation. Second, the formalism is adapted to show how the Boltzmann equation expressions for mixtures arise from the time correlation function formalism. It does not appear that either of these detailed connections have been made previously for the general case. On the basis that any physical observable is associated with individual molecular at-tributes, it follows that the flux J is an additive property of the individual molecules in the system, J = £ j K i , (4.45) whose contribution, JKj, from molecule j is a function only of that molecule's momentum, Pj . For convenience of notation it is assumed that JKJ is normalized so that «Jjn |J*r i» = E ( Q ) , equivalents ((JK1 © \JKi)) =2a+l. (4.46) Provided the distribution function for the gas N /(A° = II & (4-47) is a product of one particle distribution functions which are normalized 1, (4.48) then the inner product of Eq. (4.31) is N(2a + 1) and the normalization factor for the projector of Eq. (4.28) becomes A = 1/N. Equation (4.42) for the transport coefficient can then be written 1 V a i — = lim lim „ A T / ^ TT((JKI © £ ; ^ „ / . I J)) 70 3N(2a + 1) { { J k i ° lim lim T . —T((JR:I 0 V12 ; „. ;=rr£ I J) ^ % ^ = : ~ , B N ( 2 a + 1 ) 1 2 z - ( i - p ) £ ( i - p ) = l i m l i l T 1 r^r,n ~'' \ / ^ J / a 0 B ^ , (4.49) £ ^ o BN(2a + l)J 1 v y W/V=const . V ' on the basis, first that J is a sum with all particles contributing equally, and second, that only the potential part of £ has any effect on JKi, with particle 2 being a typical particle interacting with particle 1. In the following a closed expression for the vector valued function = / d x ^ V u — V W ( J V ) , (4-50) J z — C-p is calculated. The shorthand notation C-p = (1 — V)C(1 — V) has been introduced for convenience. Since, in Eq. (4.50) for B ^ , the pair 12 is interacting, it is appropriate to identify this interaction with a binary collision between this particular pair of particles. This can be accomplished by introducing the resolvent for the pair evolution operator £ 1 2 = Y.i + Vi2-This is the Liouville operator of an iV-particle system where only the 12 pair are interacting. After reorganization, ~B[N^ can be written in the form B[N) = [d^N-Vv12—^—[z-CP+(l-V)£(l-V)-Cl2}^—C3fW J Z — *"i2 Z — JL-p = [d^N-Vvl2-^—CJfW J z - £i2 i + J dxP-VVu — -12 -£J/ ( I V ). (4.51) k<e This is a generalization of Watson's [60] binary collision expansion. Since the two left-most operators in the integrals of this equation, V\2(z — C\2)~l, involve only particles 1 and 2, and these in a symmetrical manner, the integration over all the other particles 3 • • • N can be carried out before getting to these operators. Moreover, the equations can be simpli-fied by emphasizing only the behaviour of one particle, chosen as that labeled by 1, with particle 2 taken into account by the introduction of the permutation operator P 1 2 . Further simplification is obtained by introducing the 2-particle transition operator T12 = Via + V 1 2 — ^ V 1 2 = Vl2—±—(z - /C 1 2) (4.52) z - L 1 2 z - Li2 and the reduced pair distribution function, f$ = J dx^^f^ = <fi{p2/V2. In terms of these quantities, the first term in B<"> is / • d x ( " - l ) V l 2 _ L _ £ J / W = / d X 2 V 1 2 ^ 7 r - V 1 2 ( l + P l 2 ) J ^ A ( 2 ) J Z — L\2 J Z — L,\2 = Jdx2iTv2(l + P i 2 ) J K i ^ = ^ n 1 J K l , (4.53) 71 (4.54) where the binary (linearized Boltzmann) collision operator, IZi is denned as n f Hl = / dx2iTy2{l + Pl2)</>1¥>2-fl J Technically, terms involving a third particle, such as / dx3Vi3(Jia + JKzjf i f i f i arise in simplifying the integral on the left hand side of Eq. (4.53). It is argued that the potential between particles 1 and 3 that is incorporated in the Liouville operator C (more specifically V13) depends only on the relative position r i 3 and the integral over the position of particle 3 can be converted to that relative position whose integration then gives 0. Similarly, the definition of T12 involves a separate V12 factor that does not appear on the right hand side first in the line of Eq. (4.53) but whose integration over the position of particle 2 gives a null value, J d x 2 V 1 2 ( l + P i 2 ) J ja / i ( 2 2 ) = 0. (4.55) This same procedure is not valid for the term that has been kept because of other r i 2 dependent terms. The remaining terms in are handled in a similar manner, 1 = J d x 2 T 1 2 7 - ^ ( l + P12) [(N -2)j dx<"-2>V (JV) £ 1 •C3f (N) -I dx2Ti2- 12 -(1 + P12) (TV - 2) j d x ^ " 2 ) - J j n A ( 2 2 ) ( J V - l ) © / dxN3Kl V 13 z — Ci -C3f (N) Jdx2rl2'-(l + P 1 2 ) [(N - 2 ) B g } - f[?3Kl(N - 1) 0 jdxiJjnB^ [-0 (4-56) with a reasonably obvious definition for B ^ , which is discussed further below. The last term of Eq. (4.56) can be simplified by the use of the projection operator P 0 = VI3KI © J dpi3Ki- (4.57) This is a projection onto the one particle subspace determined by J ^ i in the momentum space of particle 1, in contrast to V which is projector onto an TV-particle subspace. On collecting the various expressions, B ^ satisfies the equation (JV) VN ^ i J ^ i - ^ / d x 2 ? ; T i 2 ( l + Pi2) [(N-2) R ( A 0 a!2 TV — 1 cp2V0 J d n B H . (4.58) 72 The function can be simplified with the following approximations B<?> = /dx<"- 2 >V 1 3 —V^J /W J Z — L,T> V J Z — L-p = ^ B ^ " 1 * « ^ B ^ . (4.59) The first approximation is based on the idea that this term inherently involves the interaction (collision) between particles 1 and 3 with particle 2 entering only indirectly as a spectator for this collision. On the basis that only binary collisions are to be retained, it is thus appropriate to ignore any interaction with particle 2, thus its distribution function can be passed through the remaining combination of resolvent and Liouville operators. The remaining integral correctly involves only N — 1 particles, which is the second equality. It is a consequence of these approximations that termolecular collisions are eliminated. Finally, if N is very large, the B ^ - 1 - * function is essentially the same as B ^ , the second approximation. Consistent with this independence of N, the system size labeling on B x is dropped in the remainder of this paper. Another simplification of the same type is (N — 1)/V & (N — 2)/V a n. Since the system is homogeneous, Bx is independent of the position of particle 1, and it follows that / diiBi = V B i and Eq. (4.58) becomes a closed equation for B x , specifically Bx = ^ f t i J j a - ^ f t i — ( 1 - Vo)Bl. (4.60) VN z ip i The object is now to solve this equation. As a closed equation for a one particle function, there is no need to retain the particle label, so this is dropped in the further development. These manipulations show the motivation for defining the B i in Eq. (4.49). A similar closed equation can not be obtained directly for l/cr. Since Eq. (4.60) contains both the full B and its projected part (1-"P 0 )B, it is appropriate to first solve for the (1 - Vo) part of B . Left multiplying Eq. (4.60) by (1 — Vo) gives (1 - V0)B = (1 - Vo)^TlJK - -(1 - V0)<pn-(l - V0)B, (4.61) which can be solved for (1 — Po)B, Substituting this back into Eq. (4.60), taking the tensor contraction with JK, and integrating over the momentum of particle 1 gives 1 dP3KeB = V N J dp3K 0 fH3K 73 - / dpJK 0 iplZ— Z J Q5 <pz + i(\-VQ)<pKHl-VQ) (1 - V0)<plUK VN [(2a + l ) ^ o , o - / r f p J K 0 ^ i ( 1 - n ) ( i _ n ) ^ ( i _ P o ) (1 - Vo)<p1UK .(4.63) The operator inverse can be evaluated if an appropriate complete orthonormal basis {Oj} of 1-particle functions is introduced. This basis must be restricted to functions orthogonal to the collisional invariants, equivalently to the eigenvectors of the collision operator 1Z whose eigenvalues are zero. Furthermore, for convenience, the basis should have JK = O 0 (p) as a particular member of the basis and all members should belong to the same (2a + 1 dimensional) irreducible representation of the 3-dimensional rotation group, with orthonormalization determined by (4.64) Matrix elements of TZ, after taking their rotational invariance into account, are usually expressed in terms of kinetic cross sections 6 [77, 78, 79], namely (4.65) wherein (v) is the average relative speed of the pair of colliding particles. The projector 1 — Vo then merely selects out the O,, ^ O 0 elements of the basis, with the consequence that / dpJK © B 2a + 1 VN {2a + l)(v V2 w o 6 0 , o ^ o , o - E Hod (^ ) n{ E ©<U (^) ©£,0 (4.66) Finally, the transport coefficient is determined by taking the appropriate limit of this expression, namely _V2 JV/V=const. — = lim lim 7 - - 7 — -a £_>o Af ,v-»oc (2a + l)B J dpJK a QB ©0,0 - E (^) &e,o (4.67) The result usually presented (see for example Ref. [52]) as the evaluation of a dilute gas transport coefficient according to the time correlation function method is the first term in Eq. (4.67), i.e., the 6 0 , 0 term. Here it is seen how correction terms to this standard approximation arise within the binary collision approximation of the time correlation function. 74 4.5 The Complete Sonine Expansion In solving the Boltzmann equation for the purpose of calculating transport coefficients, the distribution function is expanded about local equilibrium and after eliminating the time derivatives, a linear equation is obtained for the perturbation of the distribution function from local equilibrium, d> (Eq. (2.10) gives this form along with the first order in density generalizations). The inhomogeneous term in this equation is, for each of the viscosity, ther-mal conductivity and (for mixtures) diffusion coefficient, the respective flux of momentum, energy and species mass. For a given transport property, the linear (Chapman Enskog) equation [7, 62, 79] can be expressed in the present notation as JK = ndi (4.68) and the transport coefficient is given by nB f ct a = — — / dpJtf O dxp, (4.69) la + 1 J with the factor nB chosen to agree with the normalization of Eq. (2.29). There are solv-ability and uniqueness conditions that must be satisfied, the only one of importance for the present discussion being the uniqueness conditions, namely that the perturbation <p must be orthogonal to any (collisional) summational invariant ip, I dpdyiptp = 0, (4.70) these being the mass, momentum and energy. The solution of the Chapman Enskog Eq. (4.68) can be calculated in a number of different ways. A first approximation is to assume (p = bJi< is proportional to JK- In that case, a matrix element of Eq. (4.68) with JK gives b = l/n(v)e0)0 (4-71) and the transport coefficient a = B/(v)60,o, (4-72) compares with the first term in Eq. (4.67). A n exact solution can be obtained by making a complete expansion of (p and taking appropriate matrix elements with all the basis elements. A more formal method is to introduce the inverse operator 7Z~l (subject to the orthogonality condition) and formally write the transport coefficient as nB <? — 7. 2a + - J d p J * © ip JK- (4.73) 75 Introduction of a complete orthonormal basis set allows the inverse operator to be calculated. The classic orthogonal basis used in solving the Boltzmann equation are functions consisting of a product of the appropriate tensor in reduced momentum W = p/\/2mkT and a Sonine (equivalently associated Laguerre) polynomial in W2. The normalized version of this is [77], LQ S(W) V i O + l ) ! l > + l) 1 / 2 2 ( a ) ^2"+>!) 2r( S + a + 3/2) 1 J L J ' 1 ' where [W](Q) is the symmetric traceless tensor of rank a (and belonging to an irreducible representation of the 3-dimensional rotation group). For the shear viscosity and diffusion, the flux function JK = L a 0 corresponds to s = 0 whereas for the thermal conductivity, JK = L 1 1 . For future reference, the matrix elements of the collision operator 1Z with this basis set define the kinetic cross sections 6 ("*/), n(v)e(::,) = ((L*s\1l\L™')). (4.75) The standard method [7] of solving Eq. (4.68) calculates the matrix inverse and ex-presses the above matrix element as the ratio of two determinants. In contrast, if the matrix representation of TZ is divided into parts associated with and without JK, this can be most efficiently accomplished with the use of the projector Vo introduced in the last section. Since it is the projected element of the inverse operator that appears in the transport coefficient, the analog of Eq. (4.35) is 1 1 Vo^Vo = ! —jPp-Vo- (4-76) K VoKVo - VoK { 1 _ V o ) n i _ V o ) IZVo As the {0,0} matrix element of this inverse operator, the transport coefficient is then given by which, when converted to kinetic cross sections is identical to Eq. (4.67). The above formalism has inherently assumed that the molecules have only translational states. In case the molecules have internal states, the flux operators in general will involve the internal states and the complete set of functions must span the internal states as well as the translational states. It is then necessary to include a trace over the (quantum) internal states as well as the integration over the position and momentum. To correctly formulate this mix of classical and quantum descriptions the distribution function must be replaced by a quantity which is simultaneously a Wigner function [80] for the translational states and a (4-77) 76 density operator in the internal states. A further complexity is that a complete set of internal state operators includes operators that do not commute with the internal state hamiltonian. When acting on such an operator O, the z parameter in the definition, Eq. (4.52), of the collision superoperator T requires z = to + ie to have its real part equal to the eigenvalue co of K. when acting on O (that is, UJ is the difference of energies of the ket and bra Dirac states of O). This work does not address such questions. But except when treating transport properties in the presence of a field, it has been argued that terms that are off diagonal in the eigenstates of the internal state hamiltonian are rapidly phase randomized [81]. As such, the set of operators used for expansion can be limited to those that commute with the internal state hamiltonian. With this constraint, the formal development is the same as above, except that the basis elements must span this diagonal-in-internal-state-energy part of the space of operators on internal states. The collisions between molecules of the same species behave very differently from collisions between different species, thus it is not sufficient to use an average of the transport coefficients of the separate individual species in a mixture, but rather it is necessary to take into account how they influence each other. It is thus necessary, even at the lowest order of approximation, to include a separate flux function for each of the species. This section discusses the simplest case of a nonreacting ideal gas binary mixture, each species having a single basis function (that is, ignoring the role of higher Sonine polynomials or other dependences on rotational, electronic, etc. internal states). The method of formulation is to directly adapt the general treatment of Section III by considering the two species, labeled a and b, as being equivalent to two different hypothetical internal states of a molecule, with of course the mass now dependent on the internal state. The integration over molecular states now must include a summation over the species, using here a generic label s, as well as the integration over translational degrees of freedom of the corresponding species (ignoring the possible presence of other internal states). Corresponding to this, the one particle distribution function (density operator) for a typical molecule has the structure 4.6 Transport Coefficients of a Binary Mixture (4.78) (4.79) 77 involving the identification state Xs of species s, the normalized species distribution function <ps/V and its mole fraction Xs. For an ideal gas mixture, the higher order distribution functions can be considered as products of the one particle distribution functions, compare Eq. (4.47), ignoring the constraint that only X a N of the total N molecules can be of species a. To make the properties of the mixture clearer, it is useful to introduce a two dimensional species vector space, so that the species projection operators become ( 4 - 8 0 ) Finally, the flux operator also has contributions from both species, thus Jjf = E p A , (4-81) where ps are the projectors onto their respective species, namely I I ) * - ( ! ! ) • <-) The technique of introducing a vector space and reformulating the solution as a matrix problem is used a number of times in the following chapters. It simplifies the calculations whenever there is more than one type of component in the flux vector (in the present case, kinetic fluxes of species a and b). The analogous trace over molecular states becomes the row vector ^Tv3K = Y,J dp3K,xl / dp(3Ka 3Kb). (4.83) Equations (4.49) and (4.63) can be directly adapted to the calculation of a mixture transport coefficient, but now the projector Vo onto the flux function J becomes the two dimensional matrix with components (Vo)ss> = (f3K 0 J dp3K)ss> = XsXsips3Ks 0 j dpJAyxI'. (4-84) or in matrix form Vo ^ ( vT.V a ) © / dp(3Ka 3Kb). (4.85) ^ fXa<pa3Ka\ a V Xbipb3Kb There are analogous species flux projectors Vs fs3Ks 0 j dV3Ks . (4.86) Within the constraint of having two species and only the flux operators 3KS for these species, the identity is equivalent to Va + Vb, so that the projector 1 — Vo is 1 - Vo (  Xf,a " f fb) = C © e• (4-87) \— \lj0jba XaFb J 78 The factorization is recognized as a consequence of the essential factorization of (To).™1, with the two dimensional vectors defined as Xb^Ka — XaJpcb (4.88) Equation (4.87) represents, of course, the projection onto the particular vector £ in the two-dimensional space of species flux. A transport coefficient of a one particle property is given by Eq. (4.49) with the integral evaluated by Eq. (4.63). The latter depends on the evaluation of an inverse of a projected collision operator. Since in the present case this projection is one dimensional, it follows that (i-v0)cpn-(i-v0) 1 Den C © £ in which the denominator is Den 2a + 1 (p This involves the collision matrix elements 1 £ T 0 ipTZ—C, = Xb1Za,a + Xa1Zbfi - Xa7la)b - Xb7lbj0 (4.89) (4.90) 2a + - J dpJKa 0 ¥alZ3Ka = nXa{v)ae (%\°a)M + nXb(v)abe (:±) ab and Ka,b = 2J+ 1 j dpJKa © <panJKb = nXb(v)ab& (aa°0\l) ab ' (4.91) (4.92) together with the analogs obtained by interchanging species a and b. Note that when the flux operators are associated with the same species, there are two types of terms, one in which the binary collision is between molecules of the same species and another when the collision partner is a molecule of the other species. The "telephone" notation of the cross sections involves a classification of the irreducible representation for the translational (here a) and internal state (here 0) motions, the species carrying the flux function and which pair of molecules are colliding, see Refs. [78, 79]. Equation (4.63) requires the evaluation of the inner product / dp3K © ( f l l - C © ^<pTlJK 3Ka/Xa — ^Kb/Xb xn.„ xt / v a ,a (-vi) ® / D P ( X B 3 K A ~ X J K B ] * N ( JKI ) Jdp(3Ka 3Kb) 0 <pH^ I © —7latb + —Tlb,a — 7lb,b \ XaXb[lZay(l + TZajb — TZb>a — TZbyb] (2a + 1) Xh (2a + l)XaXb[7la:a + na,b - nbia - nb,b (4.93) 79 The last equality depends on the symmetry of the Boltzmann collision integral Xana,b = XbKb,a. (4.94) Also required is the matrix element fto.o = 7.—~7 / dpJK © <fKJK = Xana)a + XaTZa,b + XbHb>a + XblZbtb (4.95) la + 1 J Putting these results into Eqs. (4.49) and (4.63), together with simplifying the result by an extensive use of the symmetry of Eq. (4.94), the mixture transport coefficient becomes c _ Xb7Za!a + Xallbj — XglZgj — XbHbta ^ ^ nB TZa,aJZb,b — Ha,bR-b,a This expression is exactly the same as that obtained from the Boltzmann equation, as found for example in McCourt, et al. [79] for binary mixtures where it is expressed in terms of kinetic cross sections. 4.7 Discussion The original motivation for this work was to obtain expressions for the density corrections of the transport coefficients. As an analogous problem, it was asked how corrections to the lowest order approximation to the Boltzmann equation could be derived from the time correlation formalism. Naturally the density corrections require keeping the fluxes associated with collisional transfer, but in the low density limit, the effect of such terms in J , and the effects of spatial correlation of the particles may be neglected. Even the restricted cases considered here illustrate the elaborate coupling of terms that may arise when evaluating Eq. (2.29) to find corrections to the lowest order expressions for the gas transport coefficients. But the method introduced here deduces from the time correlation function formulation, expressions for the transport coefficients that immediately include the higher ordered Sonine polynomials and the analogous expressions for mixtures. Both derivations appear to be novel. The formalism given is completely general and can be used in the low density limit for any of the applications stated in the introduction. It only involves the assumption that J is a sum of one particle properties and the restriction to binary collisions. The implementation of the latter is the set of approximations made in Eq. (4.59). The meaning of the binary collision operator 1Z is different for each case, but the formalism is otherwise the same. For a J which includes two particle phase space functions, or for cases where the spatial correlations of particles can no longer be neglected, the manipulations presented in this 80 chapter must be changed. The nature of some of the expansions are, however, the same in these cases, and by adaptation of the formalism, a density expansion for the transport coefficients can be obtained. This method of approach is again general and is illustrated for viscosity and thermal conductivity in the following chapters. 81 Chapter 5 Bound-Free Gas Transport Coefficients via the Time Correlation Formulation based on an Atomic Picture 5.1 Introduction Bound pairs contribute to the first order density corrections of dilute gas transport coeffi-cients. These effects arise primarily through the difference between atom-atom, atom-diatom and diatom-diatom cross sections, as well as the difference in the gas density (equivalently the pressure) due to the dimerization. For the thermal conductivity in particular, the energy carried by a dimer is clearly different from that of two free atoms, so bound dimers must be taken into account for this reason as well. Another direct energy effect is the possible shift in the atom-dimer ratio as the gas changes temperature. These effects were first considered by Nernst [82] and revisited in a modern setting by Stogryn and Hirschfelder [83]. The latter used the mixture equations for the transport coefficients to treat the differing collisional ef-fects, while accounting for internal states via an Eucken method [84]. This method has been adopted by Rainwater and coworkers [8] as part of their program for fitting experimental moderately dense gas transport coefficient data. While this (molecular) method is intuitively correct from a chemical viewpoint, it does not recognize that a bound pair is truly just a pair of atoms (with of course a special chemical interaction), having the consequence that all properties of the gas should be derivable from the same fundamental formalism expressed in terms of the atoms. It is this atomic picture that is emphasized in this work, which is aimed at showing how the differing collisional effects can be obtained from a unified atomic picture. 82 A n atomic picture has been used by Kawasaki and Oppenheim [53] in the time correlation function formalism to derive density corrections for a system with an attractive potential. The results were also used [59] to obtain kinetic equations for the time evolution of the distribution function of the gas with bound pairs. To our knowledge, these expressions have not been used to obtain numerical values for the contribution of the bound pairs to the first order density correction to the transport coefficients, nor have they been compared to the corresponding formal mixture expressions which are standardly used in applications. More recently, a coupled set of Boltzmann type kinetic equations for the time evolution of the bound pairs and the free molecules has been derived and the resultant equations solved to obtain the bound state contribution to the density corrections [25]. These derivations are at times complex, and it is difficult to assess the effect of the approximations made in the derivation on the final result. The alternative of starting with the time correlation func-tion expressions for the transport coefficients [31, 32] can provide support for, or arguments against, the approximation methods inherent in the kinetic equations type approach. Unfor-tunately, the evaluation of the time correlation function also involves an assessment of how to carry out its expansion and to select which terms are important. Since the two methods are inherently different in detail, provided they result in the same estimation of a transport coefficient, they supplement each other's validity. In Chapter 4, the explicit starting point was an expression for a transport coefficient in terms of a time correlation function using a projected Liouville operator. That expression is rigorously equivalent to the standard ex-pression of Green and Kubo. In that chapter, the time correlation function expression for a transport coefficient was evaluated in the dilute gas limit, in a manner that did not for-mally reduce to solving the Boltzmann equation, but whose resulting expression is the same as that standardly calculated by the Boltzmann equation method [7, 62]. In the present chapter the iV-particle (atomic) density operator is written in a way [86] which emphasizes both unbound (free) atoms and bound pairs of atoms (dimers). The final expression for a transport coefficient in the presence of bound pairs is identical to that of a molecular mixture of atoms and dimers as derived from the Boltzmann equation. While this is to be expected, it is the unifying connection between atomic and molecular pictures that has been attained. The density operator for an iV-particle system having 2D particles bound in pairs [86] is reviewed in Section 5.2. In Sec. 5.3, the methods of Chapter 4 are used to deduce an expression for the transport coefficient of such a system from the time correlation function formalism. Kawasaki and Oppenheim [53] used what amounts to an atomic picture in the time correlation function formalism to obtain first order density corrections to the shear viscosity coefficient in the presence of bound pairs. The relation of their work to this pre-83 sentation is discussed in Sec. 5.4. The chapter ends with a discussion. The material of this chapter is to appear in a future publication [87]. 5.2 Distribution Functions for an iV-Particle System with Unbound Particles and Bound Pairs For an equilibrium system of particles with a sufficiently attractive intermolecular potential, there is a fraction that are bound together as pairs (often referred to here as dimers). The number of bound pairs D, more accurately the dimer density n#, is determined by an equilibrium constant and the unbound (free or monomer) density nM. The equilibrium constant can be calculated in terms of the molecular partition functions qp, qM a n d the volume V. The partition functions are of course dependent on the temperature T as well as the volume V. It is also assumed that the chemical binding is such that the valence forces are saturated after pair binding so that the binding of three and/or more particles is disallowed. While the equilibrium constant can be related to the second virial coefficient, see e. g. Ref. [83], and thus interpreted as a nonideality of the TV-particle system, this connection is not addressed here and the system containing bound pairs is treated as an ideal gas mixture. The inclusion of gas nonideality corrections to the transport coefficient expressions is the object of the next two chapters. The canonical distribution for an TV-particle quantum system in a volume V , having only pairwise interactions can be expanded in terms of Ursell operators [21]. If in this expansion only those terms are retained which have one and two particle Ursell operators, then this corresponds to a gas having only pair correlations, an approximation used by Snider [88] for examining the density corrections to the Boltzmann equation. Furthermore, if only the bound state part of the 2-particle Ursell operator is retained, then one has a mathematical expression which represents all particles as being free (1-particle Ursell operator) or bound in pairs. On the basis that for a large system, the number of bound pairs is very well determined, it is appropriate to restrict the sum over all products of Ursell operators to those terms which correspond to the number of bound pairs and free particles as determined by chemical equilibrium, Eq. (5.1). The resulting distribution then consists of a sum of terms, each of which has M free particles (monomers) and D bound pairs (dimers), with (5.1) N = M + 273. (5.2) 84 It is crucial to recognize that it is not known which pairs are bound, so that the resulting density operator must include all possible permutations of particle labels between being free and bound. The form of this density operator first appeared [86] when a kinetic theory for the formation and decay of bound pairs was presented. Explicitly it is p ^ = g(iV,M, JD)-1En^n7r> (5-3) with being the number of ways N atoms can be distributed into a set p of M monomers, and a set 8 of D dimers. In Eq. (5.3), pfj is the density operator for a free particle j, normalized to the number of free particles TrjPfj = M , (5.5) and pbke is the bound pair density operator of particles k£, normalized to the number of dimers, Tbkipbkt = D. (5.6) When approximately evaluating the time correlation function in the following section it is useful to select out one particle for special treatment. In particular, identifying the possible states for particle 1, { N ) = ( g(N, M, D) Y ' f P f A j N - i ) P \g(N-l,M-l,D)J \MJP g(N,M,D) Y ^ C P b i A „{N-2) , / 6V-") IYJ-I \ I ^BLJ \ + \g(N-2,M,D-l)J f \ D > P with it being understood that p(N~1^ is for the particles 2 , N and p^N~2^ is associated with the corresponding particles 2 , j — l,j + 1 , N . Note that particle 1 may be either free or part of a bound pair, with its partner in binding being any other particle in the system. This discussion emphasizes the particle aspects of the monomer dimer mixture since a dimer is just a special state of a pair of particles. But when determining, for example the pressure, it is the number of freely moving units M + D that strike the walls of the container that is of importance, namely the molecular point of view. Relevant to this, the mole fractions of monomers and dimers are Xf = M/(M + D) and Xb = D/(M + D). These quantities arise naturally in the following. 85 5.3 Transport Coefficients for a System With Bound Pairs As seen previously [31, 32] a transport coefficient a can be expressed as the limit of the Laplace transform of the time correlation function of the flux J of the corresponding physical property, iB a 1 <r= lim lim — TTTT((J © TJ»- (5-8) JV/V=const. v 1 Here B is a factor appropriate for the particular transport coefficient and the introduction of the volume factor V recognizes the difference between the extensivity of a flux and the intensivity of a transport coefficient. The bracket designates a canonical thermal average a for an A^-particle system of volume V while 0 is a tensorial contraction of order a which depends on the tensorial properties of J. Finally, z = ie and C is the Af-particle Liouville superoperator. As in Chapter 4, a quantum language is generally used in the following except when classical mechanics can make a significant simplification. In particular, the product of noncommuting operators for physical observables have not been considered with any care and should, where necessary, be replaced by their anticommutator average, see Eq. (4.27). Dynamical effects associated with the commutator with some hamiltonian occur only through their appearance as superoperators. A l l equations can be interpreted classically and easily translated into a classical formalism if that is desired. It was shown in Section 4.3 that, with the introduction of an idempotent projection operator, V = A\3))Q((3\1 (5.9) the expression for the transport coefficient could be written as, 1 A2V a i - = lim lim —-((J © C — — — C I J)), (5.10) a £->o w-+co (2a+l)B* z - (1 - V)C(1 - V) 1 " y ' N/V = const. x ' \ / \ / with the constant A, determined by the condition of idempotency, given by «J 0 IJ» This relation assumes that J belongs to a specific irreducible representation of the three-dimensional rotation group and that ((J|J)) and C are rotational invariants. In particular, the present treatment is restricted to (electric and/or magnetic) field independent transport coefficients. The general expression for a transport coefficient, Eq. (5.10), is valid for all densities and all chemical situations. In Chapter 4 this was specialized in two ways, both for gases at low 86 density. First it was considered that all particles were the same and free, corresponding to the D = 0 case discussed in Section 5.2. A complete treatment of the low density situation was carried out for this case. Second, the individual particles were assigned a species label and the properties of a binary mixture were worked out. The latter treatment was further simplified by limiting the operator space to having only one (flux) observable for each species. Clearly that restriction could be easily removed, but with the inherent addition of more involved algebra. In the present chapter, particles are allowed to form pair bound states. At low gas density the TV-particle density operator is approximated by Eq. (5.3) and it is further assumed that the flux operator for a bound pair is equal to the sum of the flux operators for the atoms making up the bound pair. The method of expansion results in a pair of coupled equations, one for each species. As a special case, these equations are further simplified by assuming that only one observable is important for each of the free and bound pairs. In this way, as is to be expected, the resulting expression for a transport coefficient has the same structure as the mixture formula of Section 4.6. But the present method of derivation of this result emphasizes the atomic structure of the diatomics and shows how molecular transport coefficients can be obtained from an atomic description, at least at low density and where only the translational kinetic contributions to a molecular flux are of importance. The flux function J in Eq. (5.10) is the sum of kinetic and potential parts. The kinetic part of a flux function is a sum of one particle functions of the momenta of each of the particles in the system while the potential part of the flux depends in general on both the coordinates and the momenta of two (or possibly more) particles in the system. In the present work on the transport properties of a dilute gas it is assumed that the potential contributions are negligible compared to the kinetic part and thus ignored from further consideration. The flux function J is thus taken as having the form J = E J ^ - (5-12) 3 where JKJ is the kinetic contribution of particle j and is exclusively a function of momentum. The flux function is, for convenience assumed to be normalized such that, ((J © J)) = T r ^ i © 3KlPn + TvL2JK12 © 3K12phl2 = (2a + 1)(M + D). (5.13) Essentially this follows from Eq. (5.7) with the notion that only the translational motion of either a free atom or a bound pair contributes, 1 for each of the 2a + 1 components of J. The factor M + 73 is the count of the number of freely translating molecules. While JKI2 = JKI + 2K2 is formally the kinetic flux operator for a diatomic, the relative motion of the atoms in the diatom should be essentially fixed and thus not contribute to the flux. 87 In this approximation only the center of mass motion of the bound pair contributes to the kinetic flux. If (as is the case for thermal conductivity), the internal motion of the bound pair contributes significantly to the flux of the property under study, a correction to the above expression needs to be introduced. The expression for the normalization constant A follows from Eq. (5.13) according to A=—i—. (5.14) M + D v ; This is a generalization of the value of 1/N which was obtained in Section 4.4 for a system consisting of only monomers. Wi th these restrictions, and the further restriction that the potential is pairwise additive, Eq. (5.10) can be expressed in a form that emphasizes the flux operator for a typical particle interacting with a second typical particle, namely (M + D)2(2a + 1)B _ ATfAT 1 W / T « V a = A ^ - I X ^ O V . — £ J > > = N(N-l)((3KlQV12 % z-C N(N - l)TvL...N3K1 0 Via 1-VC 1 z-C £ J ) z-C X CJpW - Jp{N)A © ((J£—^—£J))1 . (5.15) z — C-p J The last forms of this equation involve expanding the resolvent of the projected Liouville operator in terms of the unprojected Liouville operator for the subsequent identification of whether the selected particle is free or part of a bound pair. Neglecting the possibility of chemical recombination and decay, once a particle is identified as being free or part of a bound pair, it is required to remain so identified as far as the Liouville operator is concerned. This essentially eliminates the possibility of chemical reaction. Within this perspective, particles 1 and 2 have a special meaning in Eq. (5.15), so the appropriate form for Eq. (5.7) to select out the state of particle 1 is, ^ ) =^ l "- 1 ) + ]^(^ + | :^ ' > ( "- 2 , • <5-16' The term with pbi2 is subsequently neglected when used in Eq. (5.15) since this term would be associated with internal motion because the potential superoperator Vu is for the same pair of particles. As a consequence, the transport coefficient is the sum of two contributions, one from free particles and the other from bound pairs, namely (M + D)2(2a + 1)B a a , V M 1 -Tv13K1QBfl + Tr13JKlzQBblz, (5.17) V. a 88 where B / 1 = (TV - l)Tr2...jvV; 12" C3PflP<N-V - JPnp{N'l)A 0 « J £ z — £ 1 -CJ)) (5.18) and B W 3 = ( /V-2)Tr 2 i 4 . . . J V V ( i3 )2 : CJpmp^-JpbVip^A®((JC z-C, -CJ)) (5.19) The interaction of a dimer with another atom has been given the shortened label V ( i 3 ) 2 = V12 + V 3 2 . Note that the B^-'s have the same tensorial properties as the J R T / S . These Bfs are akin to quantities which have been named reduced time correlation functions [47]. The objective of the following subsections is to derive a pair of closed equations for By and Bb, which can subsequently be solved for this pair of quantities. 5.3.1 F r e e A t o m C o n t r i b u t i o n s The p^N~^ in Eq. (5.18) can be further decomposed in analogy with Eq. (5.7), but now selecting out the possible state of particle 2. The result allows the effects of monomer-monomer, monomer-dimer and dimer-dimer collisions to be separately identified so that Bfi can be written in the form B n = B f l f + B f l b , (5.20) with B / 1 / Tr 2 . . .jvVi 2- - C C J P f l P f 2 p ^ - J P f l P f 2 p ^ A © ({JC —CJ)) (5.21) and z - C CJpfipmP{N-3) ~ JpfiPmP{N-3)A 0 {(JC —CJ)) B/16 = TT2...JVVI(23) (5.22) In the second equation, the symmetry between the two particles in a bound state has been used to write the potential superoperator in a form suitable for a monomer-dimer collision. The resolvents in each of the above expressions involve the Liouville superoperator C of the entire TV-particle system. These need to be expanded for each case in a manner appropriate for the respective two molecule collision. For a collision between two free molecules, the resolvent (z — C)~l is expanded in terms of the resolvent for the pair of molecules '12 j>2 \ 2<j<ell (5.23) 89 Note that the 2-particle Liouville superoperator is to contain the kinetic superoperators for all the TV-particles, so that £ and £12 differ only by a combination of potential superoperators. Since the trace in Eq. (5.21) is over all particles except 1, and noting that particle 2 plays a special role via the presence of V12, the contributions in Eq. (5.23) from all Vje for j, £ > 2 vanish. Thus it follows that Eq. (5.21) can be rewritten and approximated according to Tr2.../vVi2--12 1 + E (Vy + V 2 j ) i>2 X UpnPf2P {N-2) - 3pnpf2P{N-2)A © ((JC z — C-p CJ)) I r T r 2 V i 2 — V i 2 ( J f f i + J/<2)p/iP/2 Z — *-12 •(JKl + JK2)pflpf2AQ ((JC-1 + T r 2 V 1 2 z - C 12 -(l + P 1 2 ) p / 2 ( T V - 2 ) T r 3 . . . i V V X 3 1 £ J » z-C X C J P f l P ^ - J P f l P ^ A © ((JC—^—CJ)) (5.24) involving the particle permutation operator P 1 2 . The approximation indicated in the second form of this quantity is in the treatment of the terms involving the product of resolvents. Essentially it involves treating the interaction partner, particle 2, in a manner different from the other particles in the gas. In a dilute gas, the opportunity for the two particles to recollide or for three or more (unbound) particles to be simultaneously interacting is very small, so the approximation is to implement the idea that, if particle 2 is to be involved in a collision with particle 1, it will not have been involved in any previous collision with particle 1. This is accomplished by "promoting" its density operator, pf2, through various Liouville and resolvent operators, thus 1 Tr 2...jvVi2 x Tr ,V z - £12 CJpflPf2p^ i ( l + P i 2 ) ( T V - 2 ) V ; 13 z-C a JpfiPf2P{N-2)A © « J £ 1 -£J» 2 f 12- '12 ( l + P i 2 )p/2 (A r -2)Tr 3 . . . A ,V; 1 13 X £ J p / l P ^ - 2 ) - J p f l P ^ A 0 « J £ : z-C 1 Cv CJ)) (5.25) While this has been discussed in slightly more detail in Chapter 4, it is clear that an expansion based on this "promotion" procedure could be carried out, with this as the leading term and correction terms added as the difference between the exact expression and this leading term. 90 The correction term would then account for all the recollision and multiple collision events, while the leading term that is kept here is associated with particle 1 successively colliding with a number of (different) particles. With the further approximation that all reference to particle 2 is ignored to the right of the trace over particles 3 to TV, this trace can be recognized as the appropriate B / i with particle 2 missing, namely 1 , ( i V - l ) *fl Tr3...jvV13 z _ £ ( J V _ 1 } -jx-VpjipW-QA © « J £ j ( " - l ) p / i / 0 ( J V - 2 ) 1 -CJ)) (5.26) The last approximation, which is appropriate for large TV, ignores the difference between TV — 1 and TV particles. As a consequence, B J I J can be approximated by B / i / - Pn^i JKI + -JKIA © ( T r i J j n B / ! + T r ^ J ^ B ^ ) ZPfl 'Kl i iAM iAM — B / i H VffBfi H VfbObu zpfi zpfi zpfi (5.27) Here VJy is a ^-parameterized binary collision superoperator describing the binary collision of particle 1 with another free particle [89], Pn Tr2iTi2(z)pfiPf2(l + Pi2)- (5.28) The 2-parameterized transition superoperator [90] r12(z) = vi2 z - C • ( 2 - / C i 2 ) = V i 2 + V ] 12" -V 12 12 '12 (5.29) can be expressed in several ways. In the limit z = ie —>• 0, the transition superoperator converges to the abstract transition superoperator 7i2 provided it acts on a function of mo-mentum only, and VJ * converges to the linearized binary relaxation superoperator associated with the Boltzmann equation. Vff is the projection superoperator onto the subspace of the flux operator for a free particle, here labeled as particle 1, P / / = J j n ^ 0 T r 1 J J n > being identical to the V0 introduced in Eq. (4.57). Vfb is the transfer superoperator Pfb = J J T I ^ © TriaJft-13, (5.30) (5.31) 91 describing the transfer of the polarization (nonequilibrium flux) from a bound pair to a free monomer. Finally it should be remarked that the identification in Eq. (5.27) of the unprojected B , i term assumes that the system is large so that in the definition of B , i , it does not matter if N — 1 or N — 2 particles need to be considered. Note that Eq. (5.27) reduces to Eq. (4.60) if there are no bound pairs. It is important to note that B , i , has been expressed in terms of the original B , and B 6 functions. This indicates that a closed set of equations for the latter two quantities may be possible. Similar calculations are used for the free-bound term B , i 6 , except that the TV-particle resolvent is expanded in such a way as to emphasize the dynamics of the three particles associated with the collision of free particle 1 with bound pair 23, >/16 = Tr2...jvVi(23)- i + £ (Vu + v2j + v3i) Tr23Vi(23)-C3PflPmP{N-3) ~ J p n p m p { N - 3 ) A © « J £ -[Vl (23) (J /a + J^23)P/lPfc23 - £ J » -123 -(JKl + 2K23)Pflpb23A 0 ((JC- - £ J » +Tr 2 3V 1 ( 2 3) ^ — - 3 ) ^ 4 . . . ^ ( V 1 4 + V ( 2 3 ) 4 ) X £Jp/lP623P' , ( iV-3) _ 3pfiPmP{N-3)AQ((3C-1 • £ J » (5.32) z — LV On the basis that only binary collisions are to be retained, so that neither three or more molecular collisions are of importance, nor are recollisions, it is appropriate to promote either P623 or pfi to the left of the trace Tr4...jv according to whether the V(23)4 or the V14 term is being considered. In each case the appropriate binary collision operator can be identified and the result written as iAM *fib PfiKV iAD + Jf t - l + 3K23 H (FffBfi + Vfb&b2z) ZPfl (VbbBm + VbfBfl) - - l—Bn + — I Zpb23 Z \pfi p 6 2 3 >623 where the free-bound binary collision operator, n{b = i - T r 2 3 V 1 ( 2 3 ) V" P/ l Z — L123 the projection superoperator onto the flux of a bound pair, Vbb = !~J^-3K2A 0 Tr23Ji<-23, (z - /Ci(23))P/lP&23, (5.33) (5.34) (5.35) 92 and the transfer superoperator (5.36) have definitions analogous to those given in Eqs. (5.28-5.31). Eqs. (5.27) and (5.33) can be combined to obtain an equation for the free molecules. Dropping the labels on the particles, since now only a single free particle and/or bound pair appears in these quantities, the equation for free molecules is Ji<f + pfR-fb3i<b +pf - (K" + n^) — ( 1 - XfVff) + Kfb—xbvbf B , +Pf (nff + ft/*) ^ -xfvfb - n*b^ (I - XbVbb) Bh. (5.37) It is to be noted that this is an equation that involves only two operators, namely B j and B b , rather than requiring separate treatment for each of the possible four collision types, specifically for a free particle, the operators Bfif, B j l f e , and analogous quantities for a bound pair. The analog of Eq. (5.37) for a bound pair is now considered. 5.3.2 B o u n d P a i r C o n t r i b u t i o n s The bound pair contribution to 1/a is determined by the B j 1 3 of Eq. (5.19). Its evaluation is treated in a manner analogous to the free particle contribution. First is the decomposition of p(N~2^ so as to identify whether the bound pair (13) collides in a binary collision with a free particle or with another bound pair. This leads to an automatic separation of B W 3 into a pair of terms >613 >613/ + B 6136i (5.38) where B 613/ Tr 2,4-ivV(i3)2 CJpmPf2P{N-3) - 3pbViPf2P{N-3)A 0 ((JC —CJ Z — JLj and B&13& Tr, 2,4...jvV(i3)(24) Z- Ci CJPbl3Pb2AP{N-4) - JpmPmP{N-'}A © ((JC-(5.39) -CJ)) . (5.40) 93 The bound-bound term Bbnb has the simpler structure, being by nature the analog of B , i , . Its expansion in terms of a 4-particle resolvent and subsequent identification of binary colli-sions is >6136 Tr 2 2,4-/vV(i3)(24) i + E (Vi* + v2j + v3j + v 4 i ) j>4 1 X P613^ 66 (13) Z — ^1234 £ J P 6 1 3 P 6 2 4 P ( 7 V _ 4 ) - JP613P624P ( ] V " 4 ) A 0 ((JC iAD z - C 1 CJ)) J K 1 3 + ZpblS {VbfBfl + VhhBm) ZpblZ >613 where the bound-bound collision superoperator is <r>bb — 7 <-(13) = -Tr2 4 V ( i 3 ) (24) • - ( z - / C (13)(24) ) p 6 1 3 p 6 2 4 ( l + P(13)(24)) , (5.41) (5.42) Pfcl3 ' " ' Z — £i234 involving the dimer permutation superoperator P ( i 3 ) ( 2 4 ) . This reduces in the limit z = ie —> 0 to the corresponding linearized collision superoperator of the Boltzmann equation, provided it is restricted to act only on functions of momentum. Finally, the bound-free term B f c i 3 / , can be approximated by the restriction to binary collisions to Bfci 3/ = Tr 2 > 4.../vV(i3)2-'123 1 + E ( ^ i + + V 3 i ) J>3 X Op613P/2P ( i V - 3 ) - JP613P/2P ( 7 V - 3 ) A 0 ((JC Z — Cn -CJ)) P613^({ 3 ) + Jl<13 iAM iAD + JK2 + — (P&&B&13 + VbfBfi) ZpblZ ("P/ /B/2 + VfbBm) - -zpf2 z ' 1 1 _ B / 2 H B 6 i 3 ,P /2 P613 t where / C ( 1 3 ) — -TroV, 2 l / (13 )2 _ -{z — /C(l3)2)P613P/2-(5.43) (5.44) P613 Z — £ 1 2 3 The same remarks can be made for this collision superoperator as for the former. In all simplifications involving the identification of a binary collision, there is a "promotion" of a density operator through a Liouville superoperator and resolvent to become identified with the collision superoperator, on the basis that, if a free particle or bound pair collides with the non-traced over molecule, that free particle or bound pair cannot have been previously involved in a collision that affects the non-traced over particle. It is this major assumption about the nature of recollisions, specifically the lack of any recollisions, that defines the independence of binary collision events and is the analog of Boltzmann's Stosszahlansatz for the evaluation of a time correlation function in the dilute gas regime. 94 Combining the above two quantities associated with bound-bound and bound-free colli-sions results in an equation for the bound pair contribution to 1/a, namely = Pbn^3Kf + pb(nbb + nbf)jKb +pb +pb (ftbf + nbb\ x v _ nbf ^ _ XfVff) v 1 zpb Zpf - + nbb) — ( i - xbvbb) + nb^—XfVfb V 1 zpb zpf (5.45) As for the free particle contribution, there is no longer any need for a detailed labeling of the particles. It is sufficient to retain only whether a typical free particle or bound pair is involved. Except for the inhomogeneous terms involving 3Kf and 3Kb, this equation involves only the two operators B , and B&. It remains to solve the coupled pair of equations (5.37) and (5.45) for B / and B 6 . 5.3.3 Solving for B/ and The objective is to solve the two equations (5.37) and (5.45) so as to calculate the transport coefficient via Eq. (5.17). These two equations can be written in matrix form Mff Mfb Mbf Mbb B / B 6 Yb (5.46) with somewhat obvious identifications of the M and Y components. But it is noticed that each matrix element is actually a superoperator, so inverting the matrix to obtain a solution involves inverting a matrix of superoperators. A formal solution can be written down, either in matrix form while recognizing that the matrix elements do not commute, as well as the inherent operator difference (space of operators for a free particle versus the space of operators for a bound pair) between the first and second index of the matrix, or algebraically 1 B / Mff - MfbMbblMbf 1 Mbb - MbfMjflMfb (Yf - MfbMbfYb) (Yb - MbfMj}Yf) (5.47) The simplest approximation for the evaluation of these operator equations is to restrict each species to have a single flux function, namely the appropriate 3KJ. This "two moment" approximation is equivalent to the assumption that the B j are proportional to the corre-sponding 3Kj, specifically B j « PjJxjbj (5.48) 95 with expansion coefficients br On taking matrix elements of Eqs. (5.37) and (5.45) with respect to the J's, these coupled superoperator equations become a pair of coupled scalar equations. In terms of the two moment approximation, the matrix elements of the Mjt superopera-tors become equivalent to the scalars l a % m f f " 2a + lTvf3Kf 0 MffPf3l<f = 1 + ~(Xb1Zff - Xfnfb) m f b " 2a + i T r / J x / ® MfbPbJi<b = ^(XfHfb - XbR.ff) l a % mbf = 2~aTTTVbJKb 0 M b f p f J K f = ~(Xb/Rbf ~ Xf^bb) mbb = ^ ^ T v b J K b Q MbbPbJKb=l + ^{Xfnbb-Xbnbf) (5.49) wherein it is noticed that the collision superoperator matrix elements only appear in certain combinations, namely n " = (2a + l ) A T T r j J ^ 0 fe" + ^ 3 k > = n^efe]*) +nt(v)jt6(%\>) (5.50) and (2a + l)NjTVj3K^ ° W 3 m = n ^ ^ 6 (tMhf (5-51) In these equations, Nj is either M or D, according to whichever is appropriate, and the notation for mixture kinetic cross sections has been indicated so as to make connection with the standard gas kinetic theory based on the Boltzmann equation [79]. The IZji notation for the matrix elements of the binary collision operators was introduced in Eqs. (4.91) and (4.92). In the two moment approximation and taking into account the normalization properties of the 3Kf and J # 6 functions, the projectors "P//, Vbb act as the respective identities, but a transfer operator has the detailed effect Vj(piJKi= —JKJPJ, (5.52) rij having a density ratio introduced as well as the desired transfer of polarization from one species to the other as a result of the relative concentration of each species. There remains the evaluation of the Yj and the bj within the two moment approximation. Clearly the Yj matrix elements are Vf = 2-J^T[TTf3KfYf = nff + nfb, Vb = 2a + iTrb3KhYb = ^ 6 / + ^6fc - (5-53) 96 The two moment solution is the scalar analog of Eqs. (5.47). For bf this is bf = — = 1 (yf-mfbrrhblyb) ™>ff ~ mfbmbb mbf v J nff + nfb + ^ nffnbb - nfbnbf) 1 + i [xb(nff - nbf) + xf(nbb - nfb)} HffHbb ~ TlfbTZbf . . xb(nff - nbf) + xf{Tibb - nfb) 1 ' ' and an analogous equation for bb by interchanging indices. Finally the transport coefficient is n2B nfbf + nbbb (n, + nb)(lZffllbb - nfbTZbf) xb(nff-nbf) + xf(nbb-nfb)-(5.55) This is the standard result of the lowest order approximation for the transport coefficient of a mixture derived from the Boltzmann equation [7], and is also in agreement with the transport coefficient for a mixture derived previously via the time correlation function approach in Sections 3.4 and 4.6. It should be noted that this expression arises solely from the kinetic contributions to the flux function J . 5.4 The Kawasaki - Oppenheim Treatment of Bound States Aspects of the Kawasaki and Oppenheim treatment of systems with attractive potentials and bound states were discussed in Sec. 3.5. Further discussion with reference to the work of this chapter follows. From the time correlation function formalism Kawasaki and Oppenheim [53] derived first order density corrections to the shear viscosity of a gas with an attractive intermolecular potential. Their work is based on an atomic picture with no explicit emphasis on bound pairs in the classical distribution function used for calculating the thermal averages. They make no connection to the standard expression for the viscosity of a mixture so it is difficult to compare the final expressions obtained here with their results. Within our limited understanding of their treatment, we try to make some comparison between their approach and the present treatment. Kawasaki and Oppenheim carry out a generalization of Zwanzig's binary collision expan-sion [52] of the full resolvent l/(z — C) for the A^-particle system. This produces an infinite 97 series in the divergent parameter 1/e, which is subsequently resummed (assuming that such a resummation is valid) before taking the limit e —>• 0. Since the resummation is carried out on the resolvent, the result is an operator inverse whose action on the flux function is obtained only by successively solving integral equations, each of which is of the form of the linearized Boltzmann equation. Hamer and Oppenheim [59] reexpress the integral equations derived by them in terms of kinetic equations as they might appear in gas kinetic theory. In contrast, the present approach formally avoids any such infinite series, and within the two moment approximation of Eq. (5.48), there is no need to solve an integral equation. In that way, the present procedure is directly computational. If a more accurate evaluation of a transport coefficient is desired, then matrix inversions are required. The present approach does however involve terms which are of order 1/e, so again the set of equations must be solved (or more appropriately reorganized) before the e —y 0 limit is taken. But now this involves only the inversion of a 2 x 2 matrix rather than an infinite series resummation. The first order density corrections derived by Kawasaki and Oppenheim include contri-butions from both kinetic and potential parts of the momentum flux and account for both the presence of bound states and collision nonlocalities. But their treatment is to first or-der in the density of the atoms rather than to first order in the density of the molecules. In this way their results are limited to having a small mole fraction of dimers. Moreover, their bound state contributions have only been explicitly stated for the cross effects be-tween the kinetic and potential fluxes whereas the present treatment is based entirely on kinetic flux contributions and does not address the effects of either potential contributions or collision nonlocalities. It is puzzling why their three particle collision contribution in the purely kinetic flux term has not been further decomposed to explicitly include bound free collision contributions to the viscosity. Correctly there are potential and collision nonlocal-ity contributions to the transport coefficients, but at low molecular density, the dominant contributions should be of purely kinetic origin. This is what is obtained here and discussed in the kinetic theory literature [7]. The inclusion of potential, collision nonlocality and ter-molecular collision contributions via the present approach requires a careful separation of bound and unbound motion. The emphasis of the present chapter is in understanding how to completely include bound pairs at low density. Kawasaki and Oppenheim's approach is phrased in classical mechanics, and consistent with that, their definition of a bound state is in terms of whether the isolated pair should stay forever within each other's force range. They also note the possible importance of unbound states whose lifetimes are long compared to the mean free time of particles in the gas. While the role of states with finite but long lives can be important, and difficult to know how 98 to efficiently approximate, the mathematical consequences of being bound or unbound are significant. The quantum case is especially clear with the spectrum of the (pair) Hamiltonian H12 for relative motion being either discrete and negative, eb < 0, or continuous and positive, p 2 / m , neglecting the possibility of internal motions for the individual particles. [The reduced mass of the pair is ra/2.] The corresponding states are complete in the sense that any (pair particle) state |^) can be expressed as a linear combination of them I*) = E l&XW + / dp|VPWp|*>, (5-56) where \b) is the 6th bound state and is a scattering state associated with incoming relative momentum p, all with appropriate normalization. It is the consequent structure of the resolvent l/(z — Cu) for the pair that is of importance in calculating the transport coefficients. Restricted to relative motion (for simplicity of presentation), the action of this resolvent superoperator on an arbitrary operator O is _ L _ 0 = T\b)—^—(b'\ + T [dP'\b) Q " p ' (Wl z-C12 f^l/z-eb + evx^ * 1 'z-eb + p*/m^p 1 2 j j d M ) z _ ^ + v r l l m ( M - (5.57) It is only the first term with both ket and bra states bound that has isolated poles. These occur at the energy differences (frequencies) between the bound states. This is the analog of the discussion in Appendix A of Ref. [53]. A l l such singularities need to be avoided, or if they occur, then (in transport coefficient calculations) there should be a resummation procedure that leads to a finite result. The method used in the present work is to never expand in terms of the pair Liouville resolvent when the corresponding bound state is present. Rather it was chosen to immediately go to three or four particle resolvents, to describe how the bound state collides with a free particle or another bound pair, see Eqs. (5.41) and (5.43). 5.5 Discussion To account for bound pairs in the time correlation function formalism, the density operator used in calculating the ensemble average of the flux function must reflect the presence of bound pairs. This can take the form of a mixture of two species, as done in Chapter 4, but such an approach does not reflect the atomic makeup of the bound pairs. In contrast, the present treatment includes just those parts of the pair Ursell operator that correspond to the 99 pair being bound. This "atomic" picture of the free and bound pairs gives, see Eq. (5.55), the same low density transport coefficient as obtained in Section 4.6 from the "molecular" picture of a mixture, or by solving the Boltzmann equation [7] for the pair of species. The method presented here for the reduction of the time correlation function, correctly its Laplace transform, to an expression in terms of binary collision cross sections depends on the introduction of several procedures, most of which were used in Chapter 4. First is the introduction of the projection operator V. This has the advantage of eliminating the infinite series expansions in 1/e and gives an equation for 1/a in terms of the matrix element of an TV-particle transition superoperator, Eq. (5.10), which is simplified for the dilute gas case to looking for the effect on a single particle, Eq. (5.15). Second is the binary collision expansion. This was straightforward in Chapter 4 where all particles are free, but here it requires first expanding the resolvent for C-p in terms of the resolvent for C so as to be able to recognize which pair of molecules are colliding, Eq. (5.15). It was the discovery of this obvious third procedure which allowed the various free and bound state collision processes to be identified. Fourth is the structure of the TV-particle density operator, Eq. (5.7), so as to recognize the various states for particle 1. It is at this stage that the appropriate binary collision expansion is made, with either 2, 3 or 4-particle Liouville resolvents being introduced according to the nature of the binary collision. Finally there is the assumption of "promotion", whereby the density operator for the collision partner is moved through potential and resolvent terms so as to identify an isolated binary collision operator. This allows the trace over all TV particles to be separated into that associated with the binary collision and the remainder. Its validity is based on the notion that molecules are independent before colliding and its use rules out the possible contribution of termolecular, etc. collisions and recollisions. But it is this separation into isolated binary collisions that allows the remaining pieces to be related to the primitive quantities and so to close the pair of equations and provide a formal expression of the transport coefficient in terms of binary collision cross sections. 100 Chapter 6 Moderately Dense Gas Shear Viscosity Coefficient via the Time Correlation Formulation 6.1 Introduction In moderately dense gases, the viscosity is associated with both kinetic and potential contri-butions to the momentum flux. In this chapter the effect of both of these contributions on the viscosity coefficient are studied. As in Chapters 4 and 5, the first step is to determine proper projection operators. A key point for the development of this approach was the realization that the total momentum flux requires two separate projection operators, one for each of the kinetic and potential components of the momentum flux. This is required because these two flux components have different convergence properties and must be treated in distinct manners. At these densities, it is also necessary to include spatial correlations due to the intermolecular potential in the equilibrium A^-particle distribution function of the system. The properties of the reduced distribution functions with the inclusion of spatial correlations are reviewed in Section 6.2. In this chapter, it is assumed that the intermolecular forces are repulsive in nature. This prevents the formation of bound states and makes the integrals over binary collision operators finite. In this respect, the scope of this work is similar to that of Snider and Curtiss [19], along with the added corrections of Hoffman and Curtiss [22]. The expressions for the first order density corrections that they obtain are stated for reference. In Section 6.3 the projection operators for the viscosity in the presence of poten-tial contributions to the momentum flux are introduced. These projection operators involve separate projections onto the kinetic and potential components of the Af-particle phase space of the system. In this section and Section 6.4 the calculation of the viscosity coefficient is formulated as a matrix inversion problem and the forms of the matrix elements that need 101 to be calculated are given. These matrix elements are explicitly calculated in Section 6.5. The TV-particle resolvents in the general expressions are simplified with a binary collision expansion and expressions for the matrix elements in terms of integrals over collision param-eters of a binary collision are derived. Contributions from matrix elements give rise to forms which are consistent with the Snider and McCourt corrections along with the additional Hoffman and Curtiss correction, and a density correction related to three particle collisions. The corrections are put together in Section 6.6 where it is demonstrated that within a com-parable set of approximations, the present theory gives density corrections consistent with those of the generalized Boltzmann equation. The chapter concludes with a discussion. The appendices provide details of a number of the identifications made throughout the work. 6.2 Distribution Function for the iV-Particle System and Shear Viscosity The /V-particle distribution function of the system is assumed to be normalized to unity, T w p W = 1. (6.1) The normalization convention chosen for reduced distribution functions is The one-particle and two-particle reduced distribution functions are of interest and can be determined to various levels of approximation. The one-particle reduced distribution function is given as, ,(!) py> =Ne~W IQ1 = mpi (6.3) with Q\ = V/A3 as the one-particle partition function and A = h/V2nmkT as the thermal de Broglie wavelength. The normalized Maxwellian momentum distribution is <pj = A3e~PH<1). From Eq. (6.2), p-1' is normalized to the number of particles in the system, TriPV =N = nV. (6.4) To lowest order in density, the two-particle reduced distribution function is, P{3 = + n^uif (6.5) expressed in terms of the 2-particle Ursell function, = e~0< - e - * B ? ) + H ? \ (6.6) 102 (6.7) This simplifies to, $ = n*<pm + n*A*U® a [n 2 + n2 ( e - / 3 u i 2 - l )] fif2 a n2(pt(p2e' (2) The next term in the density expansion of p\2 is required on several ocassions. This term written as, p{$ = n2ipme-^Y[0](T12) where ^ ( r ^ ) is defined as y [ 0 ] ( r 1 2 ) = l + njdv3 ( e - ^ 1 3 - l ) ( e "^ 2 3 - l ) + 0(n2). The Snider and Curtiss expression for the viscosity coefficient is, (6.8) (6.9) ry = -nkTb0 4ml- 1/ 2 cr3 , T 1 1 5 - ^ ' 2 n V 15 •(nrnkT)1'2^ - Rn) where 4nft(2>2) 1 + nB(T) Ana3 (6.10) (6.11) Along with the Hoffman - Curtiss correction, the viscosity up to first order in density is of the following form, 5kT 8fi(2>2) 4n<73 ,_, 47T 1/ 2 o . T n 1 + nB(T) - _ , , „ F n —na3Nn -R 157TV2 x " 15 " 8TT2 0( 2' 2)* (6.12) The quantities in this expression were seen in Chapter 2 to be integrals over the collision parameters, and are thus stated in the most simple form for a general potential. It is the object of the present chapter to obtain a comparable expression for the first order den-sity corrections from the time correlation function formula using the projection operator approach. 6.3 Projection Operators for Shear Viscosity The thermal average "bra" and "ket" vector notation was introduced in Chapter 4. The notation used is quantum mechanical in nature, but in the following, explicit calculations are only done using classical mechanics. Thus the trace represents an integration over phase space and the notation p(s\ usually reserved for the reduced density operator, is used for the distribution function. This allows the relations to be. transcribed into a quantum formalism with ease. 103 For a dense gas, the transverse momentum flux tensor J defined in Eq. (2.23), has both a kinetic and a potential part so that the phase function ket and bra are specifically «J| = / / w J E > f - I ^ r i « ) (6.14) Other than the momentum flux, the ket includes the TV-particle distribution function and the bra includes an integration over TV-particle phase space. The momentum and potential contributions denoted by the subscript K (kinetic) and V (potential) need to be treated separately. It is convenient to define the normalized components 3 i v s ^ E * £ > ) . I M ( 2 ) ( 6 1 5 ) chosen so that Ay j < E (kjt Vji VK\JK)) = ((M1V))=NEW. (6.16) Here W 3- = pj/W^mkT is the dimensionless momentum of particle j and E^ 2' is the rota-tionally invariant tensor of rank four which acts as the Cartesian tensor identity for the five dimensional irreducible representation of rotation group 50(3). The latter equalities follow from the rotational properties of the fluxes in the system (in the absence of external fields) along with the projector property of E^ discussed in Sec. 4.1. The numerical factors in Eqs. (6.15) are chosen in a manner so as to satisfy the normalization conditions. Thus by four-fold contraction into E ^ it is seen that, . , 5 TV = « J * : J * » = / ^ [ W x p M W x p e - ^ = dW1W14e~w? = 5 TV. (6.17) The normalization constant Ay is also determined from Eq. (6.16), by a four-fold contraction of both sides of the equation into E ^ 2 \ 4TV(TV-1)7T r (du\ 2 r4g(r) 3V J \dr J 5NnTv, (6.18) 104 where g(r) is the radial distribution function. The normalization factor Ay = ^ nTn is seen to be, \=%Sdr ffi rAg{r)- (6-19) 2 dr )Integration by parts, along with the use of the lowest order in density approximation for the radial distribution function, g(r) = e~®u relates this integral to the shear modulus at infinite frequency, derived by Zwanzig and Mountain [91], G o o = n k T + 2 - ^ r d r e - ^ Jo 4du dr (6.20) 15 dr The kinetic and potential components of the momentum are also orthogonal, <(J*|Jy» = «Jv|J*r» = 0. (6-21) In terms of these orthonormal components, the total momentum flux operator is, J = V2kTJK + Av}v. (6.22) The subspace of the momentum flux is thus subdivided into kinetic and potential compo-nents. As is explained below, it is necessary to define separate projectors onto the kinetic and potential fluxes, VK = JJ\*K))'4*K\ ^ V = ^ | J v » : « J v | (6-23) with multiplication properties VKVK = VK; VvVv = Vv; VKVv = VvVK = 0. (6.24) The combined projector V = VK + Vvi is also idempotent VV = V (6.25) and has the property that 7>|J» = ^ | J » + 7 V | J » = V2kTVK\JK)) + AvVv\Jv)) = y/2kT\iK)) +Av\iv)) = M», (6-26) 105 essentially because |J)} is an element in the 2-dimensional operator space associated with V, cf. Eq. (6.22). The standard form of the viscosity coefficient in time correlation function theory is, 77= lim lim A ( ( j : _ L ^ j ) ) (6.27) JV/V=const. where z = ie and B = l/(2/cT). From the properties of the projector V, it is possible to insert V both before and after the resolvent in this expression since only the part of the resolvent in the "P-space is needed in order to determine the transport coefficient. This has the advantage that by operator manipulations the projected resolvent can be calculated in terms of a related memory kernel, namely i V v ~ z W i v = _iz + lVCv + veil - v)7Z^Z7i)c{1_v)(1 - V)CP' (6'28) see Eq. (4.35). Moreover, it is easily proven that VCV = 0, so that in the limit z —>• 0, the projected resolvent is just the inverse of the memory kernel. Since the memory kernel is a direct product of a 2x2 matrix and the tensorial identity E^ of the 5-dimensional irreducible representation of the rotation group, the inverse must be carried out as a matrix. For this purpose it is convenient to introduce the 2-dimensional vector notation J*r» ( |Jv» <=* ( L) (6-29) . 0 7 ' 1 v " \VN and the analogous row vectors ((1K\ (y/N 0 ) ; « J V | ^ (0 VN). (6.30) In this representation the total flux operator is represented according to |J» - ^ ( 1 f ) . y/N{^/2kT Av). (6.31) By using the two dimensional vector form of the projection operators, the memory kernel is represented by the 2x2 matrix Vr * CP - ( M k k M k v ) (6 32) z-(l-V)C(l-V) N \ M V K MwJ { - > in which M6$ = \((Se 0 £—V£J*))' (6-33) 0 z — Lv 106 and 0 and $ represent kinetic (K) or potential (V) components. As in previous chapters, C-p = (1 — V)C(1 — V). The inverse of this matrix is straightforward to determine, _ i _ N ( M v v ~ U k v ) . (6.34) V - M y ^ MKK J z - C ' ' MKKMW - MKVMVK The viscosity coefficient n is thus given in terms of the components of the memory kernel lim - i ^ i ^ m ,35) £ ^ 0 + i v 7 v = 7 o n S t . v mKKmvv - mKVmVK where the factor of 5 is canceled by the trace of the identity and all matrix elements are scalars. It remains to calculate these four matrix elements of the memory kernel. Equation (6.35) appears complicated and it is appropriate to ask why the transport coefficient is treated with a 2-dimensional basis, Eq. (6.31), instead of with a simpler one-dimensional basis | J)) with the corresponding projector, ^totai = A|J»:«J | . (6.36) Such a projector was investigated, but the subsequent calculations seemed to run into conver-gence problems that could not be eliminated. This seemed to indicate that in the calculation of the transport coefficient, the kinetic and potential components of the flux |J)) should not be treated in the same manner. The 2-dimensional approach allows the kinetic and potential subspaces to be weighted differently, appropriate for their fundamentally different natures. 6.4 Reduction to Projected Liouville Superoperators Two types of terms arise in the calculation of the viscosity coefficient, Eq. (6.27). These have the interesting property that while ZKK^VKA-^IK)) (6-37) Z — JL leads to a binary collision expansion which is divergent, and so should be treated as in Chapter 4, the corresponding series for Z w = ( ( J v : | - l ^ J v » (6.38) is convergent. As a consequence of this differing behaviour, it is useful to expand the pro-jected Liouville resolvent in the M-matrix elements in terms of the resolvent of the kinetic 107 or potential projected Liouville superoperator. The Me$ involve the projected Liouville superoperator C-p = (1 — V)C(1 — V). In this case the projected Liouville superoperators CK = (i-VK)C(l-VK), (6.39) Cv = (l-Vv)C(l-Vv) (6.40) are the appropriate memory evolution superoperators if there was, respectively, only purely kinetic or potential fluxes. It is chosen to do the expansion according to associating the projection with the first index G of the matrix element, 1 1 1 z — Ci z - Ce 1 \z - Cv + Ci CQ] 1 - (CVS + VQC) z — C-p 1 Z — Cn z - CQ The final equality uses VeCV§ = 0 for all combinations of G and <E>. MKK then has the expansion (6.41) MKK = -({JK:\C-3 Z l 5 -K (CVv + VvC)-Cv\ cs K •cs K Cv UK)) ^KK + K K V — M $ K + KKV— MVK. (6.42) The notation Ke* indicates that the Liouville superoperator in the resolvent is projected onto the (1 — VK) subspace, and the two subscripts indicate the particular elements over which it is calculated. The L or R superscript indicates that the matrix element in question has only one Liouville superoperator appearing respectively on the left or the right of the projected resolvent. The MyK matrix element vanishes. By the definition of MyK, R VK « Jv :| « Jv :| z — Cv i CS K l-(CVK + VKC)-L Z — C-p i < < j v : | £ J , > > - i ( < J v : | £ J , > > ^ t o : | - ^ CSK)) UK) = 0. (6.43) 108 This uses the facts that J y £ y = 0 and the matrix elements ((Jv'\h<)) and ((JV'\CJK)) vanish. The resolvent l/(z — C-p) in M y ^ was expanded in terms of the resolvent projected onto the (1 — Vy) subspace which is associated with the first index of the matrix element. r K has the expansion 1 % MVK = g « J v : | £ 1 z -Cv i z -Cv i a K 1-(CVK + VKC) 1 aK)) + \iiv-\c—V 0 Z — i - y Z - C V i CSK = VVK + VVKJJMKK + V V K — M K K . (6.44) Clearly the quantities MKK and MVK are coupled. By an exactly parallel argument to that of Eq. (6.43), the MKK element is 5MJ R KK z — C-p i a K z-C K 1-(CVV + VVC) z — C-p\ l-((h<:\CSK)) - I ( ( j x : | £ J v > > ^ ( ( j v : | _ J — Z Z OiV z — L-p CJK) - > W ) p > , : | ^ = 0. (6.45) On this basis, the M-matrix elements are expressed in terms of matrix elements of the subspace projected resolvents according to M K K M y if 1 + ^Kf<vVVK ^VK + jf^VK^KK (6.46) (6.47) The remaining M-matrix elements are evaluated in an analogous manner. First are the relations i = V y y + VVKJJMKV + ^VK^KV 109 MKV = \((JKC:\^—av)) = K K V + K K V ^ M v - v + K K y ^ M v v , (6.48) which together with the evaluations = MyV = 0, give M K V = V L • ( 6 - 5 ° ) The use of two subspace projection operators leads to a formal symmetry in the treatment of the kinetic and potential contributions. The viscosity coefficient of Eq. (6.35), can be written in terms of the matrix elements of the subspace projected resolvents as f 2{kT)2 (Vvv + jfKKVVVK) + Av (KKK + jjKkyVVK) 1 N2B[ -V2kTAv(VyK + Kias + jjVVKKKK + jjVvvKKV) J 77 = lim lim ——— (6.51) e - * o + ^ r 0 r r t . v KKKvvv -VVK^KV It is noticed in particular that the common denominator in each of Eqs. (6.46,6.47,6.49) and (6.50) cancels out. 6.5 Binary Collision Expansions for the Matrix Ele-ments In this section, the matrix elements appearing in Eq. (6.51) are evaluated. Binary collision expansions of the resolvents are made and the order of density of the terms retained is indicated. Of the terms, those that contribute up to a first order density correction to the viscosity are explicitly evaluated. The terms K with resolvents projected onto the (1 — VK) subspace lead to divergences which are treated in a manner similar to that of Chapter 4. Evaluation of the K terms gives rise to a series of higher moment corrections. The formal expression for these corrections is given, but no explicit evaluations of these corrections are attempted. The V terms do not lead to the same divergences and are calculated directly. 110 6.5.1 Expansion of K^y The kinetic term KKV has a structure that can be solved by iteration. The binary collision expansion of the resolvent in this term gives, 5KKV = « J K : | £ — ^ - J v » = JV(iV-l)«Jifi:|Vi2—^v)) Z — JLK Z ~ L-K = N(N-l)((lK1:\Vl2 z-C 12 x { l + [(1 + PU)(N - 2)Vi 3 - VKC - CVK] 737^} Jv)) = T n J ^ i B ^ . (6.52) The tensorial operator BKV can be simplified by recognizing that ((Jjr Jy)) vanishes and separating the pair distribution function, p$ = nVi<^ 2 + Pci2 (6.53) into uncorrelated and correlated parts, the uncorrected part giving rise to the 1-particle kinetic projector Vo = J/<'i(<pi/Vr):Tr1JA-1. Thus BLKV = N(N - l ) T r 2 . . w V 1 2 — % - — ]vp{N) Z — L K ~ T r 2 V 1 2 '-r-JvnP^ ~ V^ i — ( 1 - VQ)BKV z - Cn z (px 1 1 - T r T r 2 V i 2 JT-^KI + JK2)pcl2KKV = Xv--f1nl—(l-Vo)BKV-CKKV, (6.54) z fl where the approximation involves the "promotion" of f2/V through the l/(z — CK) superop-erator. This is done on the basis that, for a dilute system the opportunity for two particles to recollide is very small. Thus if particle 2 is to be involved in a collision with particle 1, it will not have been involved in any previous collision with particle 1. This procedure eliminates all possible contributions from either three or more particle dynamics or recollision events and also assumes the absence of bound states between the particle pair 1 and 2. Thus the calculations performed are for systems with potentials that do not support bound states. After "promotion", the dependence on particle 2 has been expressed in terms of the linearized Boltzmann collision superoperator in Tli = —TT2Ti2(z)fif2(l + Pu), (6.55) fi 111 involving the z parameterized transition superoperator Tl2(z) = v 1 2 — - £12) = V 1 2 + V 1 2 — l — V 1 2 . z-C 12 ' 2 - c (6.56) 12 In the limit z = ie —> 0, the z-parameterized transition superoperator 712(2) converges to the abstract transition superoperator 7 i 2 provided it acts on a function of momentum only. This is appropriate as it is used here since the trace over the remaining particles (3...TV) gives a position independent quantity. For large TV, the difference between TV and TV — 1 particles should be unimportant, with the consequence that this trace quantity can be approximated as BKV- A more explicit discussion of these points is presented in Chapters 4 and 5. The tensorial operators Xv and C are defined by Ay = Tr 2Vi2 1 1 1 (2) J V I 2 P 1 2 ) Z - £12 1 (h<l + J*"2)Pcl2-(6.57) (6.58) Just as for B i of Eq. (4.60), the equation for BKV has the structure B ^ y = YKy - V f t l — ( 1 - WKV (6.59) z- cp1- • "• with YKV — Xv — CKKV. By iteration or by similarity to the solution of Eq. (4.60), in the z —y 0 limit, B^-y has the solution, 1 1 &KV — VKV ~ fl^l-where the projected linearized collision superoperator is (1 - VWlv (6.60) nK = {\-v*)ipjix—{\-vQ). fl This implies that K^v in Eq. (6.52) satisfies 5 K ^ y = Tv^Kl:YKy - T n J ^ : ^ i 1 1 • ( l - W x v = T r i J ^ i r X y - T r i J ^ i t ^ i — -L(l-V0)XL T r i J ^ r C - TriJtfirpxft 1 1 v ( l - P o ) C H<V T r i J K 1 : X v - T n J K 1 : ^ i j r ^ ( l - P o ) X L v 5 + Tr i J^ i rC - T T ^ K H ^ H ^ ^ I - V0)C (6.61) (6.62) 112 The final terms in the numerator and denominator are higher moment corrections due to successive binary collisions (Sonine corrections) to the terms preceding them. The order of density of the terms in Eq. (6.62) can be determined. Making use of the normalization of Pu given in Eq. (6.7) and the density dependence of the subspace flux tensors of Eq. (6.15), TnJK1:Xv = T r 1 2 J K 1 : V 1 2 and z - £ l 2 1 Jvl2p(S=0(Ny/n), TriJja:C = T f T r ^ J ^ r V ^ — N z - L12 (Jjn + }K2)PC\2 = 0(n). (6.63) (6.64) The explicit form of this term is evaluated in Appendix E. To the lowest order in density, in the one-moment approximation, KKV is given by, n 2 A 6 1, S-KV — 7 T r 1 2 J ^ ! : V i 2 5 z — L 1 l (2) J V 1 2 P 1 2 - T r i 2 . J V i V - [ r F p e ( 2 ) ^ - 7 2 - < 5 2 - / 3 « (6.65) -12 5Ay z — £ r e i Evaluation of the result is aided by expressing the resolvent in terms of the associated time evolution superoperator, TV • A 6 5A V T r 1 2 J ^ ! V 1 2 : / dte^-^liFpe^ ~s Jo (6.66) Ignoring for the moment the condition of taking the symmetric traceless part, the calculation of the time integral is carried out by following a binary collision trajectory according to lim / dtel{z'c"l)tTF / dtr(-t)F(-t) = - lim r H ) p ( - « ) l ? + f * ^ P H ) (6.67) by using Newton's second law and integrating by parts with respect to the time. This is reorganized as, lim rp — r (-. .)p(-. .)-_f*5^PH) lim < rp ,so->oo r ( - s 0 ) + s 0 P(-So) m/2 m/2 p(-so) /••so + / dt Jo P(-«o) / x P(~*) / ,x P(- SO) - - z r ^ r P l - * ) m /2 m/2 (6.68) Therefore the time integral contribution to Eq. (6.66) is, n 2 A 6 KKV(time) = ^ - T r 1 2 J ^ i V i 2 : / dt ' " > e " v 5AV Jo { m/2 m/2 !%J[P(-*O)] ( 2 ) [P(-*)] 1(2) -S 2-7 2-/3u V2NnkT 5 T T 3 / 2 A V j dvdj^V: J°°dt{[j(-s0)}M - [ 7 ( _ t ) ] ( 2 ) } e - 7 a - / » « ( 6 . 6 9 ) 113 In evaluating the contributions to KJ<V from the remaining (non-time integral) terms in Eq. (6.68), it is first useful to symmetrize the kinetic flux with respect to particles 1 and 2, J/a + J/.2 = V / 2 { ^ ] ( 2 ) + [7] ( 2 )}- (6.70) Integration over the center of mass position and momentum which are unchanged by the binary collision then becomes possible. The relative momentum is next integrated by parts and the non-integral terms in Eq. (6.68) contribute the corrections KKV(non - time) = Tc^Ffh { [ r y f > - [r'7op} e~^\ (6.71) the first part of which simplifies to !^ MTrrel[7F](2):[r7](2)e-,-/3« = Iv^ Mxr^ Fre-^ -^  =  %-^NnkTB{T) (6.72) while the second part is whose association with Fn follows from Appendix A. In summary, to the lowest order in density, T 1 KKV = iy/2NnkT AV 6.5.2 E x p a n s i o n o f KKV The binary collision expansion of KKV is carried out in a similar manner, 5KKV = ( ( J K : | £  %-^CSv)) Z — LK = N(N-l)((SKi:\V12-KKV(time) (6.74) 'z - £12 x j l + [(1 + P12)(N - 2)Vi 3 - VKC - CVK]j^-^ Uv)) = T^KI-BKV- (6-75) The definition and solution of BKv is similar to that for B ^ < V in Eq. (6.54), with the resulting expansion of KKV in the binary collision approximation as in Eq. (6.62), namely T n J / a : X y - Tv^K^n^^-il - V0)X V 114 This involves the same C operator as in Eq. (6.58) but has Xv replaced by X v = T r 2 V 1 2 L—C12lvi2p{3. (6-77) z - £12 Again the only approximation used to arrive at this result is the "promotion" of <p2/V. The last two terms in the denominator and numerator of Eq. (6.76) give successive binary collision corrections as stated in Section 6.5.1. In this case the TriJ/^irXy term vanishes, T r x J ^ r X y = Tr12JK1:V12 ^ - A a J v i a p i ? z - £ 1 2 = n 2 A 6 T r 1 2 J K 1 : V 1 2 e - / 3 < ^ £ i 2 J y i 2 ^ 0 ] ( r i 2 ) Z — M 2 = ^ n 2 A 6 T r 1 2 J A - 1 : V 1 2 e - ^ - ( -1 + —V") J vi2*[o](r 1 2 ) V Z — L,\2J -> 0. (6.78) The first integral (with -1) is odd in both position and momentum directions, so it vanishes on integration over phase space. The second integral gives a term proportional to z, which vanishes in the z —> 0 limit since both sides of the resolvent have functions which constrain the relative position. Thus KKV has no first order in density contribution. 6.5.3 Expansion of KKK The KKK term is calculated at two levels of approximation. In Section 6.5.3.1, only binary collisions are retained to obtain the approximation shown by KKK. This limitation is lifted in Section 6.5.3.2, to include collisions of up to three particles and an expression for KKK is derived. 6.5.3.1. Re ta in ing only two-part ic le coll isions The expansion of KKK follows similar lines to that of Section 6.5.1. K K * = \VK'\C-^—CSK)) O Z — JL-K (6.79) T r i J y i r X g - T r ^ : ^ j ^ ( l - V0)XK 5 + T n J ^ r C - T n J ^ i f t i ^ ^ l - V0)C This involves the same C operator as before, and XK defined as, XK = T r 2 V 1 2  %-r-Cl2(]Ki + J^pS- (6.80) z - £ 1 2 115 The structure of Eq. ( 6 . 7 9 ) includes the same successive collision corrections as the pre-vious two matrix elements. The order in particle number N and density n of the TVISKI'XK term is determined by using Eqs. ( 6 . 7 ) and ( 6 . 1 5 ) , TnJKL:XK = T r 1 2 J K 1 : V 1 2 L - £ 1 2 ( J K 1 + J A ^ P S = O(Nn) ( 6 . 8 1 ) Z — Ln With the inclusion of the equilibrium pair correlations, K2KK is identified as the modified Boltzmann collision integral that Hoffman and Curtiss [ 2 2 ] used for estimating the effect of a third particle on the bimolecular Boltzmann collision operator in their generalization of the Enskog equation [ 9 2 ] to soft potentials. Their method for the approximate evaluation of this collision integral is followed here. The evaluation is based on assuming that the factor Y[o] defined in Eq. ( 6 . 9 ) changes little during the collision dynamics so it can be moved to the left of the resolvent, and since it is also only a function of the position, to the left of V i 2 . Wi th these approximations K2KK in the one-moment approximation is rewritten as **K * ! ^ T r 1 2 J i , 1 : l [ o ] ( r i 2 ) V 1 2 e - ^ ) ^ — V 1 2 A / 2 [ 7 ] ( 2 ) , ( 6 . 8 2 ) 1U z — L,\<z which recognizes that the J/<-'s are momentum functions on which only V i 2 acts. Moreover, after using Eq. ( 6 . 7 0 ) , only the (reduced) relative momentum 7 is affected by V i 2 . A subsequent symmetrization of SKI on the left over particles 1 and 2 , conversion to center of mass and relative coordinates and integration over the center of mass position and momentum is then performed. Only the relative motion integral (trace) remains, so the particle labels can be dropped while A r is the thermal de Broglie wavelength for relative motion. Furthermore, the combination of operators lim l—r-Vi2 = fi'12 - 1 ( 6 . 8 3 ) e->0 z — £ 1 2 is identified in the limit as a restricted form of the M0ller superoperator, with the prime denoting the restricted applicability, namely that this operator identity is only valid when acting on a function of the momentum, see Eq. ( 2 . 6 9 ) . This implies that = ! ^ T r ^ [ 7 ] ( 2 ) : ^ ] ( r ) V c - ^ „ I ( n , _ 1 ) [ 7 ] ( 2 ) = _ ! ^ T r r e l [ 7 ] ( 2 ) : y [ 0 ] ( r ) ^ ' e - ^ [ 7 ] ( 2 ) . ( 6 . 8 4 ) In the final form, the " - 1 " term vanishes since it is odd in position and momentum directions. The M0ller superoperator satisfies the intertwining relation HTeln' = tt'Krel ( 6 . 8 5 ) 1 1 6 This intertwining relation has allowed the full Boltzmann factor after (to the left of) the M0ller superoperator to be replaced by the kinetic Boltzmann factor before (to the right of) the M0ller superoperator, i . e. e-W^Q! = n'e_/3K"'. (6.86) The second intertwining relation used is CQ' = fi'/C, (6.87) which implies that in the present case V Q ' e - ^ ' [ 7 p = - m e - ^ M ' 2 ) , (6.88) on the basis that the operator on which O' acts has zero JC eigenvalue. Wi th the identification that JC = —i^'-j^ and expressing the effect of fi' in terms of the precollisional relative momentum fi'7 = 7' = 70, an integration by parts of the position variable is carried out to give ,2 _ Z^L Jj dE.pd7[7] ( 2 ):e-7'2[7'] ( 2 ) -2Nn ^ K K ~ The surface integral is proportional to the Chapman - Cowling omega integral [7, 62] f^ 2- 2), and/or the kinetic cross section for viscosity 6(20) [79], KK-ji-(surf) = \ l ' / / . bdbd-fY 1 - cos2 x e 7 bmLiz Jo Jo = ^ f t ( 2 > 2 ) = JV n ( w )s(20). (6.90) 5 Finally the Y[0] dependent term is related to the Hoffman-Curtiss [22] R(T) function - ( - P , - H M ! / / r f r d 7 7 . ^ ( 7 . v , v - = R ( T , m ) Thus, up to terms including linear in density corrections, KKK is approximated by, 5 ^5mV 27r3/2' The K2KK term produces the cross section appropriate to the viscosity along with a correction for the static presence of a third particle on this cross section. The result depends on two approximations, the first being that only binary collisions are considered, and the second that the Y[0] term is assumed constant during a collision. 117 6.5.3.2. R e t a i n i n g three-part ic le collisions In the case of a collision expansion of general order, 1 i ^KK = r ( (J*:|£ 0 z — I •K -Ti:i]Ki:BKK, 5 (6.93) the projected resolvent is first expanded in terms of the unprojected TV-particle resolvent, B K K = N(N-l)Tv2...NV12 = N(N - l )T r 2 . . . J V Vi 2 z - CK i z - C U K P { N ) l - ( V K C + CVK)-c K CSKp { (6.94) A n expansion in this manner provides flexibility in choosing the number of particles taking part in a collision. This expansion was used in Chapter 5 in treating collisions among free molecules and bound pairs which involve different numbers of particles. Two and three-particle collisions are next extracted from the general TV-particle resolvent. The first step is the separation of the dynamics of the collision of the 12 pair from the TV-particle resolvent, V 12" = V Z — C ' 1 2 Z — C\2 Separating the two-particle collision contributions from the other contributions in gives, 1 + £ V ; i=3 z - C (6.95) BKK = TV(TV - l)Tr 2 . . .jvVi2-' Z - CU +TV(TV- l ) (TV-2)Tr 2 . . . i V V 1 2 1 1-VKC i 1 x V 13 z - C Z - Cl2 1 - V K C — z - CK 1 + P12) a K p { N ) CSKp^ (6.96) A further complete expansion of the TV-particle resolvent in this expression has been carried out [93]. A simplified version is presented here which emphasizes three-particle interactions without considering further higher order interactions. The linearized Boltzmann collision superoperator 7£i of Eq. (4.54) is recognized in the projected two particle term and a factor of l/(z — £ 1 3 ) is added and subtracted to the TV-particle resolvent, (2) B KK Tr 2V-2 V\2- -^—CI2(1KI + J/Y2)Pl2 + -<PlRl— VQBKK ' i-,12 z (fii +TV(TV- l ) ( T V - 2 ) T r 2 . . . i V V i 2 1 xV-13 ,Z — C Z -TV(TV- l ) (TV-2)Tr 2 . . . i V V 1 2 z - £12 1 + £13 i z - £12 (1 + P12) 1 z - £13 (V13 + V 2 3 ) U K P { N ) 1 7vKc L z — CK 118 In the part of the third term with +(z — A 3 ) - 1 , the factor f2/V is promoted through the V13 and +(z — £ i 3 ) ( - 1 ' superoperators and the structure of an TZi superoperator recognized in the remaining result. Furthermore, to limit the contributions of the remaining terms to first order in density, only triple collisions among three particles 1, 2 and 3 are retained. Within these approximations, B K K = Tr2V12 l-r-C12(JK1 + JK2)p[$ - - f { H x —(1 - V 0 ) B K K z - £12 Z ipi +Tr23V12 + P 1 2 ) V i 3 ( -r 1-7r~) £ 1 2 3 ( ^ 1 + SK2 + J ^ P S z -£12 \z-£i2z z - £ 1 3 J -Tr23V12 ^ - ( V i 3 + V 2 3 ) ^ — ( J * r i + h<2 + J/c3)p(i2)3 7l7Tr1J /,1:B^A(6.98) z - £ 1 2 z - £ i 2 3 5N This expression is simplified with the notation, BKK = YKK - -<piKi — (l - V0)BKK z fi - ^ 7 T r 2 3 V 1 2 — ( V 1 3 + V 2 3 ) ^ — ( J * r i + h<2 + J / « ) P i 3 2 ) 3 T r i J ^ i : B ^ . (6.99) 5N z - £ 1 2 z - £ 1 2 3 The YKK term contains the two-particle collision term and a three-particle collision term, YRK — Y K K + Y K K = Tr2V12 ^ - ^ ( J A T I + WPS? z - £ 1 2 + T r 2 3V 1 2— ^ ( 1 + P12)V13 Z — A-12 x ( \ 1-r~) £ l 2 3 ( J / 0 + J * 2 + JK3)P ( I2 ) 3 . (6-100) V-2 — *-123 z — L\3J Equation (6.99) is solved by iteration. Retaining only the first iteration, and neglecting higher moment corrections represented by the second term on the right of Eq. (6.99), the solution is approximated as, ^>KK — YKK - r ^ T r 2 3 V i 2 — ^ - ( V i a + V 2 3 ) ^ — ( J M + SK2 + ^ P ^ I S K ^ KK- (6.101) 5N z - £ 1 2 z - £ i 2 3 The collision operator in the three-particle YKK term is further analysed in order to reduce it to one of the familiar three-body collision operators given in the literature. With the low density approximation p[% = n3(fiif2f3 for the reduced three-particle distribution function, Y K K = Tr23V12 - r - ( l + P i 2 ) V 1 3 z - Cu ^r—(Vu + V 1 3 + V 2 3 ) ^ V 1 3 • £.123  z — I-13 119 X ( J * 1 + 1K2 + h<3)Pl23 T r 2 3 V i 2 z-C 12 (V13 + V23)-1 Cv: 13 23 z-C -V- 2.3-13 '23 x(z- Co)(JKX + SK2 + ix^PvL- (6.102) The identities, ^ ( V 1 2 + V 1 3 + V 2 3 ) = l - r - (z-C0)-l Z — Z-123 Z — L-123 -V 1 13 '13 c 13 {z -Co)-I (6.103) have been used. Recognizing the last two operator combinations of the bracket in Eq. (6.102) as transition operators 7 i3 and 723, the total operator in square brackets can be identified as IG, the triple collision operator in the form written by Green [94], YRK(G) = T r 2 3 V i 2 - C 12 (V13 + V 2 3 ) -z = T r 2 3 / G ( J / a + JiC2 + -Mpg3 -C 123 •(Z - C0) ~ 7 l3 - 72 3 (JKI + ^K2 + J iT3)p l23 (6.104) The Y K K function of Eq. (6.104) is denoted as YKK(G) to emphasize the Green form of the operator which appears in its structure. According to calculations of Kawasaki and Oppenheim [95], the Green form of the triple collision operator can be written in terms of the Bogoliubov triple collision operator [10] and a correction term, IB - AI, which substituted in Eq. (6.104) leads to YKK(G) = YKK(B) + A Y K K . The YKK(B) operator is defined as, see Ref. [95], (6.105) (6.106) YKK(B) = T r 2 3 V 1 2 — - r -z — ^12 (V13 + V 2 3 ) z z-C — z 123 Z ~ £12 -(T13 + T23) x ( J / a + h<2 + J / o ) p i 3 2 3 = Tr23/B(J/fi + J / f 2 + J /C3)P l23-The explicit form of A Y K K , with the limit of s0 -> -co implied is, (6.107) A Y K K = - T r 2 V 1 2 / dt[e~tCl Jo •it „—iCviSo ]Tr3(Tl3 + T 2 3 ) ( J / a + JiV2 + JiT3)<Pl^2<P3- (6.108) 120 Within a 1-moment approximation, the result of T r 3(7i3+ 723)(Jia + -)i<2 +•ii<3)lPi<p2(P3 is a scalar multiple of (JKI+^K2)f\f>2- Thus it is possible to insert the projection superoperator V12 = [SKX + J / f 2 ) 9 i ^ 2 : ^ T r 1 2 ( J / a + SK2) (6.109) just before the transition superoperators to give, 1 rOO T r 1 2 3 ( J i n + Jjf2)(7i3 + 7 ^ 3 ) ( J K I + J/re + ^K3)fif2fz- (6.110) The transition superoperator matrix element can be expressed in terms of the kinetic cross-section according to T r 1 2 3 ( J i a + SK2){Tx3 + T 2 3 )(J /a 4- JK2 + SKz)Vi<PWz = - 2 m 3 \ / 2 (TJ)6 (20) E ^ . (6.111) So finally, roo AYKK = m » < 5 ( 2 0 ) T r 2 V 1 2 / dt[e-tC^ - e-^s°](JK1 + J K 2 ) ^ 2 . (6.112) J 0 Substituting the results of Eqs. (6.106) and (6.112) into Eq. (6.101), the following result is obtained for the three-particle kinetic matrix element KKK, ^KK = 4 K + < K ( B ) + &KKK (6-113) with K|-a. defined in Eq. (6.92). ^K(B) is the combination of the two terms of Eq. (6.101), ^KK(B) = T r x J ^ Y L ^ B ) : T r i 2 3 J / a : V i 2 - — ( V i 3 — V 2 3 ) -5N — ™ ' ' z - £ 1 2 v ' " ^'z - £123 x(Jif i + J^2 + J*3 ) p & 3 T r i J j n : Y K * : . (6.114) As suggested by the superscript, these terms give the contribution of three particle dynamics in the Bogoliubov form to the kinetic matrix element. To our knowledge, terms of this sort have not been numerically evaluated. Further study of these terms is of interest, but beyond the scope of the present work. For AK3 ,^ it is seen that, A¥?kk = ^(v)6(20) T r i 2 j j a , V i 2 J°° dt[e^c12t _ e-icu.D]{JKI + j 2 I < ) i p i i p 2 = ^ ^ ( t , ) 6 ( 2 0 ) T r r e l [ 7 ] ( 2 ) : V 1 2 / " dt { [ 7 H ) ] ( 2 ) - [ 7 ( - ° o ) ] ( 2 ) } e iNn2 rr — roc 5? It is to be noted that the structure of this term has a similarity to that of K^y (time), see Eq. (6.69). 121 i(w>6(20)Tr r e l [ 7 ] w i2 / {[-y(-t)]™ oo)P) e~^u J 0 ^ < v > 6 ( 2 0 ) / / d r d 7 [ 7 ] ( 2 ) : V 1 2 j~dt { [ 7 ( - t ) ] ( 2 ) - [7(-oo)]<2>} C - ^ . ( 6 . 1 1 5 ) 6.5.4 Evaluation of Vy-y From the relations of Section 6.4, V w = Usy.C—^— £ J y » . (6.116) o z — Ly Using the analogy with Eq. (4.35), it is seen that, YVV = N2/Zw (6.117) The corresponding purely potential term is appropriately approximated by Z — L « ^ T r 1 2 W — ^ - W g } = 0(iV). (6.118) l Z — L\2 The lowest density approximation to this quantity describes the two particle evolution of the potential flux associated with the pair of particles. It follows that in lowest order in density VVv is proportional to N and independent of n, which is sufficient for assessing the low density expansion of the viscosity. 6.5.5 Expansion of Vyx The matrix element VVK is simplified by using a binary collision expansion 5VVK = ((lv:C—^—aK)) Z — Ly = N { N ~ ^ T n ^ J v : / : — V -P { N ) V 1 2 ( 1 K I + h<2) Z Z — Ly _ ^ ( T V - I U i T r 1 . . . N J v : £ { l + — [ ( 1 + P 1 2)(7V - 2)V 1 3 - VVC - CVy)} i x • R -P{N)V12{SKI + SK2). (6.119) Z — L12 Whenever a particle labeling on the ket is introduced, it is convenient to first move to the left of the Liouville superoperator before introducing the summation over the particle labeling. This is seen in the second equality. The expansion of if (z — Cy) is made on the right side of the projected resolvent, and the i/(z — £ i 2 ) is adjacent to the SK side of the matrix element. It is further noted that the VyC term vanishes. Continuing the simplification, 5VVK = N { N ~ 1)Tr1...Nlv:£—^—pWVn(JKi + JK2) l Z — L\2 122 + i V ( i V . 1 ) T r 1 , „ A f J v : £ - ^ ( l + P12)(N - 2 ) V 1 3 ^ 7 ^ p ( i V ) V 1 2 ( J 1 a + J* 2) +Vv-\c-Z - CV (i) + (M) + (m) 2 - CV i(N-l) 10 12 rftl...NlV>—L?-pIN)V12(JKi + J * 2 ) 12 (6.120) The first term on the right of Eq. (6.120), has the following contribution at the lowest order in density, iVnA 3 , off „ i (i) = - ^ T r r e l J y i 2 : e - ^ " ' £ 1 2 — Z Z — L\2 Vi 2\/2[7] ( 2) + 0(NnVn) = O(NnV^) (6.121) In the first term of («), the operator combination z _ l C l 2 V i 2 is the M0ller superoperator since it is operating on a function of the momenta. Using the intertwining relation, CQ, = QIC, it is seen that this term is zero. The second term (ii) involves three particles and is also 0(Nn^/n). The lowest order contribution to the third term of Eq. (6.120) is, (iii) = 5 V y y - ^ - T r 1 2 J y i 2 : %—pf2]V12(S Ki + -M = 0(Ny/n). (6.122) Thus to the lowest order VVK is approximated by (m), and both (i) and (ii) are dropped from further consideration. — ^ T r 1 2 J y i 2 : 1-^P{SV12(SKI + ha) l U i V Z — L,i2 = -Vvv = -Vvv 10AV V2nAsr Tr r e l[rF]( 2):e- 7 - ^ ( O - 1)[7](2) 1 A T r r e l - F e - 7 o 10 Av r >7o) 2 - ^ 2 7 2 V2kT An^a A v 15 * (6.123) The identification of A^ is made in Appendix B. 6.5.6 Expansion of v t L VK The resolvent in V V A - can be expanded in a manner similar to that of the resolvent in VVK of the previous section. N(N - 1) 'VK VV'-IC^J-VKI + J K 2 ) ) > = 0(NVn~) V N { Nn ^Tr^lv-.C il + — ^ [(1 + Pl2)(N - 2)V 1 3 - VVC - CVV}\ Z I Z — JLv ' 123 z - C P ( J V ) ( J * i + J* 2) 12 N { N ~ 1 W* J v : £ —W ^ J i n + ^2) N(N-l) Z - Cu 1 T r 1 . . . J V J v : £ - ^ 7 - ( l + Pl2)(N - 2)V 1 3 F Z — Ly Z — L\2 P{N\SKI + SK2) + ((JV:\C—^av))l{N^ 1 ) T r 1 . . . J V J v : - ^ p ( j V ) ( J / < 1 + SK2) Z — Ly = (i) + (ii) + (iii) 10 12 The lowest order density contribution is from (i), -p&(Jjn + J*2) . 1_ . i (2), (i) = - T r i 2 J y i 2 : £ i 2 r Z Z — L12 (6.124) (6.125) Using the commutation property of £ i 2 with the two-particle resolvent, this is simplified to, (6.126) where calculations similar to those appearing in Eq. (6.123) were used for the identification of JV The contribution of the three-particle term (ii) is neglected. The term (iii) of Eq. (6.124) is evaluated in Appendix F. The result is, ..... irmkT 4iVvvncr4 .TT , (m) = -\l— —A (Hv + Rn) 3A v (6.127) 6.6 First Order Density Corrections Within the Two Moment Approximation The general expression for the viscosity coefficient, considering the zero value of KKV, is obtained from Eq. (6.35) NnB lim lim JV/V=const. n n 2(kT)2+ A"v Yyy \ K K K + ^kTAy (^ + (6.128) Within the lowest order in density approximations of Section 6.5, the formula for the viscosity coefficient reduces to, r] « lim lim NnkT e_>0+ N,V->oo JV/V=const. 1 - AV f^VK y/2kT \ Vvv N KV + 0(n2) (6.129) 124 Within the three-particle approximation, the KKK term was found in Section 6.5.3 to be, KKK = K 2 K K + A K K K + V?KK(B) 8Nn 0 ( 2 , 2 ) x / " d t ([7(-^)](2) - [7(-oo)]^) e The second contribution from Eq. (6.129) is found from Section 6.5.5 to be — Ay VyK — /!-- ./OUT An f7^„3 ^ 7 ^ . 3 *KK(B). (6.130) V2kT Vyy V2kTV Av y/2kT Any/no-3 „ An^/lxa3 V y y — ; N„ = ^ Nn. vv 15 15 (6.131) Finally, the KKV contribution to Eq. (6.129) was determined in Section 6.5.1, B(T) Aa3 15^ 11 + KKV (time) -Av i L __ -iAv f iy/2NnkT V2kTNKliV ~ VlkTN \ l y - g ^ j / / ^ 7 [ 7 ] ( 2 ) V : / o ^ { [ 7 ( - S o ) ] ( 2 ) - frMPK^ (6.132) Substituting the results of Eqs. (6.130) - (6.132) back into Eq. (6.129), it is found that the two time integrals cancel each other and to the first order in density the viscosity becomes, V = 5kT 8Q(2>2) Ana3 _ 1 + nB(T) - , , „ F , nR(T) Arnixl2a3 157tv2 " 87r2fi(2>2)* 15 Nv-nK3KK(B) (6.133) 6.7 Discussion The density corrections of Eq. (6.133) are identical to those obtained by Snider and Curtiss from the generalization of the Boltzmann equation, Eq. (6.12). The assumptions and meth-ods of the present approach are quite distinct from those used by Snider and Curtiss and the equivalence of the final results gives greater confidence as to their validity. Assessment of the approximations is also made easier by comparison of the two theories. The formal expression (6.51) for the transport coefficient, together with the various con-tributions for the memory kernel matrix elements determined in Section 6.5 is fairly compli-cated. Inherent in the calculation of the latter has been the approximation of retaining only binary collisions, essentially that the gas density is low so that the final expressions should be interpreted as valid only over a limited range of density. It is a standard approach, and well known, that the retention of only a single moment for an unknown function gives a 125 good approximation when, calculating dilute gas transport coefficients. Here there are two types of unknown functions, one term for each of the kinetic and potential flux functions constituting a two moment approximation. Since J^i and Jyi2 arise naturally, these suggest themselves as the appropriate expansion functions. The mathematical techniques of the present theory are entirely different from the Chap-man - Enskog method and do not involve the solution of integral equations. In the Chapman - Enskog method, one begins with the Boltzmann equation and then utilizes the equations of change to derive equations that are associated with, momentum and temperature gradi-ents. Perturbative solutions to the non-equilibrium distribution function are devised which are required to satisfy a number of orthogonality conditions. The perturbations are ex-panded in terms of Sonine polynomials of different order and the expansion coefficients can be determined to different levels of approximation. By substituting the explicit form of the non-equilibrium distribution function into the expressions for the pressure tensor and the heat flux vector, formulas for the transport coefficients are obtained. The assumption of limiting the dynamics to binary collisions has been made in deriving the form of the general-ized Boltzmann equation, and the detailed dynamics of the binary collision enter in various steps of the calculation. In contrast, aspects of the equations of change have been dealt with when deriving the general time correlation function formulas, Eq. (2.29). Such relations are general and no assumption as to the nature of the dynamics have been made. The steps required for the evaluation of the transport coefficient from the time correlation formula are mostly dynamical in nature, and as such, the connection between the dynamical assumptions and the final form of the transport coefficient may seem more direct. There are complexities with regard to defining proper projection operators, but having done that, the link between dynamics and the final result is straightforward, if tedious, to implement. The density corrections of the Snider and Curtiss treatment are based on formal gener-alizations of the Boltzmann equation derived by N . N . Bogoliubov and H . S. Green. The density corrections obtained here are derived directly from the general time correlation func-tion arising naturally from the formulation. Of the terms inside the bracket in Eq. (6.133), the first to fourth appear in the perturba-tion amplitude b0 of the solution of the Boltzmann equation, and arise from the kinetic part of the pressure tensor. The b0 factor is associated with the perturbation of the distribution function from the local Maxwellian form. In the present theory, these four terms all originate from matrix elements with a kinetic projected resolvent, Ke$. These four terms also have analogs in the Enskog theory. Even for hard spheres, the fluid has a second virial coefficient and a third molecule affects the probability that two particles collide. 126 The collisional transfer contribution Nn, arises in the solution of the Boltzmann equation from the potential part of the pressure tensor. This contribution is independent of b0 and so is not dependent on the perturbation of the local Maxwellian. In the present formalism, this correction originates from a matrix element with a potential projected resolvent, VVK-Nv is also present in the Enskog theory, but in a different functional form. For hard spheres, the finite volume of the molecules and thus the possibility of momentum transfer across a boundary without mass transfer leads to this contribution. For soft potentials, it is a mo-mentum transfer of one layer of fluid to another via potential interactions (either attractive, or repulsive), again without the transfer of mass. In dense fluids, this is the major contribu-tion to viscosity since the molecules are not free to move easily between layers. Physically, it is expected that the effect of this contribution will decrease with increasing temperature (except for hard spheres where the contribution is constant), while the kinetic contributions increase with temperature. The correction for the presence of a third particle on the binary collision cross section is given in Eq. (6.91). The derivation of this term is straightforward and involves incorpo-rating first order density corrections to the spatial distribution function of the system. The derivation of this correction by Hoffman and Curtiss [22] invokes a "factorization principle" to truncate the B B G K Y equations. A similar assumption is not required in this case. Numerical calculation of terms in Eq. (6.133), other than the KKK(B) has been carried out by Rainwater and Friend [8] for the Lennard-Jones (12-6) potential. It was found to be more efficient to combine a number of the integrals involved in the corrections of Eq. (6.133) into a new set of quantities called the J's in their work. It would be helpful at some stage to separate the combinations into the original terms in Eq. (6.133) and study their individual behaviors. This may provide physical insight into these terms and their relative contributions to the transport coefficients at different temperatures. The explicit calculation of the transport coefficients via the time correlation function for-mulation, in terms of functional of the intermolecular potential has been carried to the level of including all binary collision contributions. The components of this calculation involve the derivation of the lowest order transport coefficient along with its Sonine corrections in Chapter 4, the derivation of the transport coefficient for a binary mixture of free and bound molecules using both the molecular, Chapter 4 and the atomic pictures, Chapter 5, and the present calculation of first order density corrections including the potential flux contribution. Putting the calculation of the present chapter for repulsive forces together with the additive contribution of bound pairs from the binary mixture treatment takes the calculation of the viscosity to the state where the improvement of the Rainwater theory should be introduced. 127 As discussed in Chapter 2, this involves dividing the phase space of a colliding pair into bound pair and free molecule regions. The terms in Eq. (6.133) arise even for purely re-pulsive potentials, and the phase space integrations which they involve must be limited to regions where the colliding pair are not bound. The next step in the procedure would be to incorporate bound pairs and potential flux effects in the same derivation. Although the details need to be worked out and most likely will require modifications in the structure of the theory, the method of approach introduced here should aid in carrying out such a procedure. This would require the explicit calculation of the three-body collision contributions of K3KK(B). The formation of bound pairs occurs as a result of "recombination" reactions which involve the collisions of three molecules. The cross section for the formation of bound pairs would need to be determined as would the cross section of decomposition, and rearrangement reactions. The inclusion of these reactive collisions will make a rigorous calculation of the transport coefficients much more complicated and will require much more detailed knowledge of three-particle collisions. The occurrence of bond forming and breaking reactions affects the momentum transfer in a non-equilibrium system and thus the viscosity, and also involves energy changes which affect the thermal conductivity of the gas. The magnitude of the energies involved in these reactions are such that at moderate temperatures, not many bound pair decompositions or rearrangements occur and so the assumption of a constant ratio of bound pairs to free molecules (equal to that of the system at equilibrium) is likely to be a realistic approximation. This is perhaps the reason why the Stogryn - Hirschfelder treatment of bound states is adequate for most purposes. 128 Chapter 7 Moderately Dense Gas Thermal Conductivity Coefficient via the Time Correlation Formulation 7.1 Introduction This chapter follows rather closely the pattern of Chapter 6 for the calculation of the first order density corrections to the thermal conductivity. The results of the Snider and Curtiss theory for thermal conductivity [19] are given in Section 7.2. The kinetic and potential projection operators are constructed from the heat flux vector in Section 7.3. This requires extra care in the case of thermal conductivity as the enthalpy term in the heat flux vector has to be divided in such a manner as to make the kinetic and potential contributions to the heat flux orthogonal. The two projection operators allow the calculation of the thermal conductivity to be formulated as a matrix inversion problem. The necessary matrix elements are formally determined in Section 7.4 and simplified in Section 7.5. The results are put together to obtain an expression for the thermal conductivity coefficient in Section 7.6. Appendices provide details of a number of identifications made in this chapter. The Snider - McCourt expression for thermal conductivity is composed of a number of components. The perturbation amplitude a\ is given by Eqs. (12) and (15) of Snider and McCourt 7.2 Thermal Conductivity (7.1) 129 The expression for the thermal conductivity of a system of monatomic particles interacting with a repulsive intermolecular potential, but neglecting three-body collisions is given by their Eq. (26) A 4 \ m J 1/2 ai ~d2B 4 1 / 2 nT2 T 1 / 2 dT2 3 Tx) + -n2kaA 1/2 m J To the first order in density this is, 15fc(A;T) (Hx + Rx-h). A 5 + 5nB(T) 20nu 3 ,_, 4 / 9 o , T 7^dB 5^9d2B n 3 T ^ - n S T d ^ (7.2) + ••• (7.3) 32mQ(2'2) This expression, along with the Hoffmann - Curtiss correction which approximates the effect of three-body collisions, applies only to gases with repulsive intermolecular potentials. Bound states can be taken into account by considering the system to be a mixture of free and bound pairs. If bound pairs are included, Rainwater [8] showed that the second virial coefficient and its derivatives which appear in Eq. (7.3) must be limited to the contributions from the free part of phase space to the total second virial coefficient. This thesis does not carry out such elaborations. The goal of this chapter is to obtain an expression for the first order density correction to the thermal conductivity from the time correlation function formula utilizing the projection operator approach of the previous chapters for systems with a repulsive potential and to compare the results with Eq. (7.3). 7.3 Projection Operators for Thermal Conductivity For a dense gas both the kinetic and potential components contribute significantly to the heat flux vector J. The phase space function "ket" and "bra" vectors for the heat flux are specifically ^ \ 2 m ) m 2 ^ 1 3 + u(r^ )U].5i U W ( r V ) (7.4) m « J | = // W {E ( | L - h) I + I £ [ r A + u(r„n . | } , (7.5) where U is the unit tensor of rank two. For a monatomic gas, the enthalpy per molecule is to first order in density, h = -kT + nkT B(T) - T dB_ dT (7.6) 130 The enthalpy per molecule has been broken up into kinetic hx, and potential hy, contribu-tions defined as, h K hv = nkT B(T)-T% (7.7) To simplify future formulae, it is convenient to use the dimensionless momentum W j = Pj/V2~mkT of particle j and also to define the dimensionless center of mass momentum, Q and relative momentum 7 of the pair 12, W i + w 2 = Vzg W i = -J=(<5 + 7) W i - W 2 = A/2~7 W 2 = -^(<5 -7 ) - (7.8) The formalism requires that the kinetic and potential contributions to the heat flux, denoted by the subscript K and V respectively, be treated separately. It is convenient to define the normalized flux components, and >v 2AV E V5 kT J (7.9) TjtFjt + (ujt - ^ \ U (7.10) (7.11) so that ((JK\JK)) = ((3V\JV)) = NV. The heat flux vector belongs to the three-dimensional irreducible representation of the rota-tion group and U is the rotationally invariant second rank tensor which acts as the identity in that irreducible representation. In order to make the relations more compact u'jk is defined as uj( =  UJ£ - ~jf- (7-12) Equation (7.11) for the kinetic flux is confirmed by double contracting both sides of the equation into U, 3iV = -N f d W i 5 J 25 W,6 - 5W* + —W? L 4 T T 3 / 2 5 ^ 105 ~8~ 75 4 75 ~8~ 37V. (7.13) The potential flux normalization constant Ay can be determined from Eq. (7.11) in a similar manner, 3NAV 131 (rj'f'Fj'f + u'j>i>V) * w j dvl2dQ [(r 1 2F 1 2 + u'12U) •<?] • [(r1 2F1 2 + U'12U) -G\ e'^ + 0(Nn2) Nn j du ( dus 3ul - 2ru— + r— dr \ dr, e-u/kT + 0(Nn2) = 3NnT Thus where I \ is given by dr Av = ^fnFx, du f du 3u — 2ru— + r— dr \ dr , ,-u/kT (7.14) (7.15) (7.16) The integrals with 2hv/N are all of 0(1/N) and so are neglected in Eq. (7.14). The components of the heat flux vector are furthermore orthogonal, ( J j d J v » = ( < J v | J t f » = 0, (7.17) as can be verified by observing that (3K-3 « J v J * » i l / | « ? ( ^ - | ) w , i V ( ^ " 1 ) V f ( ( (Wi - I) W r [(r12F12 + u'M -WJ; Both the momentum and position integrals vanish. In terms of these orthonormal components, the total heat flux operator is, 5 A ; T , ^ T kT A _ —kTJK + d—Av3v. 2m V m (7.19) As in Chapter 6, separate projectors onto the kinetic and potential fluxes are defined, VK = JJ\3K)H(3K\ Vv = ^ | J v » - « J v | (7.20) 132 with multiplication properties V K V K = V K - VVVV = V V ; V K V V = V V V K = 0. (7.21) The combined projector V = VK + Vv is also idempotent, VV = V (7.22) and has the property that P | J» = TK\3)) + VV\J)) = \hr-kTVK\3K)) + J—AyVv\Jv)) V 2m V m = |J» (7.23) essentially because | J)) is an element of the 2-dimensional operator space associated with V. The thermal conductivity coefficient is given in standard form as a time correlation function, A = lim lim ^ - ( ( J %— J)), (7.24) N/ V =const. where z = ie and B = \/(kT2). From the properties of the projector V, it is possible to insert V both before and after the resolvent in this expression and it is recognized that only the part of the resolvent in the P-space is needed in order to determine the transport coefficient. This has the advantage that by operator manipulations the projected resolvent can be calculated in terms of a related memory kernel, namely i V V V = (7 25) z - c -iz + ivc? + vc(i-v)z_{1_^c(1_v)(i-v)£V { - ' Moreover, it is easy to prove that TCP = 0, so that in the limit z -> 0, the projected resolvent is just the inverse of the memory kernel. Since the memory kernel is a direct product of a 2 x 2 matrix and the tensorial identity of the 3-dimensional irreducible representation of the rotation group, the inverse must be carried out as a matrix. For this purpose it is convenient to introduce the 2-dimensional vector notation o ) • | J V " ^  \VN and the analogous row vectors {{3K\ <=• (VN 0 ) ; « J V | <=M0 VN)- (7-27) 133 In this representation the total flux operator is represented according to m \ A < < J | ^ ^(^kT A v ) - ( ? - 2 8 ) By using the two dimensional vector form of the projection operators, the memory kernel in Eq. (7.25) is represented by the 2x2 matrix V C * CV 4=» - ( M k k M k v ) (7 29) z-(l-V)C(l-V) N\MVK MwJ { l > in which Me* = U(J@-C^—CJ$)). (7.30) o z — JL-p where 0 and $ represent kinetic (K) or potential (V) components. It is straightforward to determine the inverse of this matrix, V^—V — ( M v v ~ M k v ) (7 31) z-C MKKMVV ~ M K V M V K V - M V K MKK J The phase space integrals ((J£z_lc^CJ)) are rotational invariants (for an isotropic system), and thus proportional to the identity U. This allows the matrix elements (7.30) to be expressed as scalars. The thermal conductivity coefficient A is thus given in terms of the components of the memory kernel by W 3 * T A V ) \ - M V K mKK jyXy X = lim lim NnB—— , a , v „v f e->o+ N ^ l 7 0 ^ s t fn MKKMw — MKVMVK It remains to calculate these four matrix elements of the memory kernel. 7.4 Reduction To Projected Liouville Superoperators The Me* involve the projected Liouville superoperator C-p = (1 — V)C(1 — V). But the projected Liouville superoperators CK = (1-VK)C(1-VK), (7.33) Cv = (i-Vv)C(l-Vv) (7.34) 134 are the appropriate memory evolution superoperators for the purely kinetic or potential fluxes. The expansion of the resolvent (z — £ ) _ 1 in terms of the resolvents of these projected Liouville superoperators are the same to those used in Chapter 6. Intermediate steps and further discussion is given there. Only the meanings of 3K and J y and the presence of a single dot product instead of a double dot product differ from the treatment of Chapter 6. The combination of M-matrix elements that give the transport coefficient, Eq. (7.32), is written in terms of the projected resolvent matrix elements as, A = lim lim f(kT)2 (Vyy + jfKKVVvK) + A2V (KKK + jjKKVVVK) -y/lkTAv (VVK + ^KV + jfVVKKKK + jjVvvKKV) J e_>.0+ N,v-*oc mT K K K V V V - YVK^KV JV/V=const. x v x v .(7.35) Before explicitly expanding these matrix elements, it is useful to determine the order of density of each of the terms and combinations in Eq. (7.35). This work is limited to the study of first order density corrections to the thermal conductivity, and therefore a number of the terms may be eliminated from further consideration. To determine the order in density and particle number of each of the terms, the normalization conventions of Eq. (6.2) and (6.7) are used. The normalization factor of the potential contribution to the momentum flux, Ay, defined in Eq. (7.15) is density dependent and also affects the outcome. With these relations, it is straightforward to see that, KKK = O(Nn) KKV = 0(N^n) KKK = O(N^), Vw = O(N) VVK = 0(Ny/n) V%K = 0(Ny/n). (7.36) Substituting these orders of magnitude into Eq. (7.37), it is seen that to obtain the thermal conductivity up to the first order in density, it is sufficient to calculate A - lim lim ^ U (kT)2 (yvv + jfKKvVvK) -J\kTAv (yVK + KKV + jjVvvKKV) J e^o+ N ^ v ^ s t mT KRKVVV - VVKKKV (7.37) 7.5 Binary Collision Expansions for the Matrix Ele-ments In this section, the matrix elements in Eq. (7.37) are expressed, at low density, in terms of the dynamics of binary collisions. The K matrix elements have resolvents projected onto the (1 — VK) subspace, and lead to divergent terms which need to be treated in a manner similar to that of Chapter 4. The ¥ matrix element can be expanded directly. 135 7.5.1 Expansion of K^y The kinetic term K^v has a structure that can be solved by iteration. The binary collision expansion of the resolvent in this term gives, 3KKV = ((3^\C^—3v)) = N(N-l)((3KV\Vl2 % = T V ( T V - l ) ( ( J / a . | V 1 2 -'z-C 12 X 1 + {(TV - 2)(Vi3 + V 2 3 ) - VKC - CVK} z - CKI = TtxJKVBLKV. (7.38) It is recognized that ( (JR- z _ 1 C k 3v)) vanishes and that it is useful to separate the pair density operator into uncorrelated and correlated parts, = nVi¥>2 + Pc i2 , (7.39) where (fj are normalized Maxwellian momentum distributions. With the definition of the 1-particle kinetic projector Vo = 3 Ki((pi/V)-Tri3Ki and repeating the manipulations used in deriving Eq. (6.54), an expansion for B ^ y is obtained. Thus BLKV EE N(N-l)Tr2..NV12^^3vp^ Z — L.K ~ T r 2 V 1 2 %—Jvl2p<§ _ 1 ^ J L ( I _ pQ)BKV Z - L\2 Z (fii — T r T r 2 V i 2 -Z—{3KI + 3 K2) pci2KKV iV Z — L,\2 EE X ^ - V i - ( l - ^ o ) B ^ - C 4 , (7.40) Z ( f i where the approximation involves the "promotion" of ip2/V through various Liouville and re-solvent superoperators. After "promotion", the dependence on particle 2 has been expressed in terms of the linearized Boltzmann collision superoperator of Eq. (4.54). This promotion is appropriate as it is used here since the trace over the remaining particles (3...TV) gives a position independent quantity. For large TV, the difference between TV and TV — 1 particles should be unimportant, with the consequence that this trace quantity can be approximated as B^-y. A more explicit discussion of these points is presented in Chapter 4. The vectorial operators X V and C are defined by X Y = T r 2 V i 2 TT-JvviPu z - L\2 C = 1 T T 2 V 1 2 — ^ - ( J ^ i + J * 2 ) P c i 2 . (7.41) TV z — L\2 136 The equation for B ^ - v has the structure B £ y = Y £ v - —(1 - V0)BLKV, (7.42) with Y^-y = X y — CK^-y. By iteration or by similarity to the method of solution of Eq. (4.60), B^-y has the solution, with z —>• 0, *KV = YLKV - ^ - L J - ( l - V0)YLKV, (7.43) where the projected linearized collision superoperator is UK = (1 - 7>0)y>ifti—(1 - 7>„). (7.44) This implies that IKJj-y satisfies, ^ _ T T X J K X - X ^ - T r ^ r ^ x ^ ( 1 - P 0 ) X y * y 3 + TrxJja-C - T r ^ r ^ ^ a - P 0 ) C (7.45) the proof of which is similar to Eq. (6.62). The final terms in the numerator and denominator are successive binary collision polynomial corrections to the terms preceding them. Wi th a calculation similar to Appendix E it can be demonstrated that T r i J ^ r C = O(n). Making use of the normalization of p$ given in Eq. (7.39) and the density dependence of the flux vectors given in Eqs. (7.9) and (7.10), it is seen that in the one-moment approx-imation, to the lowest order in density, KKV = ^ T n J ^ . X y = ITT123KVV12 i — J v l 2 P g = 0(Ny/n). (7.46) 6 O Z — L,\2 To simplify the explicit evaluation of this matrix element, it is first noted that the potential flux contribution of a pair of particles can be symmetrized with respect to the momenta of the two particles, i r / oh„\ I (7.47) 3vu - ~k It is also convenient to symmetrize the kinetic flux with respect to 3Ki and 3K2, the sum of which is, 'Kl + JK2 = ]JI (G2 + 72 - 5) g + 2«y|(7-0)7- (7-48) Therefore, L ~ T r i 2 ( J ; a + J t f2)*Vi2 -7— J v i 2 P i 2 ^ ^KV 6 z — L 12 n 2 A 6 T r 1 2 ( J K 1 + 3K2)V V(rF + wU) • Q e - g 2 - ^ u . (7.49) bAy Z — Z-.rei 137 The contribution of the 2hy/N term has no contribution in the limit of N -» oo and so u' has been substituted with u. Evaluation of the result is aided by expressing the resolvent in terms of the associated time evolution superoperator, n 2 A 6 » roo •Tr12(3K1 + 3K2)V- / dte^-^ [r(F-Q) + uQ] e J 0 -G2-7 2 - / 3 u (7.50) The term containing the potential function u is considered in Eq. (7.53). The remaining time integral is calculated as, lim / dte^-^rfi-g) -» / dtv(-t)[Y(-t)-g) Z-+0J0 JO = - ta. { ' ( -«)[p(-*)-«l! ' + / 0 - * ^ [ p ( - « ) - f l ! } ("D which follows from using Newton's second law and integrating by parts with respect to time. This can be rewritten as, i r (p .g)-r ( - S o ) [ P N - e ] -{ Sodt^^-[p(-t).g] = lim < r(p-G) s0-+oo r ( - s 0 ) + s0 p(-so) m/2 rs0 + / Jo P ( - S Q ) m/2 (P(so)-G) ~ m/2 [P(-*O)-0] P H ) m / 2 p(-*)-0) (7.52) The time integral contribution to Eq. (7.50) is, C^-y (time) — n 2 A 6 6A y T r i 2 ( J / o + 3K2)V\2Q: x / dt J o P ( ~ S o ) p ( - » ) - ^ p ( - t ) + « H ) U m / 2 m/2 3 - 5 2 - 7 2 - / 5 « !A;TA 6 3A T r i 2 ( J x i + J K 2 ) V X 2 ^ : y /••so x / dt Jo l(-soh(-s0) - j(-t)j(-t) + -g2-f2-Pu .(7.53) The time integral over the potential is included in this term. The contribution to Eq. (7.50) from terms not involving the time integral in Eq. (7.52) is fin2 A*. 5 ~6AV x[r(p.<?)-r'(p0.<?)]e-iKV (non — time) = -Tr 12 "(e2 + 7 2 - 5 ) ^ + 2(7.^)7] v-0-g'2-72-pu [2in2AG 5 3A v T r 1 2 [ ( 7 -F )£ + (7-^)F + (F-0)7] x[r( 7-£) - r ' ( 7 o - £ ) ] e ' - S 2 - 7 2 - / 3 « (7.54) 138 where an integration by parts with respect to relative momentum has been carried out. This non-time contribution has two parts, the first part being, /2 m 2 A 6 5 3A v -Tr 1 2 [(7-F)0 + (7-0)F + ( F ^ ) 7 ] -v{vQ)e [2n2A6 1 nJA~TVl23g2 &'FKvr) + r F ^ + (7-P)(7-r)] e FliNn (3 f _n 2 rr 1 V 5 6^7 { 2 7 dTFre +^mJJ ^ 7 - F ( 7 . r ) \BkT + 11 d r rFe"* ' ^  d0 cos2 0 sin d 2 e - 7 2 - / 3 n /2 i /Vn /2 i 5 N n 5 2A kTB(T). (7.55) The second part of Eq. (7.54) is also evaluated in a similar fashion, [2n2A6 - n / ^ ^ T - T r i 2 [(7-F)0 + (7-^)F + (F-0)7] -r'(7o^)e" -7 2 —/3u 5 3Ay " M / f ^ ~ T r r 4 F [^' r)^o-0 + (7-7o)(r-r') + ( 7 O T ) ( 7 T ' ) ] 12 Nn •(2QnozkT)Fx. 5 6 7 T 3 / 2 A y The association with Fx is shown in Appendix G. Summarizing the results of this section, it is seen that, to the lowest order in density, . [2NnkT (7.56) = l\ 5 A v 5 nm 2 0 a Z F 2 B { T ) ~ 6^Fx + K K V (time). (7.57) 7.5.2 Expansion of K^V The calculation of the result of the binary collision expansion of KKV requires similar ma-nipulations, 3 K ^ V = ((3K-\C z - C K -CJv)) 7V(7V-l)((JA- r |V 1 2 z - C 12 1 + {(N - 2 ) ( V i 3 + V 2 3 ) - VKC - CVK) CJ v z - C K _ = TT^KI-BKV (7-58) The equation for BKV is similar to that for BKV in Eq. (7.40). The resulting evaluation of KKV in the binary collision approximation is as in Eq. (7.45), namely K k v ~ 3 + T r 1 J / Y 1 . C - T r 1 J / a . ^ 1 7 e 1 ^ ^ ( l - P 0 ) C -(7.59) 139 This involves the same C operator but has XV replaced by XV = Tv2Vl2—^r- £ i 2 « W f ? . (7.60) z - £12 Again the only approximation used to arrive at this result is a "promotion" of f2/V. The final two terms in Eq. (7.59) give successive collision corrections as stated in Section 7.5.1. The Tr iJ .R - i 'Xy term is shown to vanish, TTIJKI'XV = Tri2J A i 'Vi2 - — £ i 2 J y i 2 P i 2 z - £12 = ^ A ^ i a J ^ - V i a e - ^ V * 7 — i — £ 1 2 J v l 2 y [ 0 ] ( r 1 2 ) Z — M 2 = m 2 A 6 T r 1 2 J K 1 . V 1 2 e - < / f c T ( -1 + ^ 7 ^ ) J y ^ o ] ^ ) \ Z — L\2J 0. (7.61) In this contribution, the resolvent has been combined with £12 to give a term proportional to z, which vanishes in the z —>• 0 limit since both sides of the resolvent have functions which constrain the relative position, and the remaining integral (with -1) is odd in both position and momentum directions, so it also vanishes on integration over phase space. 7.5.3 E x p a n s i o n o f KKK 7 . 5 . 3 . 1 Retaining two-particle collisions The evaluation of KKK a t the binary collision expansion level follows similar lines to that of Section 7.5.1, ^KK = o((3K'\£- -p—^K) T h J j q . X j r - T r ^ - y ^ ^ a - Vo)XK 3 + T r i J K 1 . C - T n J K i - ^ i ^ ^ - ( l - V0)C" This involves the same C operator encountered previously and has XK defined as, 1 ''Z-C12 . K EE T r 2 V 1 2 — — £ 1 2 ( J ^ + JK2)P{3- (7-63) The structure of Eq. (7.62) includes Sonine corrections as in the previous two matrix elements. The order in particle number N, and particle density n of Tri J ^ ' X ^ - is determined by using Eqs. (7.39), (7.9) and (7.10), TTXJKVXK = T r 1 2 J K 1 . V 1 2 ^ £ i 2 ( J / n + JjraU? = O(Nn). (7.64) z - £12 140 With a correction due to the equilibrium pair correlations, KKK is identified as the modified Boltzmann collision integral used by Hoffman and Curtiss [ 2 2 ] for including an estimate of the effect that the presence of a third particle has on the bimolecular Boltzmann collision operator. Their method for the approximate evaluation of this collision integral is followed here. They approximated the integral by assuming that the factor Yjo] defined in Eq. (6.9), which describes the spatial correlations of the other particles in the system with pair 12, changes little during the collision dynamics. As a result, this factor can be moved to the left of the resolvent, and since it is also only a function of the position, to the left of V12. Wi th this approximation KKK is rewritten as ^KK « ^4-Tr123Ki-Y[0]Vi2e-pH» ^ 7 - ^ 1 2 ( ^ 1 + ^ 2 ) z — L12 Tr12JK1'Y[o]Vl2e-0H» \($2 + 7 2 - 5) 0 + 2(7-0)71 • (7.65) Z — L\2 LV ' J The second form is obtained by recognizing that the Jj<-'s are functions of the momentum so only V12 acts on these, and moreover that it is only the (reduced) relative momentum 7 that is affected by V\2- The transformation from individual to relative and center of mass momenta is done as in Eq. (7.48). The combination of operators z_lCl2 Vi2 is identified in the limit as a restricted form of the M0ller superoperator fi'12 — 1, with the prime denoting the restricted applicability. This implies that <2 ^ ^ T r [(g2 + 7 2 - 5) 0 + 2(7^)7] •Yltil(T)Ve-HW<«r(Q! - 1) x [(e2 + 7 2 - 5) 0 + 2 (7 .0)7] = - ^ ! T r [(g2 + 7 2 - 5) g + 2(7-0)7] -Ym{v)Kn'e-K{2)lkT x \(p2 + 7 2 - 5) g + 2(7-0)7] • (7-66) In the final form, the "-1" term vanishes since it is odd in position and momentum directions, while the M0ller superoperator satisfies the intertwining relations H^ti = Q'K^ (7.67) and CQ' = fi'/C. (7.68) The first intertwining relation allows the full Boltzmann factor after (to the left of) the M0ller superoperator to be replaced by the kinetic Boltzmann factor before (to the right of) 141 the M0ller superoperator. The second relation implies that in the present case m e - K ^ l k T [^2 + 7 2 _ 5 ) g + 2(7.^)7] = -m'e-K(2)?kT [(g2 + 7 2 - 5 ) g + 2(7-0)7], (7-69) on the basis that the operator on which Q.' acts has zero fC eigenvalue. Wi th the definition JC = a n o - e x P r e s s m g the effect of Q,' in terms of the precollisional relative momentum fi;7 = 7' = 7 0 , an integration by parts is carried out on the position variable to give x e - 7 o 2 - ^ [^2 + 72 _ + 2(7o.g)7o] + UJ drdpdg [(c72 + 7 2 - 5 ) C? + 2(7.0)7] P - ^ 15m7r3/2 JJJ ^ ~ LV- ' ' 1 -w - " J - £ r x c - ^ - ° a [ (^ 2 + 7o - 5) g + 2(7o-S)7o] • (7-70) After integrating over the center of mass momentum, the surface integral can be identified as being proportional to the Chapman - Cowling Omega integral [7, 62] fi(2,2\ 4r*(surf) = 1 5 m i / 2 y 0 J0 ™ 7 7 7 ( 1 - c o s 2 x ) e 7 = - ^ - H ( 2 ' 2 ) = —(*;>6(20), (7.71) and/or the kinetic cross section [79] 6(20), with (v) the average relative speed. Finally, after the dot product is explicitly evaluated, in the second term of Eq. (7.70), the only non-zero integral that remains is the one in which p is multiplied by another angle dependent term. The remaining Yj0] dependent term is related to the Hoffman-Curtiss [22] R function Thus, up to terms quadratic in density, KKK IS approximated by, 167Vn ( 2 i 2 ) J i V n V ^ _ 16JVn ( 2 , 2 ) f JWna2R(T)\ SZ + 1 5 m W / 2 [ k l ) H [ I ) ~ 1 5 S 2 1 1 + V nm 4TTQ(2.2) f i K K ~ 15 The KKK term produces the binary cross section appropriate for the thermal conductivity along with a correction from the static presence of a third particle on this cross section. 142 7.5.3.2 Retaining three-particle collisions The inclusion of three-body collisions in KKK involves the same type of calculations as in Section 6.5.3.2. The difference is related to the presence of the kinetic part of the heat flux vector instead of the momentum flux tensor, and corresponding to this fact, dot products instead of double dot contractions. Numerical factors also differ in the general relations. The expression for KKK with a general order of collision is given as 1 i KKK = O((JK-\C- T 6 z — L K £ J K ) ) = gTriJ/a-Bft-K-. (7.74) Calculations to determine B K K are of the same form as Eqs. (6.94) to (6.113) of Section 6.5.3. The final result also has the form, K K K = K K K + K*r K (£) + &K3KK, with K 2 K K defined in Eq. (7.73). KKK(B) is the combination of two terms, KKK(B) = THJKI-YKK (7.75) 3/V Tri23J/ci'Vi2 z-C •(V 1 3 -v 2 3) 12 x  t ^ ( J K i + JK2 + JKz)p{%rTr13K1:YKK, (7.76) Z ~ *-123 which give the contribution of true three-particle dynamics to the matrix element. Further discussion of these terms is beyond the scope of this work. For AKKK, with the limit of s 0 —>• —oo implied, it is seen that, in3 2 (v) 6 (20), AK3KK 3 3 w 32(w)6(20) [2 Vs 3 3 roo T r 1 2 J K 1 . V 1 2 / dt[e~M - e-TC"S0](JK1 + J2K)<PX<P2 Jo Tri 2 J /aVi2* x / dt[e~lCl2t Jo E-«A2«o][(02 + ^ _ ^g + 2(g.y)i\<Pm in3A6 /2 2(w)6(20) T r 1 2 ( J ^ i + 3K2)V12Q: roo x / dt Jo 7 2 ( - t ) M 7 2 ( - ° O ) -U U + 7(-*)7(-*) - 7 ( -oo )7 ( -oo) 3-72-S2-/3u m 3 A 6 /22(v>6(20) Tr12(3Ki + Jtf 2)Vi20: f oo x / dt f Jo 2kT U + 7(-*)7(-*) - 7 ( - o o )7 ( - ° o ) -72-e2-/3u (7.77) The equality of 7 2 ( - £ ) - 7 2 (-oo) = -u(-t) was used in the relation above. 143 7 . 5 . 4 Evaluation of Vy-y-From the relations of Section 7.4, Vyy = hiJv-C-^CJv)). (7.78) o z — L Using the analogy with Eq. (7.25), it is seen that, VVV = N2/lw. (7.79) The corresponding purely potential term is approximated by %>w = ((Jy ~p. Jy)) z — Ly ~ j T r 1 2 J y i 2 i _ J y i 2 P g = 0(iV), (7.80) Z Z — L\2 from which it follows that Vvv = O(N) and is therefore an extensive quantity. The intensiv-ity and low density behaviour of the final expression for the thermal conductivity is obtained by assessing the combinations which appear in Eq. (7.37). 7 . 5 . 5 Expansion of VVK The matrix element VVK is simplified by using a binary collision expansion 3vVK = ((Jv.\£—^-—£JK)) Z — Ly N(N-l) 2 T r 1 . . . „ J y £ { l + [(N - 2)(V 1 3 + V 2 3 ) - CPV}] x — L - ^ ) V i 2 ( J / f i + J* 2 ) . (7-81) Z - Lu It is convenient to first move pW to the left of the Liouville superoperator before introducing the summation over the particle labeling. 3vVK = N(N~ 1 ) T r 1 , . . / v J y £ 1-7-P{N)VI2(3KI + 3K2) A Z — L\2 + N(N- 1 ) ^ nJ^c_L_(]V _ 2 ) ( V i 3 +V 2 3 ) Z Z — Ly x - ^ T - p w V 1 2 ( f e + fe) Z - Li2 + 3 V V y ' ( i V ~ 1 ' ) T r 1 , „ J v J v i 2 ^ - p ( J V ) V i 2 ( J K 1 + JK 2) 6 z - Lu = (i) + (n) + (m). (7.82) 144 The (i) term on the right of Eq. (7.82) is, r? 2A 6 i (t) = -^-Tr123vl2-e-^Cl2 rz-Vl2(3 K 1 + 3 K2)Y[0](v12) l Z — L\2 = y f ^ - T r 1 2 J v l 2 . e - ^ » £ 1 2 — % — V 1 2 [(G2 + 7 2 - 5) Q + 2(7-<5)7] + 0(NnJn) = 0(Nny/n). (7.83) The operator combination Z2C V i 2 in the lowest order in density term is identified in terms of the M0ller superoperator 0 — 1 since it is operating on a function of the momenta. Using the intertwining relation, CQ = QIC, it is seen that this term vanishes. The second term (ii) involves three particles and is also 0(Nriy/n). The lowest order in density contribution to the (iii) term of Eq. (7.82) is obtained by using a binary collision expansion, (iii) = 3 V v y - ^ T r 1 2 J v l 2 1—fj$v12(3 K l + jK2) = vvvO(Vn~). (7.84) OiV z — L\2 The lowest order correction term coming from YVK is thus V V K « - v v y - i - T r i 2 J y i 2 l—— p$ V12(3Ki + 3K2) 6N z — L\2 = - v ^ ^ / | ^ T r [ ( r F + „ O j ) . 0 ] . e - ^ ^ - L - V [(<?2 + 7 2 - 5) Q + 2 ( r 0) 7 ] = - V y y y / | ^ T r [ ( r F + u\)).g].e-^(Cl - 1) [(O2 + 7 2 - 5) Q + 2(7-0)7] = - v v v ^ / | ^ T r [ ( r F + v\S)-Q\e-^ KG2 + 1 2 - 5 ) g + 2( 7 o-0) 7 o] The identification of the last equality is made in Appendix H. 7.5.6 E x p a n s i o n o f NyK From the result that KKV = 0 and Eq. (7.37), it is seen that VVK no longer contributes to the first order density corrections of the thermal conductivity. Nonetheless, the expansion of this term is presented, as it leads to a second order density correction that is identical to a term presented by Snider and McCourt. Expanding in the manner of the previous quantities, Z — Ly 145 N { N " 1 ) - T r 1 . . . i v J v ^ - ^ - p ( J V ) ( J K i + JK2) H ^ i r i . . . j v« Jy*-1 - ( i V - 2 ) ( V 1 3 + V 2 3 ) -'12 P W ( J ^ 1 + J K 2 ) * ~ * (2) +3Vyy—Tri . . . ]vJyi 2 -6 AT z — £12 P12 (JJVI + J / T 2 ) = (i) + (ii) + (m). (7.86) The trace in the first term of this expansion is identical to that of Eq. (7.84) and so the result is, (7.87) dT2 J ' The contribution from the three-particle term is not considered. The (iii) term of Eq. (7.86) is evaluated in Appendix J . It contributes a second order density correction to the thermal conducivity that involves the H\ + R\ + I\ correction of Eq. (26) of Snider and McCourt. 7.6 First Order Density Corrections Within the Two Moment Approximation Within the lowest order in density approximation of Section 7.5, in particular the zero of KKV and the nature of Vyy, Eq. (7.37) for the thermal conductivity becomes Nn k2T A ~ lim lim JV/V=const. A n 2 Av (VVK 12 V 5 & T VVyy ' N K V The contributions from KKK where determining in Section 7.5.3, (7.88) K k k = !6Nn n<2,2) i x + 15 nR(T) in / 2 _ _ _ 8 7 r 2 ^ ( 2 , 2 ) * ^ 31/ V 5 roo x / dt Jo 2kT U + 7(-*)7(-*) - 7 ( - o o )7 ( - o ° ) 72_g2_0u + KKK(B).(7.89) The matrix element VVK contributes to Eq. (7.88) with the terms determined in Section 7.5.5, FlnkT ( A 1/2 3 A r nrrdB r:rr,2d2B\ Finally, the K^-y contributions were determined in Section 7.5.1 to be, 2 NnkT -Kv = m-5 A v 5 7?m 2 0 ( j 3 2 B ( T ) " 6 ^ ' + KKV(time), (7.90) (7.91) 146 where n 2 & T A 6 lKV(time) = — T r 1 2 ( J i f i + Jftr2)Vi20: 3A y /••SO x / dt Jo 7 ( - * o ) 7 ( - * o ) - 7 ( " * ) 7 ( - 0 + (7.92) Substituting the values of these matrix elements into Eq. (7.88), the thermal conductivity is found to be A 15fc2T 32mfi(2>2) 5 + 5nB(T) 20na3 "37V2" n R(T) An-Kll2a3 87r2Q(2,2)* 7 dB 5 ^2d2B r T - 3 " T d T - 3 " r ^ + 5 n K - ( S ) (7.93) The A K ^ and K ^ y (time) contributions to the thermal conductivity cancel. 147 Chapter 8 Discussion and Conclusions The calculations of the previous chapters have verified that within a similar set of approx-imations, the detailed forms of the first order density corrections to the viscosity and the thermal conductivity from the generalized Boltzmann equation are equivalent to those of the time correlation function formalism. The different pathways used in obtaining these results increases our confidence in their correctness. The comparison of the forms of the different approximations made in each of these theories helps to clarify the physical content of the final formulae. The method of projection operators introduced in this thesis allows the dynamics of the system to be expressed in analytic form in terms of projected resolvents. For each transport coefficient, the projection operator reflects both the functional form and the tensorial properties of the relevant flux function. In cases where the flux function includes more than one type of contribution, for example, both kinetic and potential con-tributions, it becomes necessary to define more than a single projection operator for the system. In these cases the calculation of the transport coefficient is expressed in terms of an equivalent matrix inversion problem. Both the calculations of the transport coefficients for a binary mixture and for the density corrections were solved as matrix problems. There are a number of directions that can be followed for future work. The method can obviously be extended to other transport coefficients like the diffusion coefficient and bulk viscosity, or more interestingly perhaps to the coupled-flux transport coefficients such as the thermal diffusion factor. In the latter case, the nature of the projection operators will have to reflect the different flux functions that appear on the two sides of the resolvent in the time correlation function formula. There also appears to be no reason why the method can not be extended to non-thermal transport coefficients and to other dynamical problems as well. Another possible extension for future work would be the study of second-order density corrections to the transport coefficient. The explicit study of the three-particle and higher 148 collision terms may reveal logarithmic singularities [96, 97] in the second order corrections. Other than a great amount of calculational detail, no extra difficulty is anticipated to occur if this is carried out. 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Rev. 139, A1763 (1965). [97] E. G. D. Cohen, Am. J . Phys. 61, 524 (1993). 155 Appendix A Identification of The integral in Eq. (3.66) is defined as / . The collision parameter r' has been written in terms of RKR + s0go, 1 du I = U d r d 7 ^ { [ 7 r ] ( 2 ) + [ r 7 ] ( 2 ) } :{ i?[K«7o] ( 2 ) + *o[go7o] (2)K 7 e _ / 3 u . (A.l) This is a six-fold integral over the relative position and momentum. In the following, it is reduced to the form of a standard integral from the work of Snider and McCourt. This facilitates the comparison of the results of the present thesis with previous work. The first set of contractions gives (7'«fl)(r-7o) + ( r ' K « ) ( 7 - 7 o ) - g (7- r ) (7o" K «)- ( a - 2 ) { [ 7 r p + [ r 7 ] ( 2 ) } : [^7o ] ( 2 ) The dot products are given in terms of the collision angles in Chap. 2. Using these relations, { [ 7 r ] ( 2 ) + [ r 7 ]( 2 )} : [ K f l 7 o ]( 2 ) = - 7 7 o r [cos pRcos(r]R -9) + cosrjRcos(pR - 9) 2 --cos#cos(?7R - (3R) - 7 7 o r cos(pR + r)R-6) + - C O S ( % ~ PR) COS 9 (A.3) Using the definition, r]R = PR + TT - (f> with (f> = arcsin[L/i?(m£') 1 / ' 2 ], the right hand side of this expression simplifies to, cos(nR + PR-6) + ^  cos(r)R - pR) cos 9 = - cos(2/?fl. - 9 - cos 6 cos 9. 3 Y (AA) The value of PR is related to its value at the limit of R -» co, /?, by the relation, pR = P + > Substituting this relation, and recognizing that, cos</> = 1 -0(1/R2) sin</> = C/R(mE) 1/2 (A.5) 156 gives: 1 cos(2/3 - 0 + 0) - -cos0 = - cos(2/3 - 0) + -3 R^mE)1' L — s i n ( 2 / 3 - 0 ) - i c o s 0 - cos(2/3 - 0) + TJEtl sin(2/3 - d ) - \ cos 0. (A.6) i?7o 3 With /3 — 0 = a; ± C the first set of contractions becomes i?77 or[cos(2a; + 0 ± 2C) + 1/3 cos 0] - 7 2 r 2 sin 0 sin(2a; + 0 ± 2 0 • The second set of tensorial contractions in Eq.(A.l) are [7r] ( 2 ):[7o] ( 2 ) = [r7] ( 2 ):[7, ] ( 2 ) = (7'7o)(r-7o) - 3(7-r)7o> (A.7) (A.S) which can be written in terms of angles defined in Section 2.5.1, {[ 7r] ( 2 ) + [r7]( 2)}:[ 7 o] ( 2 ) - -2 7 7 0 2 r[cos^cos(r ? i ? . - 0) - l/3cos0] = -77o^[cos(2r7B - 0) + 1/3 cos0] = -7 7 2 r{cos(2o; + 0 ± 20 + 1/3 cos 0}, where terms of order 1/Rov smaller have been neglected. Adding the two expressions from Eqs. (A.7) and (A.9) gives, (A.9) [7r] ( 2 ):[7 0r'] ( 2 ) = -jj0r[s09o - R] r'l(2) - cos(2u; + 0 ± 20 + cos 0 - 7 2 r 2 sin0sin(2o; + 0 ± 2 O - 7 V s i n 0 ( X ± Y) cos(2w + 0 ± 2 0 + COS0 7 V sin0sin(2a; + 0 ± 2 O , (A.10) where the definitions of X ± Y as given in Eq. (9) of Snider and McCourt were used with + for the incoming and — for the outgoing part of the collision trajectory. Substituting this result into Eq. (A.1) gives, I = -8TT r 3 g - 72 g - / 3 « 7 4 x + Udrd^'' / " ^ { ( X + Y) cos(2o; + 0 + 2 0 + Jo I fn {(X-Y) cos(2o; + 0 - 2 0 + Jn/2 COS0 COS0 + sin(2w + 0 + 20} sin 2 8d9 + sin(2w + 0 - 2 0 } sin 2 0d0J . ( A . l l ) 157 Changing variables in the last integral to 9' — TT - 9, and then renaming the dummy variable 9' as 9 once again gives, du I . - 8 T T dr jj drd'y x | ^ ' r / 2 { ( X + Y) cos(2o; + 8 + 2C) + + / ^ { ( Y - X) cos(2w -9-2Q + Jo cos 9 3 COS0 + sin(2w + 9 + 2C)} sin 2 0d0 + sin(2o; - 9 - 2()} sin 2 . (A.12) Adding the two integrals together gives, - 1 6 7 T 2 rrr/2 du I = jjj" ^ r 3 e - / 3 « e - 7 2 7 4 s i n 2 9^y c o g e + 3 Y c o s 2 L J cos(9 + 2C) - 3 X sin 2LO sin(0 + 2() + 3 cos 2LV sin(0 + 2Q\d9drd^. (A.13) The triple integral in this expression is the same as that appearing in the definition of Fn as given by Eq. (25) of Snider and McCourt [20]. Thus - 1 6 T T 2 / W V \ _ - 8 T T / V 3 V / _ 3 I 2TT V ~ 3 *" (A.14) 158 Appendix B Identification of Nq 2 2 2 r To e" 7 e _ / i u . (B.l) The integral in Eq. (3.72) is denoted as / I=SSdvdit\[T'ia)Simplifying, this becomes, / = // dvd^r^ cos2 (r,R - B)<r*<r* - \ j dt^re^ j d ^ e ^ = (i) + (ii). (B.2) Evaluation of (i) involves simplifications of the angles, (i) = 8TT2 //r r 3 ^ e - 7 o 7 2 7 2 cos2({3R - <f> - 9) sm 6d6drdj. (B.3) JJJo dr If PR = (3 + (f> is used in the limit R —> oo, then cos2(/5ji -<j>-6) = cos2(/3 - 0) = cos2(u; ± C)- (B.4) This expression is substituted in Eq. (B.3). The resulting integral over 9 is written as two parts, one for the incoming and the other for the outgoing trajectory, rn/2 cosz(o; + () sin 9d9 + / T/2 r cos 2 (LU ± C) sin 9d9 = f'2 2 (to in  T cos 2 (w - C) sin 9d9 JO Jo JTT  I + I C 0 S ( 2 a ; + 2C) TT/2 / • T / 2 = / ( 1 + c o s 2 w c o s 2 C ) s i n 0 d 0 . JO 1 + I c o s ( 2 a ; - 2 C ) sin 0ci0 sin(7r - 9')d9' (B.5) Putting terms of (i) back together and using the definition of N\ given in Eq. (31) of Snider and McCourt [20] gives, (i) = 8TT2 J[£/2 r 3 | j V 7 ° 2 7 o 2 7 2 ( l + cos 2LU COS 2() sin 9d9drd-f = 47T2kTa3Nx. (B.6) 159 The term (ii) is simplified by integrating over relative momentum, (ii) = ~}j drr±e-»> J d 7 ( 7 2 + fiu)e~^ = / dr-re^ (3- + (3u), dr and writing the second integral as, { n ) = — {2-TdTj Recognizing that the second virial coefficient B(T) can be written in the form, du B 6kT J dr the (ii) term becomes, - T T 3 / 2 /3 m d (ii) [l+T-^j [-6kTB(T)) = n*'*kT 5B(T) + 2T dB(T) dT Adding the simplified terms gives an expression for the integral in Eq. (3.72) I = 2n3/2kT \B{T) + T^p- + 2^2a3Nx 2ir3/2kT 4aW2 3 / 5 n / m TdB(T)\ 3 A r ' -B(T) + — + -Nx ° 1 ; dT j 2 X AaH1/2 \ 2' 8a3ir2kT A 7 Nn. 3 " (B.7) (B.8) (B.9) (B.10) ( B . l l ) 160 Appendix C Binary Mixture Viscosity Cross Sections The following notation is standard in the theory of the viscosity of a binary mixture [7], 5kT r , 5kT [Vf] 8fi (2,2) [Vfb] A * H ff Xf ] 2Xfxb rnfmb / 5 | mb\ [Vf] [Vfb] (mf + mb)2 \3A*fb mf) 8 Y 2 Q ( 2 , 2 ) 3 2 X / X 6 m / m f c n ( 1 | 1 ) 16 XfXbm2b (22)' . 5 * ' " " + T ( m / + m&)2 / 6 + T ( m / + m b ) 2 ^ -2XfXb mfmb I - 1 M (m/ + m 6) 2 V3A}6 -32 XfXbmfmb 16 XfXbm2b ( 2 ; 2 ) [ 3 (m/H-mft)2 / 6 5 (m ; + m 6) 2 / 6 By permuting the / and 6 indices, the other two H's can be obtained ,2 # 6 6 = Hbf = 8 32 XfXbmfmb 16 XfXbmf ( 2 ; 2 ) 3 (m, + m 6) 2 6 / 5 (m, + m 6) 2 fe/ -32 XfXbmfmb 16 XfXhm) (2|2)' 3 (mj + mft)2 6 / 5 (m7 + mfc)2 6 / /ArT. (C.i) /JfeT (C.2) (C.S) (C.4) (C.S) 161 Appendix D Determination of the Fourth-Rank Identity Tensor The identity tensor of rank four must be a linear combination of the three fourth rank rotationally invariant tensors defined in Eq. (4.12), E ( 2 ) = a|JU+bliU+cy. (D.l) The coefficients of this expansion can be determined by utilizing the idempotency property of the identity, E (2 ) . E (2 ) = E ( 2 ) j ( D 2 ) along with the fact that must be symmetric traceless with respect to the the first pair of indices and the last pair of indices. This symmetry implies that b = c. Therefore, the idempotency condition is written as, [aUU+b(OJJ + y)]:[aUU+b(l[JJ + y)]=aUU+b(llJJ + y ) . (D.3) It is straightforward to show that U U UU y uu = 3UU uu U U = U U ILU U U = U U (UJ UJ = uu uu uu = y ILU UJ = uu y =y = u u =y = uu, :(UJ = y . (D.4) Using these equalities in Eq. (D.2) gives the following set of equations for the unknown coefficients a and b, 3a2 + 4ab = a 2b2 = b, which have a = —1/3 and b = 1/2 as their solution. Therefore, E ( 2 ) = J[uu+y]-^uu-(D.5) (D.6) 162 It can be verified that 0 E ^ = 5 which is the dimension of the irreducible rep-resentation of the three dimensional rotation group of J. This makes use of the following contractions, uu©uu = 9 uu©uu = uu©y=3, ILJoWJ = 9 UJoUU = UU©W = 3> y©y = 9 y©uu = y©UU = 3. p.?) 163 Appendix E Evaluation of C The quantity C is defined as, C = — T r 2 £ 1 2 — N z - £12 (E.l) Since the two-particle Hamiltonian commutes with the £12 combination, C is written as, C = ! ^ T r 2 e - ^ ) 1 N - £ i 2 ( J * i + JK2) - ^ T r 2 — L - ^ J ^ + SK2)e-^\ (E.2) 12 N z - £12 In both terms, the functions to the right of the operator combinations are only functions of the momentum and the operator combination can be identified as the M0ller operator, z - £12 The effect of the M0ller operator is, •12 z - £12 V12 = ft' - 1. (E.3) C TV ! A 6 N n 2 A 6 N Tr, e - ^ « (ft - l ) ( J / A + ]K2) - (ft - l ) e - ^ " ( J K i + J A T 2 ) 2nA 3 Tr 2 ( e - ^ i ? - e - ^ » ) ( J ^ + J * 2 ) = ^ . B ( T ) J K i V i . (E.4) 164 Appendix F Determinat ion of Hf] + Rrq The integral in Eq. (6.124) has the following form, ™ = OTTRI2J"2:E^"^WI£"{LPIRIL<2,+LP2R2,(2,} = '^^^vn : e-"<. 'n + i {[(r, + r 2 ) ( P l + p 2)]W + [(n - r 2 ) ( P l - p 2 ) p } = (cm) + (rel) + time integral corrections + • • •. (F.l) The centre of mass term does not contribute and the M0ller operator changes the relative position and momentum = -iVvynA*T ,HrJ_^u (2) ; (2) V ' 2V2kTAv rdr[ J / fn iVvvn r _^_8uldu \ , 1 2 , = - * t e 2 i 7 l w + ( i ) 1' ( R 2 ) For the first integral, utilizing the definitions of Section 2.5.1 gives (a) = 8TT2 j drd9d1r sin 0 7 V 7 o ^ r 7 o cos(r/fl - 0)r[flcos(/?R - 0) + s0<7o cos(r7R - 9)]. (F.3) Substituting 77A = + ,7r — ^ in the trigonometric functions above, and making use of the fact that in the limit of R - » 0 0 the angle + and finally making use of the limiting forms, cos 4> = 1 - 0 ( l / i ? 2 ) sin4> = C/R(mE)^2 (FA) then gives, Rcos(f3R - 5) cos(/?R + 7T - 0 - 0) + s0g0 cos2(pR + ir - (p - 9) 165 = {sogo — -R)[cos(/3 — 0) coscfi — sin(/l_0) sin</>]' (s0g0 - R) cos2(/3 - 0) + -2-rsin0cos(/3 - 9) sin(/3 - 9) + 0(1/R) To J_ To r sin0[(.X ± Y) cos2(o; ± C) + cos(o; ± Q sin(o; ± £)]. (F.5) This expression is substituted back into (a) to give, (a) = 8TT2 d9drd-frA s i n 2 0 7 3 e - 7 o ^ [(X ± y ) c o s 2 (a;± C) + cos(a; ± C) sin(a; ± C)] • (F.6) o a r The integration over 9 gives m/2 10 / d0 s i n 2 9[(X + Y) cos2(u + C) + cos(w + C) sin(o; + C)] >/ o+ / d9 sin2 9[(X - Y) cos2(to - C) + cos(o; - £) sin(o; - ()}. Jir/2 r/2 Changing the integration variable in the second integral from 9 to 0' = n — 9 gives, d0 sin 2 0{[(X + Y) cos2(o; + C) + cos(o; + C) sin(o; + C)] JO + [(X - 1') cos2(a; - () + cos(a; - C) sin(a; - £)]} = d9 sin 2 ei(X + Y) + l cos(2a; + 2C)1 + I sin(2a; + 2() Jo I L2 2 J 2 + ( X - Y) [1 4- ^cos(2u; - 2C)] + ^ s n u > - 2C) | = d9 sin 2 0 + ]• (X + y ) (cos 2a; COS 2( - sin 2LO sin 2() Jo 1 2 + - sin 2a; COS 2( + - cos 2a; sin 2£ + i ( X - y ) (cos 2a; COS 2( + sin 2LO sin 2() 1 1 1 H— sin 2LO COS 2£ cos 2a; sin 2£ f = / d0s in 2 0(X + X cos 2a; cos 2£ — Y sin 2a; sin 2£ + sin 2a; cos 2£). J o Thus overall we obtain, (a) = 8TT2 / / d 9 d r d ' j r A s i n 2 0 7 V J J Jo dr x [ X ( l + cos2a;cos2Q - Ysin 2a; sin 2( + sin2a;cos 2(]. Identifying this integral as Eq. (33) of Snider and McCourt gives, (a) = 4n2kTa4(Hx + Rx). .~2 du '0 (F.7) (F.8) (F.9) (F.10) 166 For the second integral, /du dvd7r- d U - ^ ( r ' - T o ) 8TT2 jjde sin 2 Odrd^'h3 ^ r 4 (X±Y). Integrating over 9 for the incoming and outgoing parts of the trajectories gives, r*72 fn dO sin 2 9(X + Y) + [* d9sin29(X -Y). JO JTT/2 thus the second integral of Eq. (F.2) is r ir /2 (b) = 16TT2 Jd8sin2 edrd-ye-^^^X = 4ir2kTa4(HK + Substituting these results into Eq. (F.2) gives, (iii) = — m iVwn 2kW 2AV 1 An2kTa\Hx + i ? A ) - —4n2kTa4(HK + RK) *mkT4iyKKno* f3 _ 1 ' 2 3 A y [2V A X l 2 7rmA;T 4iVyyner4 3 A ( # 7 7 + -#7))-( F . l l ) (F.12) /2 Changing the variable in the second integral from 9 -± 9' = ir — 6 gives, d0 sin 2 + Y)+ T / 2 d0 sin 2 0(X -Y) = 2 d0 sin 2 (F.13) Jo Jo Jo (F.14) (F.15) 167 Appendix G Evaluation of F\ After integrating over the center of mass momentum, / , the integral in Eq. (7.56) becomes, ( G . I ; / = A r 3Tr r e l-F[(7•r)(7o•r ,) + (7'7o) + (7oT)(7-r')]e Carrying out the dot products of the r terms gives 1 = ~Tpl jj drd^e~y2~/}U^: -7cos0(r '-7 o) + 7ocos (%- 0)(r'-7) - ^770 cos ry^r-r') (G.2) With the definition of r' = RKR + sogo, the terms inside the brackets become, {...} = 77o{- cos 6{R cos(r]R - pR) + s0g0] + cos(rjR - 9)[-R cos pR - s0g0 cos nR] - cosr)R[Rcos(pR -9) + s0g0 cos(rjR - 9)}} = -77o {R[cos9 cos(rjR - pR) + cos pR cos(rjR -9) + cos r]R cos(f3R - 9)} +g0s 0[cos9+ 2 cos rjRcos(r]R - 9)}} = (i) + (ii). (G.3) The factor involving R in Eq. (G.3) is simplified by applying various trigonometric identities, (i) = -jj0R{cos9cos(nR - pR) +-[2 cos(r)R -9 + pR) + cos(nR - 9 - pR) + cos(r]R - pR + 9)]} = -77o-R[cos(7?R + PR - 9) + 2 cos(nR - pR) cos 9}. (GA) Using rjR = pR + IT - <fi with <fi = arcsin[£/ i?(mE) 1 / 2 ] as given by Snider and McCourt, the above expression simplifies to: (i) = 77„.R[cos(2/?fl -</>-9) + 2 cos(-</>) cos 9} « 77o# cos(2/i - 9) - 7 r p S i n < 9 sin(2/3 - 9) + 2 cos 9 Rio (G.5) 168 The so^o factor is also simplified, (ii) = -77o30So[2 cos nR cos(nR -8) + cos 8} = -'y)'0g0s0[cos(2r]R -8)+ 2 cos 8} = -77o3oso[cos(2/fo - 20 - 8) + 2 cos8] « - 7 7 oc/oSo[cos(2/3 -8)+ 2 cos 0], (G.6) where terms of order l/R have been neglected. Adding the two parts (i) and (ii) gives (i) + (ii) = - 77o(p 0so - R)[cas(2p - 8) + 2cos0] - 7 2 r sin0sin(2/3 - 0). (G.7) Finally substituting this into Eq. (G.2) gives, / = _ L /" d r d 7 ^ e - 7 2 - ^ 7 2 r s i n 0 { ( X ± y)[cos(2/? - 0) + 2cos0] + sin(2/3 - 0)} dw du dr x{(A: ± y)[cos(2/3 - 0) + 2 cos0] + sin(2/5 - 0)} = - S v r 1 / 2 1 / / drd 7d0 sin 2 0 r 3 ^ e - ^ 2 7 4 2 0 ^ T F A . (G.8) 7T 1/2 169 Appendix H Evaluation of N\ The dot product between the two brackets in the integral of Eq. (7.85) (which is denoted as /) is first evaluated, TT- 3 / 2 Jdge-g2[(rF + uU).0]- [(G2 + To - 5) Q + 2(7o'0)7o] +u (G2 + 7o - 5) G2 + \uGh rr-3/2 / dg +G2 -<?2 \M (~rdu , 1 \ G { - T r + U —rdu0 5 du 2 du , x 9 9 _ 2 9 T ^ 7 ° + 3 r Tr ~ 3 7 ^ ( r ' 7 o ) + U 7 ° " 5 M + 3 U 7 ° (H.l) Substituting this back into the expression of Eq. (7.85) and completing the integration over g gives, 15 / rdu dr u 3 + 2 r du 0 5 du 2 du, . 9 5 9 •3d7 7 o + 3 r ^ - 3 7 ^ ( r ' 7 o ) + 3 M 7 ° - 5 U = TT 3 ^ 2 j drd-ye' -y^—pu 5 du 15 r du y 1 du. n 2 5 2 4 r ^ - T U - 2 ^ 7 ° - ^ ( r ' 7 o ) + 2 U 7 ° (H.2) Using the conservation of energy for the relative motion, 7 Q = 7 2 + @u, and integrating over the relative momentum, this becomes, I 5- [drr^e-* - [ d r r ^ - ± / d r r / f c ^ e " * 4 J dr 4 J dr 2 J dr 2/? 2 e-/3n 170 du _a 5 u—e p u + dr 1 du, ,2 J dvr((3u) 2e-Pu - T r " 3 / 2 ft drdye-^-^-^ir-ro) JJ r dr (H.3) The last integral in Eq. (H.3) is related to Nx. If the limit of R —> oo is used, ,du TT"3/2 fj dvd1(fr1lcos2(r}R-e)e-^-pu = STT1/2 JJJ* d9 sin 9 drdjr'^e^hfo2 cos 2(a;±C). (H.4) The integral over 9 is written as two parts, one for the incoming and the other for the outgoing trajectory, d6 sind cos* (u + Q + I T / 2 / d9 sin 9 cos2 (a; ± C) — \ d9 sin 9cos2(u + () + / d9 sin 9 cos2 (a; — Q JO Jo JTT/2 = / d0sin0 Jo l- + l- cos(2a; + 2C) tit 2. + d0 ' s in(7r -0 ' ) Jo r-K/2 = / d0sin0[l + cos2a;cos2C]. Jo i + i c o s ( 2 W - 2 C ) (H.5) Putting terms back together gives, p 7 r / 2 . . . . . . . o d w 87T 1/ 2 f[r,Z d9sin9drdyr3^e-'*'hh2(l + cos 2a; cos 2() = 8^2^^NX, (H.6) J J J 0 CJ/ ^ where the definition of iVx given in Eq.(31) of Snider and McCourt is used. Using Eqs. (1.4), (1.5) and (1.3) of Appendix I, the integrals in Eq. (H.3) are expressed in terms of the second virial coefficient and its temperature derivatives, / = -kT U^2a3Nx + 7 T ^ + 5 T 2 ^ § ) . (H.7) 171 Appendix I The Second Virial Coefficient and its Temperature Derivatives Some useful identities with the second virial coefficient: B(T) = z l | d r ( c - " « - l ) , (1.1) = - * r § - y < W » ) ' e - > " . (L3) In terms of an alternate representation of the second virial coefficient, these relations are: 172 Appendix J Determination of H \ + R \ — I\ The integral in the last term on the right hand side of Eq. (7.86) has the following form, I = n > A 6 5 kT(mkTy/2Av i J dndrsdpidps [ ( r 1 2 F 1 2 + ul2\J)-Q] - e "^ 1 x n 2 A 6 5kT(mkT)l/2Av "1 (v\ J drKlridptdpie-P1112 [ ( r i 2 F 1 2 + ul2\J)-Q] xfi(12) Pi (J.l) The centre of mass part of the above expression vanishes, so that l~2 Nn r ' = Vi ( m k T Y I V A v J + ""XI -«-""n(12)(g-7)r = Iww, / " l(rF + "u)'gl («) Retaining only contributions with even powers of the center of mass momentum components gives, I = vf2w",3 / M v[(^' +(reI)--]. —du rdr (r.7 0)(r.r') + M ( r ' . 7 o ) r-7o)( r- r ') + «(r'-7o) (J.3) Similar to the work of simplifying (6) in Appendix F, the (rel)" term may be simplified as, (rel)" = 16TT2 Jj j*'* dd sin 2 Odrd^^u^X = An2kTa4Ix. (J.4) 173 The (rel)' can also be simplified as ! du (rel)' = -87r 2 J drd6djr sin6>72e 7° ^ l o cos(r)R-6)r[Rcos(PR-9)-s0'yo cos(r)R-6)]. (J.5) Substituting rjr = (3R + ir — (fi in the trigonometric functions above, and making use of the fact that in the limit of R oo the angle (3R /3 + </>, and finally using the limiting forms cos(Tj = l - 0 ( l / i ? 2 ) sin0 = C/R((mE)1/2 (J.6) gives, Rcos((3R - 9) cos(pR + TT - 0 - 9) + so7o cos2(/3K + TT - </> - 9) = (s07o - #) cos2(/3 - # + <£) = (s07o - i?)[cos(/3 - 9) cos 0 - sin(/3 - 0) sin^] 2 = (s07o - R) cos2(/3 - 0) + —r sin 8 cos((3 - 0) sin(/3 - 0) + 0(1/7?) 7o = sin 0[(X ± y ) cos2(o; ± Q + cos(o; ± C) sin(o; ± ()]• 7o This expression is substituted back into the (rel)' expression to give ! du, (J.7) (rel)' = -8TT2 jjj* d9 s i n 2 9d1drrA1\-^-^[{X±Y) c o s 2 ( a ; ± C ) + c o s ( w ± C ) sin(a;±C)] (J.8) Integrating over 9 then gives r 7 r /2 r d9 s i n 2 9[{X + Y) cos 2(w + C) + cos(w + C) sin(w + C)] Jo + d# s i n 2 e\(X - Y) c o s 2 (a; - Q + cos(a; - C) sin(a; - () ] . JTT/2 (J.9) Changing the integration variable in the second integral from 9 to 9' = ir — 9 gives, r-Tr/2 '0 r c $ s i n 2 9{{(X + Y) cos2(u> + C) + cos(u; + C) sin(a; + C)] J O + {(X - Y) c o s V - () + cos(a; - C) sin(a; - C)]} d9 sin2 9 {{X + Y) Jo ^ + i c o s ( 2 a ; + 2C) + - s i n(2a; + 2C) + (X-Y) i + i c o s ( 2 a ; - 2 C ) 1 + - s i n ( 2 a ; - 2 C ) } = r d9 sin2 9 X + -(A + r ) ( c o s 2 w c o s 2 C - s i n 2 a ; s i n 2 C ) Jo L 2 + ^ (X — Y) (cos 2a> cos 2 ( + sin 2u> sin 2£) sin 2u cos 2£ = r / 2 d 0 s i n 2 0 ( X + s i n 2 w c o s 2 C + A r c o s 2 a ; c o s 2 C - y s i n 2 w s i n 2 C ) . (J.10) J O 174 Thus overall, (rel)' = -8TT2 jjf^ d.6 sin 2 edjdrr^e^^ x[X(l + cos2a;cos2() + sin2c<;cos2( - Ysin2a;sin2(]. (J.H) This integral agrees with the form given in Eq. (33) of Snider and McCourt, that is (rel)' = -4ir2kTa4(Hx + Rx), (J.12) So that all together, !=-^^Sw^2kT^+R^-h)- (J-13) The contribution to the thermal conductivity from this term in Eq. (7.35) is = \n2ka\l^(Hx + Rx-Ix). (1.14) 3 V ?n This agrees exactly with the second order in density correction given in Eq. (26) of Snider and McCourt. 175 List of Symbols Ay normalization factor for the potential flux functions b impact parameter bf^ jth coefficient in the Sonine expansion of B°(pi) B scale factor in transport coefficient expression B°(pi) unknown function which arises in the inversion of Q(e) B(T) second pressure virial coefficient Bv second viscosity virial coefficient B\ second thermal conductivity virial coefficient B ^ reduced flux evolution vector Be$ reduced 0 $ matrix elements of the kinetic projected resolvent for thermal conductivity Be$ reduced 0 $ matrix elements of the kinetic projected resolvent for viscosity E relative energy of a colliding pair, E = pgl/2 Cartesian tensor identity for the irreducible representation of 50(3) of weight a (v, r, t) one-particle distribution function FJC force between particles j and £ Fff a first order non-locality density correction to the transport coefficient a in Snider-Curtiss theory g0 relative speed of approach of the 12 pair prior to collision g12 relative speed of approach of the 12 pair G(z) resolvent with an TV-particle Liouville operator i/(z — £) Go resolvent of a system with non-interacting particles i/(z — K) C72(12) resolvent of a system with only particles 1 and 2 interacting i/(z — £12) Q dimensionless center of mass momentum for particles 1 and 2 Q (e) TV-particle resolvent expanded in powers of 1/e 176 h enthalpy per molecule, divided into kinetic hx and potential hv parts H12 total hamiltonian of a pair of particles Htf binary collision cross section for a mixture IB the Bogoliubov form of the triple collision operator IG the Green form of the triple collision operator J x correction to the Boltzmann equation J 2 correction to the Boltzmann equation J flux tensor of general rank JK kinetic contribution to the flux tensor J y potential contribution to the flux tensor J symmetric traceless momentum flux tensor of rank 2 Kyi kinetic energy of the pair of particles K, the kinetic contribution to the Liouville operator Ke$ €>$ matrix elements of the resolvent of CK £ x Kawasaki - Oppenheim binary collision operator C TV-particle Liouville operator Cv (l-V)C(l-V) CK (1-VK)C(1-VK) Cv (l-Vv)C(l-Vv) M©$ 0 $ matrix elements of the resolvent of C n number density TV the number of particles in the system, N —>• oo Na a first order collisional transfer density correction to the transport coefficient a in Snider - Curtiss theory p relative momentum of particles 1 and 2 Po relative momentum of 1 and 2 prior to collision p^ relative momentum for a general point £ on a 2-particle trajectory P the center of mass momentum of particles 1 and 2 CX. V projection operator, |J)) © ((J| Vo projection operator onto the flux of particle 1, | Ji)) 0 ((Ji| VK 1/N\JK)) © « J * | Vv l/N\Jy)) 0 ((Jv| ps species projection operator r relative position of particle 1 with respect to particle 2, ri — r 2 r' H 1 2 r 177 R center of mass position of particles 1 and 2, (ri + r-2)/2 1Z linearized Boltzmann binary collision operator 7lK projected linearized collision operator, (1 — Vo)ip\R,i^(l — VQ) 7Za^ ab binary collision operator matrix element R(T) Hoffman - Curtiss correction integral R cutoff distance for the intermolecular potential s 0 time from the start of the collision interaction to the reference point of the trajectory 6 binary collision kinetic cross section T transition superoperator u(r) intermolecular potential u' u — hy Ui2) two-particle Ursell function U 2nd rank unit tensor XR b~l times the projection of r ' n on the asymptotic collision direction Xs mole fraction of species s (v) average speed of relative approach Ve$ 0 $ matrix elements of the resolvent CV V the potential contribution to the Liouville superoperator W j dimensionless momentum of particle j Y collision parameter related the projection of r' on the asymptotic collision direction Yjo] density expansion factor for the 2-particle reduced distribution function Ze$ 6 $ matrix elements over the resolvent of C a weight of the irreducible representation of 50(3) 0 l/kBT /?£ angle of the general relative position vector £ with respect to —p 7 dimensionless relative momentum of particles 1 and 2 7o dimensionless relative momentum of 1 and 2 prior to collision integrals over the potential function found in determination of Ay A correction term to the Boltzmann equation A i correction from displacing the particle positions from r i and r 2 to R e m and time to to t in the generalized Boltzmann equation 178 A 2 correction from displacing the center of mass position to r x in the generalized Boltzmann equation e a small, real convergence parameter, e —»• 0 + c angle between r m and the reference point r V shear viscosity coefficient Vo low density limit to the shear viscosity coefficient Vi angle of the general momentum vector p^  with —p binary collision cross sections for a mixture 6 angle between r and —p at the reference point of the trajectory unit vector in the direction of the general relative position vector £ unit vector in the direction of the general relative momentum vector A thermal conductivity coefficient reduced mass a general position on the collision trajectory s-particle reduced distribution function p ( iV) iV-particle distribution function cr position scale parameter for the intermolecular potential Vj the normalized Maxwellian momentum distribution for particle j X deflection angle in a binary collision identification state matrix for species s angle of r m and the asymptotic direction of approach i i 1 2 M0ller superoperator n(2'2)(T) Omega collision integral Q(2>2)*(T*) reduced Omega collision integral 179 

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