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On the quantum statistical theory of thermal conductivity Griffin, Peter Allan 1961

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OK THE QUANTUM STATISTICAL THEORY OF THERMAL CONDUCTIVITY. fey PETER ALLAN GRIFFIN B. Sc., University of Bri t i s h Columbia, i 9 6 0 . A THESIS PRESENTED IN PARTIAL FULFILMENT- OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER, I 9 6 I . 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 i t 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allox^ed without my written permission. Department of PI-} VS I C S The University of British Columbia, Vancouver 8, Canada. Date UJUyCuU^ £LO,\cM$\ ABSTRACT A c r i t i c a l survey of the present state of the quantum s t a t i s t i c a l theory of thermal conductivity i s given. Recently several attempts have been made to extend Kubo's treatment of el e c t r i c a l conduction to other irreversi-ble transport processes in -which the interaction between the driving system and the system of interest i s not precisely known. No completely satisfactory solution of the problems involved is contained in the literature. In this thesis, a detailed derivation of a Kubo-type formula for thermal conductivity i s given, using essentially the concepts and methods of Nakajima and Mori, with no pretense that i t settles the problem completely. Some general remarks are made on the evaluation of a Kubo-type expression, in particular, the use of Van Hove's master equations and the reduction of the usual N-particle formula to a single particle formula, An explicit calculation of thermal conduc-t i v i t y i s made for the simple model of elastic electron scattering by random-l y distributed, spherically symmetric impurities. - i i i -TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i l l ACKNOWLEDGEMENTS iv INTRODUCTION 1 CHAPTER 1 . NON EQUILIBRIUM PROCESSES IN ISOLATED MANY-PARTICLE SYSTEMS 8 CHAPTER 2 . THE PROBLEM OF TRANSPORT PROCESSES IN OPEN SYSTEMS AND THE STANDARD "BOLTZMANN EQUATION" SOLUTION .23 SECTION 2.1 :TRANSPORT PHENOMENA IN OPEN SYSTEMS .23 SECTION 2.2 :OUTLINE AND CRITICISM OF THE STANDARD. METHOD OF FINDING ELECTRICAL AND THERMAL CONDUCTIVITY " .28 CHAPTER 3 . KUBO'S DERIVATION OF FORMAL EXPRESSIONS FOR THE TRANS-PORT COEFFICIENTS DESCRIBING MECHANICAL DISTURBANCES 33 CHAPTER k . ATTEMPTS TO FIND KUBO-TYPE FORMULAS FOR TRANSPORT COEFFICIENTS CONNECTED WITH THERMAL DISTURBANCES (THERMAL CONDUCTIVITY) A? SECTION k.l :INTRODUCTION AND CHOICE OF OPERATORS ^5 SECTION k.2 :DERIVATION OF A KUBO-TYPE FORMULA FOR THERMAL (X5NDUCTIVITY 5 5 SECTION U.3 : CRITICAL REVIEW OF OTHER METHODS .65 CHAPTER 5 . EVALUATION OF KUBO-TYPE EXPRESSIONS .755 SECTION 5.1 :RELATION TO VAN HOVE'S MASTER EQUATIONS AND THE REDUCTION TO SINGLE PARTICLE EXPRESSIONS 0 .75 SECTION 5.2 :EXPLICIT EVALUATION OF THERMAL CONDUCTIVITY FOR ELECTRONS SCATTERED BY IMPURITIES £2 BIBLIOGRAPHY .85 - i v -ACKNOWLEDGEMENTS I wish to express my gratitude to Professor W. Opechowski for suggesting this problem and for his continued interest and valuable advice throughout the performance of this research* In addition, I am indebted to Mr. K. Nishikawa for many useful conversations as well as for considerable assistance i n the preparation of the f i n a l draft of this thesis. I wish also to acknowledge the financial assistance of the National Research r Council of Canada. (1) INTRODUCTION Experimentally, i t i s well known that when a weak external dis-turbance i s applied to an Isolated system, the resulting "currents" are proportional to the external "forces". The constants of proportionality enable us to describe irreversible transport processes by a few experimental parameters. I t i s a fundamental problem to give a theoretical basis for the existence of such " f i r s t order transport coefficients" (as they are called by the theorist) and to show how to calculate them in terms of the micro-scopic properties of the physical system involved. Examples of these c o e f f i -cients are e l e c t r i c a l conductivity, thermal conductivity, viscosity and the diffusion constant. In this thesis, we shall be mainly interested in the f i r s t two coefficients. Unti l very recently, the only method of calculating these c o e f f i -cients was by the use of Boltzmann-type kinetic equations. These are assumed to be sat i s f i e d by single particle distribution functions of the particles and quasi-particles involved. Except for certain special cases, the validity of these transport equations i s highly doubtful. In addition, these complex integro-differential equations are not easily solved and, i n most situations, a universal relaxation time must be introduced before calculations are feasi-ble. The use of a distribution function i s suspect from the point of view of quantum mechanics and at best gives only p a r t i a l information, since i t corres-ponds to a diagonal matrix element of a reduced density matrix. Over and above any theoretical objections, the whole transport equation method does not give too much physical insight into the processes involved. Neither does i t involve any consistent manner of making approximations. The extreme d i f f i c u l t y of extending the standard treatment in order to f i n d corrections of higher order ( 2 ) i n the interaction between the particles and quas i-particles i s well known. Recently there has been considerable success achieved in the attempt to give a rigourous quantum s t a t i s t i c a l theory of f i r s t order irreversible transport processes. Same authors have put the Boltzmann-type equations on a sounder footing. A more interesting step has been a whole reformulation of the problem, i n i t i a t e d mainly by the Japanese physicists Mori (1956), Nakano (1956), Nakajima (1956) and Kubo (1956). Following the practice of the current literature, we shall c a l l this new approach Kubo's method (or Kubo*8 formalism). I t i s based on the use of the density matrix whose equation of motion i s solved to the f i r s t order in the "external disturbance". It i s a quantum s t a t i s t i c a l theory applicable to any transport process where the weak interaction between the isolated system and the "driving" system can be expressed by some appropriate quantum mechanical operator. The main result of this theory i s that we are able to express certain transport coefficients by means of a double integral involving a time dependent correlation function evaluated over an equilibrium ensemble. These formal expres-sions are derived without any important restrictions being introduced. The evaluation for specific physical models generally involves a f a i r l y complicate ed calculation, although the form of these expressions enables us to develop general approximation methods. ' Now there exists a whole class of irreversible transport processes to which the above-mentioned Kubo's formalism i s not directly applicable,since, in these processes, the interaction between the "driving" system and the system of interest cannot easily be expressed by means of an unambiguous quan-tum mechanical operator. In these processes, the precise mathematical descrip-tion of the external disturbance (for example, a temperature gradient) i s not obvious, other than in a macroscopic sense. Several authors have attempted to derive Kubo-type expressions for the transport coefficients occurring in this ( 3 ) class of transport processes. The major aim of this thesis i s to give a c r i t i c a l review of these derivations, taking thermal conductivity as a con-crete example. In our opinion none of the discussions in the literature i s completely successful. Chapter 1 i s entirely devoted to a discussion of the dynamical evo-lution of a large system of interacting particles, with particular reference to the work of Van Hove (1955, 1957) on quantum mechanical transport equations. At f i r s t sight, the study of the non-equilibrium behaviour of an isolated system has l i t t l e to do with the main topic of this thesis, which i s the i r r e -versible transport processes occurring i n open systems. However, one of the important characteristics of Kubo-type expressions for transport coefficients i s that their expli c i t evaluation involves only a knowledge of the dynamical evolution of an isolated system. Indeed, we can look upon the derivation of closed, formally exact expressions for transport coefficients as simply a transformation of a complex problem to a more basic one. That i s , we are originally dealing with-an open system (an isolated system under the influence of an external disturbance); but, i n the end, the transport properties are incorporated into a Kubo-type expression which involves only the properties of an isolated system. In chapter 2, we come to transport processes in open systems. After the problem has been stated mathematical 1y, the phienomenological description of irreversible processes i n open systems i s touched upon. In particular, the macroscopic description of electric and thermal conduction i s outlined. In the second half of chapter 2, we give a brief analysis of the disadvantages of the standard method of evaluating e l e c t r i c a l and thermal conductivity by means of Boltzmann-type transport equations. An account of Kubo's formalism (1957&) i s contained in chapter 3 . This formalism deals successfully with mechanical disturbances such as an ( h ) external electric f i e l d . Detailed discussions of Kubo's work have been given by several authors - for example, Van Hove ( i 9 6 0 ) and Montroll ( 1 9 5 9 ) . This chapter contains nothing new, except possibly a more c r i t i c a l treatment of the physical assumptions necessary in order that the response current be independent of when and how the external f i e l d i s applied. There i s some ambiguity on this point in the literature, for example, in the treatments given by Lax ( 1 9 5 8 ) * Kubo ( 1 9 5 7 a ) and Van Hove ( i 9 6 0 ) . One of the main reasons for including the material of chapter 5 is to provide an example of what we mean by a satisfac-tory quantum s t a t i s t i c a l theory of linear, irreversible transport processes. It may be pointed out that i t has been the great success of Kubo's treatment of mechanical disturbances which has encouraged attempts to extend the treat-ment to external disturbances which are, physically, entirely different. That we are able to calculate the effect on an isolated system due to an electric f i e l d , for example, does not Justify the expectation, a p r i o r i , that we can do likewise for a temperature gradient. With chapter k, we come to the main part of this thesis, a c r i t i c a l survey of the attempts to find a closed, formally exact Kubo-type expression for thermal conductivity. The f i r s t problem to be considered i s the proper choice of a quantum mechanical operator whose ensemble average gives the energy current. The usual definition Involves a summation over the products of the energy and momentum operators of the particles and quasi-particles composing the system. Ideally the proper definition should follow from a rigou-rous theory of heat conduction. The manner in which we define local density operators is another problem often skipped over in the literature. Both Mori ( I 9 5 8 ) and Nakajima ( 1 9 5 8 , 1 9 5 9 b ) simply take the quantum mechanical operators corresponding, via Weyl's rule, to the cl a s s i c a l functions involving delta singularities in the positions of the particles. As a consequence of these singularities, the content of these theories i s basically semi-classical. ( 5 ) In the second section of chapter 4, we give a derivation of a Kubo-type formula for thermal conductivity by making use of the concept of the l o c a l equilibrium density matrix. The f i n a l formula has been written down by several authors including Mori (1956) and Kubo, Yokota and Nakajima (1957b), but the concepts and methods used in the literature db not as yet provide a satisfactory proof. Our discussion follows the work of Mori (1958, 1959) and Nakajima (1959). It i s based on the use of an equivalent isolated system relax-ing from a weak fluctuation. This fluctuation i s chosen so that, i n i t i a l l y , i t leads to the same energy current flow which exists i n the open system involving a small temperature gradient. Essential use i s made of the smallness of the temperature gradient. As a consequence, the analysis cannot be generalized to deal with quadratic and higher order effects, i n contrast to Kubo's original method. As in chapter 3* we pay particular attention to the importance of the dissipative properties of the many-particle system. In our opinion, the l i t e r a -ture does not place enough stress on the fact that any satisfactory derivation of Kubo-type formulas for the transport coefficients presupposes that the dynamical evolution of the isolated system has been investigated. This i s quite apart from the evaluation of these abstract formulas. In section 4.3, same brief remarks are made on other methods of deriv-ing Kubo-type formulas. Of special interest i s the work of Montroll (1959) who attempts to expressly , w t r i c k " boundary conditions, the interaction between the driving system (the heat reservoirs in our case ) and the system of interest by means of some quantum mechanical operator. Lebovitz (1957, 1959) also attempts to e x p l i c i t l y f i n d the interaction, although his analysis makes use of a kinetic equation. Although we are mainly interested in quantum mechanical systems, i t should be mentioned that M. S. Green (195*0 developed a theory for classical r systems which in many ways i s the classical analogue of the treatment i n section 4.2. A recent paper by Kirkwood and F i t t s (i960) results i n the thermal ( 6 ) conduct!vity tensor (among other transport coefficients) for class i c a l sys-tems being expressed as a time integral over a correlation function. Indeed, in a certain sense, Kubo's method i s simply a quantum generalization of Kirkwood1 s pioneering work ( 1946) . In the f i n a l chapter, we consider the general problem of evaluating Kubo-type formulas. The role of these abstract expressions i n linear transport processes Is similar to that played by the formal expressions used i n e q u i l i -brium s t a t i s t i c a l mechanics which give the average value of some operator by means of the trace over the product of the operator and the quantum mechanical partition function. By means of the latter, we can express a l l thermodynamic variables, the only problem being that of explici t evaluation. Except for certain investigations into the foundations, most of the recent literature on equilibrium s t a t i s t i c a l mechanics i s a search for better methods of approxima-tion of these formal expressions involving the partition function. Similarly, once we have derived Kubo-type formulas for transport coefficients, the only remaining problem i s to evaluate the complex, N-particle, time-dependent corre-lation functions involving the partition function. Only a beginning has been made in the literature in finding general methods of approximate evaluation of Kubo-type expressions. The cluster integral method of Montroll and Ward (1959b) and the use of Green's functions (see, for example, the review a r t i c l e by Zubarev ( 1 9 6 l ) ) both enable us to evaluate these transport coefficients to any stage of approximation, in principle at least. No calculations are made using either of these methods in this thesis. In the second part of chapter 5* we discuss the relation of Van Hove's work on master equations to the evaluation of the correlation functions involv-ed i n the exact expressions for transport coefficients. As Chester and Thellung (1959) and Klinger ( 1 9 6 l ) have stressed, Van Hove's work i s one of the most f r u i t f u l methods of evaluation, i f not as general as those mentioned at the end of the above paragraph,, After a few remarks on the assumptions < 7 ) necessary to reduce the usual N-particle Kubo-type formula to a single particle formula (see, for example, Verboven (i960) ), we make use of the latter to e x p l i c i t l y evaluate the thermal conductivity for a simple model of a metal in which electrons are scattered e l a s t i c a l l y by static, randomly dis-tributed, spherically symmetric impurities. This calculation i s simply an extension of the work of Chester and Thellung (1959) on el e c t r i c a l conductivity. To conclude this introduction, we should l i k e to add two points. F i r s t , Dresden (I961) has recently published the f i r s t part of an exhaustive review a r t i c l e on the recent progress i n transport theory. While this f i r s t part only emphasizes the d i f f i c u l t i e s of the standard methods, the published summary of the second part indicates that i t w i l l cover similar topics as are discussed i n this thesis, although attention w i l l not be restricted to thermal conductivity. Lastly, though the work of many authors i s quoted i n this thesis, no attempt at completeness has been made. Only those papers which have been used directly i n writing the text have been e x p l i c i t l y referred to. ( 8 ) CHAPTER 1 NON EQUILIBRIUM PROCESSES IH ISOLATED MANY-PARTICLE SYSTEMS. Consider an i s o l a t e d p h y s i c a l system composed of a large number of i n t e r a c t i n g p a r t i c l e s enclosed i n a volume _TL . A fundamental problem of t h e o r e t i c a l physics i s , given the kinds of p a r t i c l e s and the various i n t e r -actions among them, t o derive the macroscopic properties of the p h y s i c a l sys-tem. In recent years, considerable progress has been made i n c a r r y i n g out t h i s program f o r various n o n - t r i v i a l cases. The advances i n t h i s so c a l l e d N-body problem have been l a r g e l y due to the improved mathematical techniques which were o r i g i n a l l y developed i n quantum f i e l d theory, as f o r example, the use of diagram techniques. An i s o l a t e d p h y s i c a l system (through out t h i s t h e s i s , t h i s w i l l always r e f e r t o a quantum mechanical many-particle system unless otherwise s p e c i f i e d ) may be i n an "equilibrium" or "nonequilibrium" s t a t e a t any p a r t i c -u l a r time. Before g i v i n g a mathematical formulation of t h i s d i s t i n c t i o n , we s h a l l make a short digression i n order t o introduce the density matrix formal-ism used i n quantum s t a t i s t i c a l mechanics. Let the normalized quantum mechanical s t a t e f u n c t i o n (rv,t) of the p h y s i c a l system be expanded i n t o a l i n e a r combination of a complete set of orthonormal functions denoted by |TJ.(^-)J * that i s , ( i . i ) f u , t ) « ^ Cj(t)Y.(/t X We have used a coordinate representation with ft representing the coordinate degrees of freedom, and t denoting the time dependence. For s i m p l i -c i t y , we have assumed tha t the functions 4n( A* ) are s p e c i f i e d by a d i s c r e t e index " h n « Otherwise, we would have i n place of (1.1), the f o l l o w i n g expan-sion (1.2) ¥(yi,t) - J c j W t ( * > ) d * We now introduce an ensemble of systems, each system i n the ensemble ( 9 ) being denoted by the set of functions ^ Cn( t )} • The ensemble average of C m ( t ) C n *( t ) v i l l be denoted by p m n ( t ) . That i s , U . 3 ) p m n ( t ) E < C m ( t ) C * ( * ) > , p m n ("t) being an element of the so called density matrix p(-t ) describing the ensemble. The ensemble average, except when a l l the systems in the enseal-ble are identical ("pure case"), introduces an additional statistical averag-ing apart from any averaging inherent in quantum mechanics. As is well known, the expectation value at time "t of some dynamical observable A of the system in the state ^ ( A j t ) is given by (i.M A (-fc) - f ¥ K t ) 4 ¥ u , t ) c U where the integration Is over the volume SL of the system. As a matter of notation, a cap A over a letter always denotes a quantum mechanical operator. The only exception to this convention will be the density matrix p ( t ). Now, i t can be easily shown that the ensemble average of a dynamical operator at time "t is d.5) <AX=<A(t)> =ZA„mpmn(t), vith AMm=(y*U)AH^)c{/r ,M Since the choice of I^T^CA. )] can be any complete orthonormal set of wave functions describing the many-particle system, we can write (1.5) as (1.6) <A> t= Tji(Ap{t)) ; where T / t stands for the trace. It should be noted that | ( /i) ] must satisfy any periodic boundary conditions imposed j. on the isolated system. The evolution in time of the system with a time-independent Hamil-A tonian f< is described by the solution of Schrodinger8s equation, d.7) i u , t ) = e ~ L * ( t - t e ) i W o ) . Using the density matrix formalism, Schro dinger's equation is transformed into Von Neumann's equation, (10) (1/8) - r i [ A , p ( i ) ] « O , where [ A , B] E AB - B A and- units are chosen so « j . The cl a s s i c a l analogue of (1.8) is Liouville's equation. One can easily check by differentiation that the solution of (1.8) Is given formally by W9) P ( t ) - e-^-VVctoe*1**-*-'. Both (1.7) and. (1.9) express the determinate quality of the dynamical evolution of the isolated physical system. If at time " t 0 , the properties of the system are known, then for a l l values of "t > "t o , the state Is completely determin-ed. We assume that the ensemble average given i n (1.6) i s the average observed value of the dynamical observable A0 In principle, a l l that remains i s to evaluate the right hand side of (1.9), substitute this expression for p ("t) into (1.6) and calculate the trace in order to f i n d the average value of A at time "t . The d i f f i c u l t problem l i e s In evaluating (1.9) since this indirectly involves the solution of a N-body problem. It may be appropriate to make a few remarks on the Heisenberg repre-sentation which w i l l be used i n later chapters. In .the discussion so .far, the Schrodinger "picture" or representation has been used. The complete •dynamical properties of the system have been incorporated i n p ( "t), with the Hermitian operator A constant i n time. On the other hand, in Heisenberg representation, the state functions (and^ consequently }the density matrix) are constant i n time, while the dynamical observables are functions of time. The new operator A("fc ), associated with the old operator A , satisfies the following equation of motion (l.io) dAl*L - i f t f t . Ait)] where X i s assumed to have no explicit, time-dependence. As before, the formal solution of (1.10) i s given by ( i . o i ) A ( t ) = e l A t . ) e^ *""**1 (11) The obvious physical interpretation of ( l . l l ) is that i t describes the "natural" behavior of the operator A ( t ) . If we define A(O) = A 7 then the relation between A and A (-fc ) is (1 .12) Act) = elk± Ae'iHt Using the formal trace identity T/i(AB) = 7/L (BA) , where A and B are any two operators, we can rewrite (1 .6) in the form ( i a 5 ) < A > t = T u ( / \ w p c o ) ) . .. The advantage of the Schrodinger "picture" is that i t puts the emphasis on the dynamical properties of the system, rather, than on the dynamical obser-vable A . It separates the problem of the time evolution of the system p ( t ) and the problem of what the dynamical observables A are in any particular case. For isolated systems, the main problem is the former one. For open systems resulting in transport processes, the latter problem is not so easy, as we shall see in later chapters. Let us return to the question of equilibrium and non equilibrium states of the system. If p i t ) 1 8 independent of t , then <&7i. is independent of "t also and the system is said to be in equilibrium. The equilibrium density matrix CJ B Q describing the ensemble of systems is a solution of the equation U.110 C « , p E 0 > o We state, without any further discussion, that we can take for the equilibrium state a microcanonical ensemble i f the energy is fixed, a canonical ensemble i f the temperature is fixed and a grand canonical ensemble i f the temperature is fixed but the number of particles is not constant. For future reference, the canonical ensemble, in particular, has the following density matrix (1-15) where k is Boltzmann*s constant and T is the absolute temperature. As is well (12) known, the canonical ensemble describes a small system i n contact with a larger system which acts as a temperature bath. That i s , we s p l i t a very large system i n equilibrium into two unequal parts and concentrate our atten-A tion on the smaller one, whose Hamiltonian i s denoted by H . For the rest of this thesis, the equilibrium density matrix p M w i l l be that given by (1.15). The above paragraph deals with the quantum analogue of the e q u i l i -brium s t a t i s t i c a l mechanics as developed by Gibbs and others at the turn of the century. The principles are clearly formulated in a manner which Is inde-pendent of any model or special postulates about the particular system being considered. A l l macroscopic "thermodynamic*1 properties are directly related to the interaction forces and int r i n s i c characteristics of the particles of system by the relation (1.16) <A> R A= T/t (APBO) = T A (A e-**) = £ TA(e-0«) As mentioned before, In certain cases the mlcrocanonical or grand canonical ensemble may be more appropriate. Once A and H are known for the particular physical problem at hand, the evaluation of (1.16) i s essentially a mathematic-a l problem. Any ex p l i c i t calculation of A may become quite a complicated blend of approximations made on physical grounds, but this does not alter the useful-ness of (1 .16), which is a sort of half-way house between the general p r i n c i -ples of quantum s t a t i s t i c a l mechanics and the explici t formulas an experimen-t a l i s t uses. As w i l l be seen later, some transport coefficients can be express-ed by formally exact expressions different from (1.16) but having the same usefulness and generality. In case p(-fc) i s not independent of the timet, then obviously <A)t -i s not, and the system is said to be in a non equilibrium state. The task of finding the evolution of the system in time involves evaluating,. (0(t) 7 (13) (1.17; p(t) = e p(o)e ( pio) is assumed to be known) Now from physical experience, most isolated systems in a non equilibrium state approach an equilibrium state after a "long enough" time. This time interval, of course, depends on the properties of the system, especially on the strength of the coupling or interaction forces. In terms of our formalism, this irreversible approach towards equilibrium can be expressed by the follow-ing relation, t ~ » O O where for any particular system, the limiting process really means, (1.19) p ( t ) * p e f t , with t » T R and T R being same appropriate relaxation time. The fundamental conceptual problem is to somehow arrive at irreversi-b i l i t y , starting from a framework of reversible equations. Over the years, many discussions have dealt with this difficult problem without trying to describe in any detail the time-development of the system. We may briefly mention ergo die theory. Quoting from a recent article by Farquhar ( 1 9 6 1 ) , the purpose of this theory "is to establish the necessary and sufficient conditions for a system to approach equilibrium from an arbitrary non equilibrium situation", that Is, "to derive conditions for the equality of time and ensemble averages...of expec-tation values of dynamical observables". A much more specialized, as well as more ambitious, line of attack lies in trying to evaluate ( 1 . 1 7 ) for particular systems and thus follow their total time-development. Equation ( 1 . 1 8 ) would follow as an asymptotic result for long times. It may be noted that ergodie theory gives no estimate of Relax-ation times, while this method definitely does. In any discussion along these (lk) l i l i e s , we must introduce i r r e v e r s i b i l i t y i n some natural manner as well as successfully approximate (asymptotically) the equations of motion of a system with an enormous number of degrees of freedom (of the order 10 ^ ) . It turns out, as we might expect, that the two problems are intimately related, i r r e -v e r s i b i l i t y appearing as a characteristic property of macroscopic systems, -. . . . . . Van Hove's f i r s t paper ( 1 9 5 5 ) on the lowest order Markovian master equation must be considered to be the f i r s t rigourous resolution of these d i f f i c u l t i e s , although Kirkwpod and Bogoliubov both made significant contributions i n the case of cl a s s i c a l systems. To do justice to the considerable progress that has been made i n this f i e l d in the last few years would require a large monograph. Indeed, i f we consider the important gains in understanding, the time i s ripe for a modern version of the Ehrenfests* femous Encyklopadie a r t i c l e . We shall give a brief account of Van Hove's work not only as an excellent example of recent work on non equilibrium processes but also since i t w i l l be used later in chapter 5 . The mathematical techniques used by Van Hove enable us to evaluate the time dependent correlation functions occurring in certain formal, expressions for transport coefficients. Van Hove assumes that the system of interest is such that Its total Hamiltonian ii can be s p l i t into two parts. More precisely, we have ( 1 . 2 0 ) A = -Ho + XV where the "free Hamiltonian" ^ 0 describes N/ free excitations and the A "perturbation" X V represents the interactions among the excitations. The term "excitation" i s used by Van Hove to denote both particles and quasi-particles, while A i s a dimen3ionless parameter measuring the strength of the interactions among the excitations. The eigenstates and eigenvalues of A A ~Ka are assumed to be known and A V is treated formally by perturbation (15) theory even though i t i s not necessarily small. The basic representation used throughout Van Hove's work i s that i n which f<o i s diagonal, that i s (1.21) ft0 /*> = €« lot) where both the eigenstates and eigenvalues are labelled by a complete set of quantum numbers ot • Some of these <X quantum numbers are always discrete, such as the ones denoting the spin and polarization indices of the excita-tions. Each elementary plane wave excitation is also labelled by a wave vector. The components of this wave vector* i n the l i m i t of a large system, are regarded as continuously varying quantum numbers. In general, we may adopt the normalization (1.22) <<*U-> - cf where £(ot -<*') is.the product of Dirac delta functions for the continuous quantum numbers and Kronecker symbols for the discrete ones. To simplify the analysis, Van Hove (1955, 1957, 1958) often assumes that a l l thec< quantum numbers are continuous. In a recent a r t i c l e (I960), he always keeps i n mind the fact that real physical systems are f i n i t e i n size and consequently a l l the c< quantum numbers are discrete. In our outline, we sh a l l follow the presentation of this a r t i c l e , and therefore we adopt the normalization (1.23) <<*lcx'> = SKI**' -CO oi 4 aL* 1 I oL^oH and use summation notation throughout. I t i s essential to Van Hove's analysis that the unperturbed energy spectrum formed by the £ « ^ i s continuous (or, at least, quasi-continuous). If the system of interest i s large, the unper-turbed eigenvalues have this property (see, for example, Landau and Li f s h i t z (1958), Chapter 1, for an interesting discussion of this point). The entire discussion i n this chapter deals with the whole system and therefore i s i n T-space.'' Much of Van Hove's work i s valid for a (16 ) single system, but in many other portions of this and other chapters, we often speak of an isolated system when we are really dealing with an ensem-ble of such systems. The problem of making a transition to a description involving a single excitation by integration (or summation) over the degrees of freedom of the other excitations w i l l be commented on In chapter 2 . The lat t e r description i s said to be i n ytx-space'* and w i l l be used i n chapter 5. The aim of Van Hove's investigation i s to f i n d the time-dependent A expectation value or ensemble average of an operator A diagonal i n the l<*>-representation. The eigenvalues AW are given by (1.24) AI ay = A(ot)l*> . and, as we shall see, Van Hove's analysis requires that they be smooth func-tions of the a quantum numbers characterizing the state. I f the system Is In the quantum state | at t-o, then at time t, i t i s In the state given by d . 2 5 ) »<K> = e - " ? - ^ * / * ^ = 0(t)i<Po/. The expectation value of A at time t i s consequently ( 1 . 2 6 ) ^(t)= <<pt\A\<P+> Expanding the i n i t i a l state J&>in the lot)-representation, * ( 1 . 2 7 ) |4>.> = TLct4)\ot> and using (1 .24), we may rewrite Atfiin ( 1 . 2 6 ) as (1.28) A(+) = £ H c U ) C U ) < o M U ( - t M UL-t)lott> =2Z \ccoit)\iZl A(oi}) ko^ / u c o i o ^ r V"r^cc^o^/4c^)<^ju(- t )u J><^/L)( t)Wz> (17) The prime' i n the double summation means that <*,#yr . We introduce the following definitions: (1.29) Pt (<*>i = \Q(-t)\*z><'*3lUHr)\ol,> . (1.30) ItC^,,cfi;u3)=, \\J(-t)\d3><^i ( U t e , ) M i 7 • (1.31) P ( ^ / t ) = Z Pt (•«*.;a/*) Ic6x.)|* ( P ter-m) . (1.32) I ( ^ 3 , t ) = j y It (*. ,^;«? 3 ) C*GOci<*J (I t e r m ) . (1.33) PtM= P(*3rt) + I(ot?,t) Using these new quantities and (1.28), we have (1.34) A(t) = Z A W , ) P f ^ i ) So far, everything we have done i s completely formal since no important assumptions have been made. ' Van Hove assumes that the perturbation operator V sat isfies the so called diagonal singularity condition. Physically, this analytic property of V i s connected with the fact that the perturbation extends over the whole system. I t may be mentioned that another consequenoe of this character of the perturbation i s that even though the interactions among the excitations may be quite small, the perturbation energy of the whole system i s quite large. Therefore f i r s t order time-dependent perturbation theory may not be va l i d . The deep relationship between the diagonal singularity condition and the dissipative properties of the system i s discussed i n the series of papers by Van Hove (1955, 1957, 1959). In the l i m i t of a large system, the diagonal singularity condition can be expressed by the equation (1.35) <C<X I v Vl<*'> - c f O -*') W(o() + <oilYloC> where the second term i n the right hand side of (1.35) contains a weaker singularity than the one contained i n the f i r s t term. (18) Van Hove (1955, 1958) was able to evaluate 2Z A (ot3) pt by a.straightforward but lengthy method i n the weak coupling, long time approximation. He expanded the evolution operator , U u ; s c , by the use of the well known Schwinger formula for an ordered exponential, into a power series in A • Substituting this series into the right hand side of (1.29), the formal expression for At(<*-;°<j) was systematically simplified through the use of the diagonal singularity condition (1.35) and asymptotic time integrations* Van Hove ended up with an infinite series with terms contributing as yy\ ~ o, \t 2}... ) • In the long time, weak coupling approximation, in which A—)0;"t-*<x> such that (A21) remains finite, only terms with h = 0 need be kept. Van Hove found that P± (<* 1,0(3) was given by iterated solution of the equation (1.36) ± d P, q.;-,) = £ r p t ( _ , j ( + f o r t > 0 J - |or t<o) with the i n i t i a l conditions Po (rf,;<rfj) = and where W/Grf,^ )^  <2 7T A2 J(f«rf1-f^ )fC°<.lvU,>|1. This latter quantity is the same as the transition probability per unit time as calculated by fi r s t order time-dependent perturbation theory. We should add that the solution of (1.36) is a valid approximation for P-t (<*,', 1*3) only when i t is used in YL A tol3 ) P+ ; o^ 3) and /U^ 3) is a smooth function of o(3 . In addition, Van Hove's analysis only gives us the various probabilities for times "t which are sufficiently large. (19) A very similar analysis concentrates on the so called P term ( 1 . 3 1 ) P(c< 3,t) = I C C o O l * P t(o(,;oC 3) where the C(oO specify the initial quantum state of the system. In using the previous method of finding Pt(<*M°<3), Van Hove must assume that the initial probability amplitudes C ( ° 0 are appropriately slowly varying. We refer to Van Hove's papers ( 1 9 5 5 , 1 9 5 8 ) for the precise meaning of what is meant by saying the C(d,) (or, the eigenvalues A(«(,) ) are slowly varying when consi-dered as functions of the unperturbed energy £0(i . Van Hove has shown ( 1 9 5 5 ) by direct calculation that for slowly varying C(o<.)ii,the I term d - 3 2 ) I(c(3,-t) = Z' I t U , ^ « 3 ) cU)CM is negligible when compared to the P term. This is, as in the whole dis-cussion so far, for the long time, weak coupling approximation. As a result, the occupation probability ^(o^)of the system being in the state l ^ a t time t is ( 1 . 3 7 ) P t ( * 3 ) = P ( « * 3,-0 and satisfies the equation ( 1 . 3 8 ) ± <LEg&= Z WCo^foCoO-P-tCoO} with the initial condition p 0 ( o < 3 ) = I C(o( 3)| 2 . The mathematical properties of the linear first order time differential equations given in ( 1 . 3 6 ) and ( 1 . 3 8 ) are well known since a similar equation describes the evolution of Markovian processes. It may be mentioned that the so called lowest order master equation in ( 1 . 3 8 ) for a "fine-grained" occupation probability was first derived by W. Pauli in 1 9 2 8 . He was forced to make the so called "random phase assumption" repeatedly in his derivation. (20) To avoid t h i s u n s a t i s f a c t o r y assumption, Van Hove iritrbduced a "coarse-grained" t r a n s i t i o n p r o b a b i l i t y 22 i £ i where the denominator i s the number of <x -values i n a small neighbourhood Ao( around o<3 . I f we use t h i s coarse-grained t r a n s i t i o n p r o b a b i l i t y , we can apply Van Hove's mathematical method,based upon the use of the diagonal s i n g u l a r i t y c o n d i t i o n ? t o the evaluation of i t . This "coarse-grained" t r a n s i t i o n p r o b a b i l i t y s a t i s f i e s formally the same equation as the " f i n e -grained" t r a n s i t i o n p r o b a b i l i t y P+Wi,'0^) . I n contrast to Pt fai,'*^) , Pt (<* i '> °* 3) » as c a l c u l a t e d by Van Hove,, has a meaning by i t s e l f since a summation over has already been made. We may s i m i l a r l y introduce a "coarse-grained" P term (1.1*0) P(o<3,t) H H P U 3 y t ) <<*3'< *3+4* 2 T 12 1 2 a Using these "coarse-grained" q u a n t i t i e s , we may discuss the d i s s i p a t i v e p roperties of the system independently of the i n i t i a l state | <J>„> or the eigenvalues of the diagonal operator A* As to the question of when I t i s possible to neglect the I term, i t i s quite obvious that t h i s i s not p o s s i b l e f o r a r b i t r a r y i n i t i a l s t a t e s . I f so, then a l l systems ( f o r which the weak coupling, long time approximation i s v a l i d ) would evolve according to the i r r e v e r s i b l e master equation given (21) above and I t i s easy to f i n d exceptions to t h i s . For example, l e t the i n i t i a l s t ate be an exact eigenstate of the t o t a l Hamiltonian of the system. As with some other topics discussed above, only a b r i e f o u t l i n e o f Van Hove's (1957) penetrating remarks on t h i s subject w i l l be given. I f we consider an ensemble and make the so c a l l e d random phase assumption (R.P.A.) i n i t i a l l y , then the I term w i l l vanish. The repeated use of R.P.A., as i n P a u l i ' s d e r i v a t i o n , implies that the system i s i n equilibrium at a l l times and consequently i s untenable. This r e s u l t follows from the symmetry of the fundamental equations of motion under time r e v e r s a l . In a l a t e r paper (1957)» Van Hove succeeded i n d e r i v i n g a general-i z e d non-Markovian master equation describing the evolution of c e r t a i n quantum mechanical many-particle systems t o any order i n A • No b r i e f note can describe the highly s o p h i s t i c a t e d mathematical technique (resolvent operator formalism) used i n a r r i v i n g a t t h i s generalized equation of d i s s i p a t i v e motion. The mathematical i n v e s t i g a t i o n of the properties and consequences of t h i s complicated equation i s no easy task e i t h e r . Other than showing that i t describes the approach to equilibrium, few e x p l i c i t a p p l i c a t i o n s have been made using i t . For the weak coupling case, i t reduces to the previous lowest order master equation and i n a recent paper (1961) with Verboven, Van Hove solved t h i s generalized master equation f o r extremely crude models of electron-impurity and electron-phonon s c a t t e r i n g i n a metal. Although much more work on s p e c i f i c models i s needed, there i s no doubt that Van Hove's work i s the f i r s t r e a l l y s u c c e s s f u l i n v e s t i g a t i o n that deals with the d e t a i l e d dynamics of systems with an enormous number o f degrees of freedom. I t shows how the l a t t e r play a c e n t r a l r o l e i n the r e l a x a t i o n to equilibrium and enables us to describe t h i s i r r e v e r s i b l e behaviour i n some d e t a i l . (22) Prigogine and his co-workers have also developed their own approach to the rigourous study of irreversible processes (in clas s i c a l and quantal systems) i n a series of papers i n "Physica" and other journals i n the last few years. Although different i n method and detail, the main aim i s the same as Van Hove's, that i s , to obtain the equations which describe the i r r e v e r s i -ble behaviour of macroscopic systems from f i r s t principles. In Prigogine's case, this again involves the solution of a N-body problem, not an exact one, but an asymptotic solution for a large number of degrees of freedom and long times compared to characteristic microscopic times. He uses a complex, diagram technique i n his analysis. We may note that Prigogine's group has mainly worked i n the "Schrodlnger picture" and, consequently, attempts to describe the dissipative behavior of a system by means of the density matrix. In principle, i f we could find the solutions of their complicated equations, we could use these results to find the expectation or ensemble average values of any operator. In contrast, Van Hove's work uses the "Heisenberg picture" and never loses sight of the particular dynamical observable he has i n mind. Of course, Van Hove's analysis only works for diagonal operators i n the \ <*•> -representation and therefore i s not as general as the work of Prigogine. In a recent paper (19$9) Van Hove has generalized his analysis to a class of operators which are "almost" diagonal, and states that this class includes a l l the physically important operators* In a recent paper (I960), Prigogine and his group have stated that "non equilibrium s t a t i s t i c a l mechanics has been now brought to a stage similar to that of equilibrium s t a t i s t i c a l mechanics." This i s certainly over confident and many authorities might disagree, but no one can doubt that considerable progress has been made i n this f i e l d by Prigogine's group, Van Hove, and others• Much remains on the computational side, of course* CHAPTER 2 : THE PROBLEM OF TRANSPORT PROCESSES IN OPEN SYSTEMS AND THE STANDARD "BOLTZMANN EQUATION" SOLUTION.' SECTION 2.1: TRANSPORT PHENOMENA IN OPEN SYSTEMS. Throughout chapter 1, the many-particle system was assumed to be a perfectly isolated system, and described by some unique Hamiltonian vH .As a matter of fundamental physical principle, no system can be rigourously isolated from i t s surroundings. Consequently no system has a rigourously defin-ed Hamiltonian. This f a i r l y subtle point w i l l be largely disregarded in the rest of this thesis, although this weak interaction with the rest of the uni-verse must enter i n any complete theoretical analysis of i r r e v e r s i b i l i t y . From physical experience, as we have mentioned, most systems in a non equilibrium state approaches irreversibly towards thermodynamic equilibrium after some time interval. In the last decade, a detailed description of this process, starting from fundamental principles, has been attempted with consi-derable success. Of course, once the system i s in equilibrium for some time, there i s always the poss i b i l i t y of spontaneous fluctuations resulting in the system returning to same new non equilibrium state. The.equations derived by Van Hove and Prigogine t e l l us nothing about the time or frequency of these equilibrium fluctuations. Now transport phenomena proper arise when.an isolated system is placed under the influence.of external "forces" or "constraints". This prevents the system from reaching equilibrium and various transport or "flow" processes take place in the system. Any dynamical observable JB( the tiJ,de denotes a vector) which i s defined in terms of the microscopic variables of an isolated system corres-ponds to a "current" i f the following c r i t e r i a . hold, (24) (2.1) <|>rQ*T/t(pEQ| ) = 0 (2.2) <|>{ = T/t,(p<tV§ ) * O i f p ( - t ) * pes where B - ( B , B, , B , ) and pit) i s the density matrix of the isolated system., as this system approaches equilibrium. Of course i f the system start-ed from a non equilibrium state quite far from equilibrium, < B > t would i n i t i a l l y be quite large, but i t would decay to zero according to the equations of motion discussed in chapter 1 . In the presence of external "constraints", the density matrix of the system must incorporate the effects of these distur-bances and consequently, so must < B>.. The theoretical problem before us i s to give a rigourous quantum s t a t i s t i c a l formulation of the effects of these external "disturbances". We might attempt to include the "driving systems" producing the external "disturbances" and the many-particle system of interest in one com-A posite, closed system. We assume that the total Hamiltonian K Tof the compo-site "supersystem" can be s p l i t as follows ( 2 . 3 £ T - + £ D . S .+ V A A where "ft i s the t o t a l Hamiltonian of the isolated system of interest, "K o.s. A i s the total Hamiltonian describing the "driving systems" and V represents the interaction between the system of interest and the "driving systems". We shall now reformulate the problem. We consider an open system consisting of the system of interest acted upon by the external disturbances A A through the perturbation V . Actually the Hamiltonian W D s used in describ-ing the "driving systems" i s a l i t t l e ambiguous. In transport theory, the "sources" are usually assumed to be " i n f i n i t e l y large" (batteries produce a constant voltage, heat reservoirs stay at a constant temperature, etc.). Thus the "sources" have an open character and cannot be expressed in the Hamiltonian formalism rigourously. If the "sources" are large systems (25) compared to the system of interest, the difference between a closed com-posite system and a quasi-closed composite system i s negligible and we shall neglect I t . As emphasised, what i s important i s to be able to f i n d an appro-> A priate quantum mechanical operator expressing V . For some "driving systems", this Is a well defined operator, while i n other cases, an unambiguous expres-sion for V i s not so obvious. This distinction w i l l become basic i n the follow-ing chapters of this thesis. From an experimental or phenomenological point of view, in many cases the behavior of Isolated systems under the influence of external "disturbances" can be simply described by means of a few parameters known as transport c o e f f i -cients. Up to now everything has been f a i r l y general. For concreteness, we shall consider a metal specimen under an applied electric f i e l d and a "tempera-ture gradient". This thesis w i l l concentrate on these two external "disturbances" and w i l l review recent attempts to f i n d theoretical expressions for the trans-port coefficients experimentally connected with them (in particular, e l e c t r i c -a l and thermal conductivity). In the remaining paragraphs of this section, we shall outline the experimental or phenomenological description of these trans-port processes. Assume that we have a homogeneous metal specimen in the shape of a cylinder of length L. f completely insulated from i t s surroundings except for the ends. Apply a sufficiently small electric f i e l d E x along the axis (X-direction) of the cylinder and assume that the temperature i s constant throughout. An electric "current" J x w i l l arise along the length of the cylind-er, and i s found to be proportional to £ * . (»3k) J x = c r x x E x where <TXx * s the e l e c t r i c a l conductivity, the transport coefficient describ-ing this electric current flow. The equation (2.4) holds once the system has (26) reached a steady state with a constant current flow. When the external f i e l d i s f i r s t switched on, the resulting transport situation i s more complex due to "transients" but these are soon damped out. We can immediately general-ize ( 2 . 4 ) to a more general situation in three dimensions, ( 2 . 5 ) J » CT E or J=E(T v.£u K / i * 1* 2 , 3 or x, y, z) with CT the e l e c t r i c a l conductivity tensor (nine components). Macroscopically, the case of a "thermal gradient" i s similar. Assume that the two ends of the cylinder are i n contact with heat reservoirs (temperature baths) at different constant temperatures, the difference AT being sufficiently small. An energy "current" Q w i l l flow along the cylinder. We further make the restriction that the electric current JK i s zero. Since the energy current w i l l indirectly produce a charge flow, we must have effect-ively, in addition to the temperature difference, an electric f i e l d E xwhieh cancels out any electric current associated with the energy current. Mathematic-a l l y the experimental situation involved in thermoelectric processes can be expressed as follows ( 2 . 6 ) j x = cr x x E x • B ( 1 *2 L ( 2 . 7 ) Q x » B 2 1 E X + B I 2 . A T L where B , 2 ; B 1 ( and B 2 1 are experimental parameters. Now i f we make the assumption that JM s O i n ( 2 . 6 ) , express E x i n ( 2 . 6 ) In terms of A T , and insert the result into ( 2 „ 7 ) , we have (2:8) Q = {Bu(-Bn) + B I 2 | 4 I - - kl A T where k"Xx i s known as the thermal conductivity. Generalizing ( 2 . 8 ) In an analogous fashion to ( 2 . 4 ) , a more general expression i s ( 2 . 9 ) Q = - K ^ T ( A ) o r Q y = - £ C27) where kl i s the thermal conductivity tensor with J= O • I f the energy "current" Q i s constant i n a steady, state and K. i s independent of the s p a t i a l coordinates. T(A) must be of the form ( 2 . 1 0 ) TlA) - T c + a -A. with T c some constant temperature and & = (Qi.>a*.,a3) some constant vector. Since the temperature gradient must be small, we must have i n a d d i t i o n ( 2 . 1 1 ) |fl£Z£|<3U Tc 1 For the macroscopic function TCA) to have any p h y s i c a l meaning as a tempera-ture v a r i a t i o n i n the specimen, we must have some s o r t of " l o c a l equilibrium" i n regions o f the system macroscopically small but microscopically l a r g e . I n t h i s case, the concept of a " l o c a l temperature" may be introduced. As i s usual with t h i s type of macroscopically varying f u n c t i o n of the coordinates, the s i z e of the region i n which the function i s approximately constant i s l e f t f a i r l y vague. We s h a l l r e turn t o t h i s " l o c a l temperature" concept l a t e r i n chapter U. We emphasize that the above analysis i s completely phenomenological* The term "current" i s so f a r only a name o f a' quantity appearing i n the equa-ti o n s which experimentally describe the various i r r e v e r s i b l e processes discussed. As a l a s t remark, even though the present discussion has no pretense of any t h e o r e t i c a l background, the fundamental d i f f e r e n c e between the two external "forces" i s noticeable. The e l e c t r i c f i e l d i s p r e c i s e l y defined i n terms of electromagnetic theory and has a d e f i n i t e c a l c u l a b l e e f f e c t on a charge c a r r i e r such as an e l e c t r o n . The so c a l l e d temperature gradient i s e a s i l y defined macro-s c o p i c a l l y but on an atonic s c a l e , i t i s quite d i f f i c u l t to give i t any pre c i s e meaning, l e t alone c a l c u l a t e i t s e f f e c t s on the atomic constituents of a metal* Before discussing the method of using a transport or Boltzmann equation i n the transport processes mentioned above, a b r i e f mention may be made of the semi-macroscopic theory of i r r e v e r s i b l e processes developed by (28) Onsager (1931). A very clear discussion of this type of theoretical analysis i s given i n several monographs (for example, see S.-R. de Groot (1952) ). In this theory, other than the standard procedures of s t a t i s t i c a l mechanics, two assumptions are used. The f i r s t i s the fundamental microscopic r e v e r s i b i l i t y of the dynamical equations of motion ( the laws are invariant under a change of the time variable "t-*-•£«). Because of this use of some microscopic prop-erties of the system, irreversible thermodynamics i s not entirely phenomenolo-gical i n content. The second important assumption i s really a hypothesis, as emphasiz-ed by Casimir (19^5). On the average, the decay of a fluctuation (which has caused the deviation from an equilibrium state in an "aged" system) i s assum-ed to follow the ordinary phenomenological linear laws governing irreversible processes. By introducing appropriately defined "forces" and "fluxes", we can express the "fluxes" as a linear combination of the "forces". The constant co-efficients play the role of transport coefficients and are known as "kinetic coefficients". Onsager*s theorem i s basically a criterion of when same of these kinetic coefficients are equal to one another. It should be emphasized that no information i s given as to the evaluation of these kinetic c o e f f i -cients i n terms of the microscopic properties of the system. SECTION 2 . 2 : OUTLINE AND CRITICISM OF THE STANDARD METHOD OF FINDING  ELECTRICAL AND THERMAL CONDUCTIVITY. ~ A well known "recipe", using Boltzmann type equations for jU.-space distribution functions, i s used almost exclusively i n any discussion of trans-port phenomena in solids. Detailed accounts may be found i n Wilson. (1953) or, for a sl i g h t l y more modern version, Ziman (i960). Although the theoretical structure of this "recipe" was never too rigourous, i t did and does provide a reasonable f i r s t approximation to the solution of some f a i r l y complex physical (29) problems in the theory of metals. The basis of this method l i e s i n the use of -space distribution functions. In the theory of metals, i t i s almost always assumed that i t i s possible to define a electron distribution (^vk) which gives the probable number of electrons (conduction) i n a single particle state with wave vector k, and a phonon distribution n ( ^ ) which gives the probable number of phonons in a single particle state with wave vector (jj>. The interactions between the free electrons and phonons, the interactions with impurities, and the presence of the external electric and thermal gradients are a l l considered as causing "transitions" between the unperturbed states labelled by Jk and <j . That this procedure enables us to effectively describe a whole system of interacting particles i s not at a l l obvious u n t i l a f u l l scale many-body treatment, taking into account the collective behavior of the particles, has been undertaken. We must show that the A and ^ single particle states define the appropriate "good" quantum numbers. Recent work by Bohm and Pines on the plasma theory of metals i s an example of what we have in mind. By guessing appropriate collective coor-dinates, the system's to t a l Hamiltonian may be s p l i t into a part depending on the collective coordinates only and a part depending on the particle or quasi-particle coordinates only, the interaction being negligible. In the case of metals, the quasi-particles are fermions that behave as nearly free fermions. That i s , they axe electrons "surrounded by a charge cloud" which produces a screened Coulomb f i e l d . This result explains the success of the free electron model. A rigourous development as sketched above i s especially important in. that i t may lead to the occurrence of new carriers of energy. In the standard theory, i t i s simply assumed that only the electrons are charge carriers, while only the phonons and electrons are energy carriers. In terms of the distribution functions ^ (k) andn( <fy), the electric current density '^ and the energy current density <j> are assumed to be given by the "natural" expressions. (50) (2.12) j =-/e v f(A)d3Jk, ;(2.i3) j - j£CA)v^(A)^.^J^(^)cn(<j-)^^ where e is the electronic charge, V and E( ) are the electron's velocity and energy in state k, and C and U/ (<fy) are phonon's velocity and energy in state Even i f we could f i n d the values of £ ( X ) and rj ( <^), i t must be remembered that these are the analogues of the diagonal elements of the reduc-ed density matrix, and that the information contained in the off-diagonal matrix elements i s completely ignored. The next problem is to set up gain-loss Boltzmann-type equations for both electron and phonon distribution functions, using the transition probabilities for the various processes. The manner in which this i s usually done i s highly intuitive and really can only be called a "recipe" which works, at least to a f i r s t approximation. When there exists electric and temperature gradients, and the system is assumed to be in a steady state, the transport equation for each distribution function may be formally given as \ ax/ -fiei^ terms * o w scattering terms The f i e l d terms contain the external gradients while the scattering terms describe the complicated interaction mechanisms between the atomic constituents of the metal. Assuming, as i s usually done, that the right hand side of (2.14) is given by a Boltzmann gain-loss term ( generalized to take into account the st a t i s t i c s obeyed by the particles and quasi-particles), we end up with a coupl-ed set of extremely complicated integro-differential equations for £(Jt) and n ( c j ) . Theoretically, the only remaining problem i s the mathematical one of solving these equations, but before any usable solution may be obtained, many further physical approximations are necessary. Since our goal Is to evaluate jr and f u s i n g the previous definitions, it some obvious simplifications are perfectly Justified. The linearization of the c o l l i s i o n term i s acceptable since we are only interested in non equilibrium states which are not too far from the equilibrium state. If the solution i s expressed as a series in the external gradients, we need only keep the linear term. Other assumptions are not easily j u s t i f i e d , except on the ground that otherwise the problem is almost hopelessly d i f f i c u l t . The most important example of this i s the almost universal use of a relaxation time in simplify-ing the scattering term. S t r i c t l y speaking, this use of a universal relaxa- , tion time Is j u s t i f i e d only in the elastic and isotropic scattering case. A recent a r t i c l e by Dresden (l Q6l) gives a detailed account of the assumptions, methods, d i f f i c u l t i e s and shortcomings of the transport equa-tion approach very b r i e f l y outlined above. We should l i k e to emphasize two points. . F i r s t , in recent years, attempts to derive transport equations from the general principles of quantum s t a t i s t i c a l theory have met with considera-ble success. We mention the work of Kohn and Luttinger (1957) in which the usual Boltzmann equation was f i r s t successfully derived for a simple model of a metal. The work of Van Hove, Prigogine and others mentioned i n chapter 1 «> / / i s of interest. Although these discussions are often i n I -space, i n contrast * // to the^C-space transport equations mentioned above, by appropriate summations and integrations over the degrees of freedom connected with the rest of the system, a Boltzmann-type equation for a single particle may be obtained. Van Hove, in particular, has carried this out in the case of an electron-phonon system (1958) and the case of an electron-ion system (I960). I t should be emphasized that when we speak of a distribution function for the number of particles in a certain single particle state, this i s the same as a occupa-tion probability for a single particle, except for a normalization factor. The second point i s that even i f a sounder theoretical justification could be given, the whole transport equation procedure is unsatisfactory as a (32) mean of finding transport coefficients. Except for the simplest examples, any use of this method ends up i n a hopeless tangle of physical and mathe-matical approximations. There i s no consistent manner by which we can compare various approximations or extend the theory to the case of strongly interacting particles. The study of metals (or more generally, solids) i a a complicated problem at the best of i t , and the introduction of transport equations and distribution functions, as an intermediate step i n finding transport coefficients, simply adds to the confusion. We shall see that another method i s possible which enables us to give formally correct N-particle expressions for certain transport coefficients by a direct quantum-statistical arguement. The problem of evaluation of these Kubo-type formulas i s of course d i f f i c u l t , but they have the same advantage as the partition function has ia equilibrium s t a t i s t i c a l mechanics in that consistent procedures of approxi-mation may be found. I t may be mentioned that the physical insight into irreversible processes and transport phenomena gained by the standard methods i s almost n i l . Thus even though an electric and a temperature gradient are really entirely different from a physical point of view, almost no distinction i s made i n the way they are treated. The method to be described i n the next three chapters removes seme of this hypocrisy. (33) CHAPTER 3. KUBO'S DERIVATION OF FORMAL EXPRESSIONS FOR THE TRANSPORT COEFFICIENTS DESCRIBING MECHANICAL DISTURBANCES, In this chapter, we shall deal with transport situations where the A isolated system has a well defined time-independent Hamiltonian K including a l l interactions between the particles and quasi-particles composing the system and where the interaction operator v , describing the effect of the 'Striving systems" on the system of interest, i s same known quantum mechanical operator. We shall , for generality, assume that V may have some ex p l i c i t time A dependence and therefore denote i t as V^. Any external disturbance which can be A expressed unambiguously by means of a perturbation V t w i l l be referred to as a "mechanical disturbance", a terminology introduced by Kubo (195Tb). The example which we have in mind i s an electric f i e l d E ( "t). For "external A disturbances" such as a temperature gradient, where the form of V t i s unknown or undefinable, Kubo uses the term "thermal disturbance". We shall discuss these latter "disturbances" i n chapter 4. A The effect of V t on the density matrix of the isolated system w i l l be now investigated. We assume that the mechanical disturbance i s turned on at time "t = 10, and that before this, the system of particles was i n equi-librium. Therefore, we must solve (3 . D - - i C K ^ V t f o r t * t 0 with the i n i t i a l condition (3.2) p^m p E Q ? t < ± 0 The equation (3.2) can be transformed into the following equivalent integral equation, as can be v e r i f i e d by direct differentiation, ( 3 . 3 ) p t - P ^ - i f t e - i w - t ' " i C v t . , p t Q e + ; " - v > " c t t ' t o O ) We can solve ( 3 . 3 ) formally by an iteration procedure but since we are interested i n weak external f i e l d s , the following linear approximation w i l l be. sufficient^ t o where we shall denote the second term on the right hand side by . The equation ( 3 . 4 ) gives P^^o the f i r s t order in VV, and i t i s sufficient for calculating f i r s t order transport coefficients such as el e c t r i c a l conductivity, as we shall see. If we were interested in second or higher order transport coefficients (such as the Joule heat), we should have to use the neglected higher order terms in ( 3 . 3 ) . More information on the latte r subject may be found i n a paper by Bernard and Callen ( 1 9 5 9 ) . Let us denote by B some quantum mechanical operator describing A some sort of "current". We have given the conditions which 8 must satisfy in ( 2 . 1 ) and ( 2 . 2 ) . As in our discussions of isolated systems, we assume that the ensemble average ( 3 . 5 ) < §>T -T/v(&B ) i s equal to the actual observed value i f the system of interest i s large enough. We shall omit any reference to the effect of the observations on the quantum mechanical system, although this i s a delicate point worthy of atten-tion. Using ( 3 . 4 ) , ( 3 . 5 ) and ( 2 . 1 ) , we "find ( 3 . 6 ) <B>t =~MApt B ) . Kubo's response method ( 1 9 5 7 a ) can be summarized as follows. Assuming that both B and are precisely defined operators, we evaluate A p^ . and calculate the trace i n the right hand side of ( 3 . 6 ) . The resulting expression i s then assumed to give the observable value of the macroscopic current corresponding to the operator B . We then try to separate the external f i e l d (incorporated in Ap«) from this expression, leaving a factor which w i l l be Identified as a transport (35) coefficient. Although i t is not the most general possibility, we are usually A concerned with mechanical disturbances where the perturbing operator can be expressed as a scalar product of the form (3-7) = — M • • (- sign chosen for convenience) where F Ct) is a c-number and A is a, Hermitian operator, both of them being vectors in the general case. . FCt) may be an explicit function of the time and is related to the external disturbance while A is not explicitly dependent on the time and describes some property of the system of interact-ing particles. What A and F(t) are in any particular example" is the result of a separate investigation. The well known answers for an external electric f ie ld denoted by E Ct) are (3.8) ^ = 2 a - A ; (3.9) p(t)= g(-t) where and are the charge and position operator. , respectively, of the th ^ charge carrier, and the summation is over the whole system of N charge carriers. We may write Aptin (-3.4), using (3.7), as (3.10) & e ± = L(* e'^'^C&Petle^-^ott'.FU1) J ~to . ~ -~ Using (3.6) and (3.10), the value of the current, produced by the external f ie ld , at time "t is given by a n ) < § > t = T/ i [ i^e - i ( t ^' 1 *rA,p„]e* £ l t ' 1 : ' , *-F ( . t ' ) e t t 'g} = L P<tf£tt , ) -TA.{e - l « - * , > * c a , P l J e * « * - t - M 5 g } r l C F(f)cU'- TA,{ [AjPe*] B C t - f ) } where the cyclic property of the trace was used and B(i) is in the Heisenberg representation } (3.i5) B W ^ e ' ^ S e ^ ' with s ^ ) = | be) The time dependence of F ("t) should not be confused v i t h the Heisenberg representation notation used for operators. Particular attention should be paid to the dot • appearing in these and future equations. I t denotes the scalar product of the tiro vectors which are separated by the dot. The response current < § > t I n the last equation of (3.11) i s s t i l l not i n the desired form. As a further step, we introduce a new integration variable X = t -1 ' • By means of this transformation, we f i n d (3.13) <!>.= I {x"Xodx fU-Z) . T / v ( [ A , p ^ ] B ( T ) } . In order to simplify £ A; pt^J * we shall make a short digression. A l i t t l e b i t of manipulation shows that Now the Heisenberg equation of motion i s (3.15) A = t [ K , A ] - ct 6 where A i s not ex p l i c i t l y dependent on the time t . Remembering that p E < a i s the canonical distribution (3.16) p e a = ^ . e - ^ A with z ~ T a ( e " p M ) . we may combine (3.11*), (3»15) and (3.l6) and arrive at the useful relation The la s t result Is known as Kubo's Identity, and i s a very useful formula. Inserting (3.18) into (3*13), we have '0 (3.19) <B> t = r f TA(PECI ^ C - i ^ g U ) ) (37) where in the last equation, the cyclic property of the trace and the faett that p E ( 5 commutes with G were used. .Summing up the results of the previous paragraph, the response of a dynaiaical observable B , due to a mechanical force which introduces a perturbation V t = - A • F(-fc) , is given by I (3.20) < B > t = J*"Sr F ( t - r ) . <C ( o ) B C C + i A ) ^ where T / L - { P B Q = < - - - > e q and / j = iC^jA] S C .This result lias been derived by several authors, the f irst being Kubo, by the same method as used above. Following Kubo, we introduce the response or after-effect function f c g ( t ) , so named since i t gives, the response at a time X after the application of a unit pulse of a mechanical disturbance. The response function (a tensor) is defined as ( 5 * 2 1 ) T^eCt) = \[d\ <cCo)gCr+i'A)>^ In terms of the response function, the response current is (3.22) <B\= (^'dr F(t-t)4c BCt) . By a change of integration variable, T —> t'-(t-t) , we have in place of (3.22), (3.23) < B > t - r t c a , f ( t ' ) . $ c e ( t - t ' ) in which the physical meaning of the response function as the response to a unit pulse is more evident. If F("t7 — 6(t'-t,)U. , where to^ t, <t and U. is a unit vector, then (3.24) < i>t = r "W ^ ( t H ) ^ - ^ C B c-t-f) Where ( - f r - t , ) is the time elapsed since .the pulse xra.s applied. Making use of the assumption that the response current is linear in the external f ield, the most general equation for the response current (38) at time t" Is ( 3 . 2 5 ) <§> «= r ° ° c ( r F a - T ) . f C ( Y ) •where K ( T ) (a tensor) is some function of the time which depends on the properties of the system. We have written ( 3 . 2 5 ) in a form compatibles with the "principle of causality" which, in this case, states that <B>+ can only depend on F ( t ) where -00<T< "t . On comparing ( 3 . 2 5 ) and the equation ( 3 . 2 2 ) previously derived using Kubo's formalism, we see that the only differ-ence lies in that Kubo's method enables us to find an explicit formula for K ( r ) , namely that given by ( 3 . 2 1 ) . If we assume that p (-t) = F c e " i w t that i s , the external f i e l d has a harmonic time dependence, we can rewrite ( 3 . 2 5 ) as ( 3 . 2 6 ) < | > t = £ ( f ) . { n c ( X K C T ) e i w r ] = F ( t ) • OC(UJ) where (X (LU ) is known as the generalized •susceptibility tensor. The well known fluctuation-dissipation theorem uses C<(u)) to connect the equilibrium fluctuations of physical quantities and the dlssipative properties of the system when external disturbances act on i t (Callen and Welton (1951) )• A glance at ( 3 . 2 2 ) w i l l show that the response current < 8 ) Is ~ t dependent on "to, the time at which the external f i e l d was turned on. We shall now show how to get rid of this i n i t i a l time dependence, following Van Hove'3 discussion ( i 9 6 0 ) . Consider <^  C(0) §\tjy , an expression usually known as a "time-relaxed correlation function". It gives a measure of the "correlation" between one operator C at time Z = O and another operator B at time T « T , the value of the latter being governed by the "natural" motion of the system of particles independently of the external field. Now i t w i l l be assumed that, due to the dlssipative coupling between the parti-cles of the system, (3.27) Aw, < C(0) 0 ( T ) > _ = Q t--»oo ~ (39) The validity of (3.27) can be checked in certain cases by means of calcula-tions very similar to those used by .Van Hove in his derivation of the master equation. The problem of evaluating correlation functions will be discussed in chapter 5. The limiting relation given in (3.27) means that < c'(o) B'(x)ye(k is negligible whenever t" » } where TN is appropriate relaxation time. Now we shall take B '= A , where A - LCH.AJS § , and c'~ c : we may rewrite (3*27) as (3.27) Mm < C(o)A(?)>eei = 0 a result, we conclude that (3.28) £<*n, ^ K C ( o ) 0 ^ f U ) > f f t = O T-*GO since we can easily show that (3.29) r ' c fx < c ( o > f i C r + t A » e a « t < C ( o > A ( T ) > ^ - L < A ( t ) C ( o ) > ^ -/ O -Ay With (3*28) in mind and assuming that (3.30) ( t - to )»Tn we can effectively replace the upper limit ( t - t 0 ) in (3.22) by oo . Thus <f> tis independent of the time t 0 at which the external disturbance was switched on, as long as the time elapsed Is large enough so that (3.30) holds, and i s given by (3.31) <0> t= i*F(t'Z)' §ceCt) dt Thee physical meaning of this result is clear, for we would intuitively expect that, even though the behaviour of the perturbed system might be complicated and highly dependent on the i n i t i a l conditions when the field was fi r s t applied, as time passes, these "transients" would be. "damped" out. We shall now specialize the problem to the case of an electric field Bit) • More particularly, we assume the following time dependence (3.32) Bit)- Ec e~iu,i: where the subscript c denotes a time independent quantity. The electrical current operator J" is given by (3.33) J~£e±Pi 1 /YY\ L(ao) where €y, pi and are the charge, momentum operator and mass, respectively of the charge carrier, and the summation i s made over the whole system. For simplicity, we assume that we are dealing with electrons i n a metal, and therefore (3*33) simplifies to (3-3U) J" - ® £1 p-The dipole moment operator X - A was given i n (3.8). In order to find C = A , we take the time derivative of (3.35) -A = S eAi ' t 2. p: = J and consequently c - J in this special case. Substituting (3»35)» (3«3lO and (3.32) into (3.31) and (3.28), we have (3.36) <J\^?dt^ei"L'-*)-$,r(T) with (3.37) - tfd\ < £<•> J ( r + t > ) > « Rewriting (3.36), we take the electric f i e l d Fit) outside the integral (3.38) C J>t= F ( t ) . { > c i w t t„W . Introducing the complex conductivity tensor VU)(UJ) , we may rewrite (3»38) as (3.39) < J > t - E(±) • cr 0 ,(uJ) or i n component form (3.U0) < JMy = 21 0~*Uw)gyU;) - 1, 2, 3 or x, y, a) where J = ( J , , J M j,) . The complex conductivity tensor i s e x p l i c i t l y defined as (3.UD CT" ,(w)= J 7 V T e ' u , T $,.r(t) or i n component form (3.U2) CT^>^)=;*^te'w rX^(X < J , f o ) £ < T * f A ) > e f t It should be noticed that the indices ^ u, V are permuted i n going from the l e f t hand side to the right hand side of (3<>U2). A well known equation in electromagnetism gives the total "miGroscopic current" i n terms of the obmic conductivity tensor ° 0~O)(w) and the medium (41) dielectric tensor A £ ( u> ), that i s , (3.^ 3) <5> - J ( t ) • <> P ( - f r ) where the macroscopic current j (f ) i s defined by (3.44) j ( t ) . ^ Ct).'cr w(^) and the polarization vector P ( -fc ) i s defined by OM) P C t ) - E ( t ) * AE( co) By combining (5.43) and (3.39), we have, assuming £ ( t ) = £ c e~ i W* (3.46) CT0,(u,)= [ °crc,)(w). - i w A £ ( o d ) ] where, in particular, the ohmic conductivity i s given by (3.47) C o a . u i T f < i y Co)^(.-r+iX )> p c tX . . Generalizing once again, we shall summarize the results of Kubo's A analysis. Assume that we have a "current" operator B rigourously defined and a perturbation, due to a mechanical disturbance, with V t = - A * £c e l W Then the current "response" < 8 > ^ linear i n the external f i e l d is given by (3.48) < B> t = F( -t) • L A S with the tensor given by (3.^ 9) L* 8 - $™dz eiu,X ^d\ Tn{pe* A(0) BU+<X)} or i n component form (3.50) < §J>t = ^ Fu(~t) with the tensor elements given by (3.5D L i * - "i'di^tfdxT^lpci.to B^CT-JAI] . The formally exact expressions given i n (3.**2), (3.47) and (3.51) are known as Kubo-type formulas for the components of f i r s t order transport coefficients. They are admittedly quite abstract i n appearance. As we mentioned i n the introduction, the main result of K U D O ' S analysis,. 4S that the study of linear transport processes involving open systems (with mechanical disturban-ces) can be reduced to the study of non equilibrium processes in isolated sys-tems. The link i s the response function <^ c 8( t ), which i s an integral over the time-relaxed correlation function, (3.52) < c(o)B(z- i^xf) = T/uj e £<°>e. gcoe J-A where -is the Hamiltonian of the isolated system. As we shall see i n more detail i n chapter 5, the evaluation of the right hand side of (3.52) is very closely connected with the study of the dlssipative properties of an isolated system as discussed i n chapter 1. It w i l l be seen that one of the most f r u i t -f u l methods of calculating transport coefficients i s a combination of Kubo's general formalism as discussed in this chapter and the work of Van Hove on the master equations which describe the irreversible approach towards equilibrium. It may be noted that i n passing from the f i n i t e upper l i m i t ("t — ~tD) i n (3.22) to tiie effectively i n f i n i t e upper l i m i t i n (3.31) , the ju s t i f i c a t i o n given was that ^ c 8 ( ) i s negligible as long as f » tn. , where f e Is some appropriate "relaxation time" related to the dlssipative properties of the system. In doing this, we have followed Van Hove (I960); but, i n the literature, other methods of Justification have been used, A few remarks w i l l be made below on the relationship between these various procedures. Lax (1958) assumes that the system of interestj undergoing an external perturbation represented by V t , is not completely isolated but may weakly inter-A act with the rest of the "universe" in such a way that when Vt = O , the density matrix pt approaches the equilibrium density matrix p e G ). The weaker the interaction with the surroundings, the longer the "relaxation time" (denoted by Tfi) i s . To describe this state of a f f a i r s , Lax assumes p t satisfies the following equation of motion ^ y • • • (3.53) ijh^ + [ f e , - H + V * ] + j(p^-pBQ) _ Q Karplus and Schwinger (1948) and others have used this sort of equation previ-ously. Carrying through a similar analysis as was given i n the f i r s t part of this chapter for (3 .1 ) , we are led to the result •This relaxation time T R has, of course, a different physical meaning than the relaxation time t R introduced before. If T » T i , then # c e ( t ) e _ T / t c i s negligible. Consequently f i f t - t 0 » Te then the upper li m i t in (3.5*4 can be effectively replaced by OO .Lastly ve take the l i m i t with the result (3.55) < g > t = (°° F(-t-T) e'^dt or, i f we introduce a new parameter £ ± ' (3.56) <8>, . J*n r e - £ t E ( i - T r V $ c B ^ ) ^ r o o where (3.56) defines the limiting procedure used to evaluate the integral denot-ed by j> . I n contrast to our original method (following Van Hove) in deriving (3.31), Lax's analysis makes no use of the vanishing of § ra. it) as a result of the internal dlssipative forces. On the other hand, Lax i s forced to introduce i r r e v e r s i b i l i t y through a weak interaction with the "universe". This results i n the convergence factor Q appearing in the integrand of (3»55). However, introducing a convergence factor becomes redundant whenever the condi-tions (3.27) or (3.28) are valid. In addition, although Lax's procedure is bas-ed on a definite physical assumption, It has the appearance of a mathematical "tric k " , especially when the l i m i t £ -»o i s taken. Since (t - ~t0 )» t , equa-tion (3.52) i s only v a l i d for i n f i n i t e times, in contrast to (3.31) which-holds for (t - " t 0 ) > > te where VK i s some specific, * calculable relaxation time depending on the properties of the particular system in question. In contrast to the methods used by Van Hove and Lax, Kubo (1957a) and others use a procedure which depends on how the external mechanical disturb-ance was "turned on". Here we assume that at ' X - - 00 > the system was i n thermal, equilibrium and the external f i e l d i s introduced adiabatically, that Is, extremely slowly. This can be done by Introducing an extra time dependence factor 6 6 * i n the external f i e l d with £ always being a very small positive (kk) number. In the end, the l i m i t 6-»o i s taken. Again we find, after the usual calculations, (3.57) <|>- i U ( ; ' ^ e - " F ( t . t ) . e + ^ M l t ) where "t 0 = — CO .' The l a s t equation i s formally identical to that obtained by Lax i n (3.51). Kubo's result i s definitely dependent on how and when the f i e l d was switched on, in contrast to the procedure used by Lax. On the other hand, no physical assumption such as a weak interaction with the rest of the "universe" was necessary. As we have seen, neither Kubo nor Lax ex p l i c i t l y use any vanishing property of the response function; but both introduce, rather a r t i f i c i a l l y , a convergence factor into the integrand of the i n f i n i t e integral in (3.31), mak-ing the latte r integral convergent and well defined. By considering the long-time behavior of the response function, Van Hove succeeds i n giving a more direct j u s t i f i c a t i o n of (3.31) starting from (3.22). To verify the va l i d i t y of the important criterion (3.27), of course, involves some approximate calcula-tion of the correlation function. This i s not easy task i n general, but this i s no criticism since the whole of Kubo's formalism i s not too much help un-less we can evaluate the correlation function (or equivalently, the response function) i n some approximate manner. (45) CHAPTER 4 ATTEMPTS TO FHiD KUBO-TYPE FORMULAS FOR TRANSPORT COEFFICIENTS  CONNECTED WITH THERMAL DISTURBANCES (THERMAL COITOUCTIVITY) SECTION 4.1: INTRODUCTION AND CHOICE OF OPERATORS. In chapter 3, we have o u t l i n e d Kubo.'s method of f i n d i n g exact general expressions f o r c e r t a i n transport c o e f f i c i e n t s . In the f i r s t place, i t was assumed that the quantum mechanical current operator B could be un-ambiguously defined. The p a r t i c u l a r example we used was J , the e l e c t r i c current operator of a system of N electrons i n a metal undergoing various i n t e r a c t i o n s among themselves, with impurities, with the l a t t i c e , e t c . Following Kubo, we next c a l c u l a t e d the change ( l i n e a r i n the external f i e l d ) i n the density matrix describing the system, the change being due t o an exter-n a l mechanical disturbance. In the case of an e l e c t r i c f i e l d J=(t) , the perturbation energy i s V t = - X • E(t) f where X i s the e l e c t r i c d i pole moment operator of the system. We found the density matrix p± to the f i r s t order i n the external f i e l d , Now there i s a whole range of i r r e v e r s i b l e processes f o r which the above procedure cannot be e a s i l y c a r r i e d out, i f at a l l . There are two reasons f o r t h i s . F i r s t of a l l , the quantum mechanical operator corresponding to the current flow i n these transport processes i s not known as easily found. Secondly, A the external disturbance may be given phenomenologically, but the operator V t , which gives the i n t e r a c t i o n between the system of i n t e r e s t and the " d r i v i n g system", nay not be e a s i l y defined i n any precise microscopic manner. For the r e s t of t h i s chapter, \re s h a l l concentrate on i r r e v e r s i b l e transport processes i n which these d i f f i c u l t i e s are present. For concreteness, we s h a l l consider the s p e c i f i c example of thermoelectric phenomena. A f t e r making a few remarks i n t h i s section on the proper d e f i n i t i o n of an operator Q which would des-c r i b e the energy flow, we s h a l l then review, i n section 4.2 and 4.3, recent attempts to generalize Kubo's results to thermal disturbances . . In particular, we shall work out the details for a macros-copic temperature gradient and attempt to find a Kubo-type formula for the thermal conductivity K. It may be pointed out that the literature on this subject is distinctly unsatisfactory from a rigourous quantum statistical standpoint* The physical problem should be stated precisely, and emphasis be given to the difficulties which must be surmounted. Instead, much of the literature boils down to an intuitive Justification of results written in analogy to Kubo's formula for electrical conductivity given in chapter 3« In the literature, there is considerable variety in the choice of an observable corresponding to the energy current. If a satisfactory quantum statistical theory of heat conduction was available, the appropriate dynamic-a l observable would be given as a result of the theory. In the absence of this, we are forced to make a reasonable "guess". Suppose the total Hamiltonian can be written as the sum of the energy operators of free noninteracting excitations (particles or quasi-particles) and the velocity of each excitation is unambiguously defined. Then we may write the energy flow operator Q as (4.1) 0 = Eu Yn th where Em and V i , are the energy and the velocity operator of the L type of excitation in the state labelled by £(there being M kinds of excitations), and nlje is the occupation number of the state specified by t and X . The operator in (4.1) is a well defined quantum mechanical operator and comple-tely describes the energy flow in the system. In the absence of any inter-action between these excitations, though, the energy flow Q is a constant of the motion. Its expectation value depends entirely on the i n i t i a l conditions . As soon as we introduce the interactions between the excitations, the very concept of the energy carried by a particular excitation loses its precise meaning. In terms of an electron-phonon system, for example, we can speak about the total energy of the system but not of the. energy of the electrons or phonons separately. It Is the old question of which "part" of the interaction energy belongs to the electrons and which part to the phonons. What is often done is to consider separately the energy flow of the type of excitation which makes the dominant contribution to the transport of energy. In ordinary metals, for example, the conduction electrons, as carriers of energy, usually completely overshadow the other .'excitations. Therefore we may, to some approximation, only consider the energy flow-due to the electrons; the phonons, impurities, etc., enter.: in so far as" they effect the dynamical behaviour of the electrons. On the other hand, in an insulator, the heat conduction by phonons is dominant as long as the tempera-ture (and temperature gradient) is not too large. Even for semimetals, where the heat conduction by electrons and phonons is of comparable order, we may treat them separately to a fi r s t approximation. In this case, • the effects of the interactions with each other are taken in account through scattering processes and the total energy flow is simply taken as the sum of the electron and phonon energy flows. The sort of physical approximation discussed in the above paragraph A A ' consists of concentrating on the part H,, of the total Hamiltonian H twhich contains the degrees of freedom associated with the "dominant" type of exci-tation. It is usually assumed, in addition, that H, may be split as follows (4.2) H, = H ( 0+ V, where Hlorepresents the energy operator of the dominant excitations considered as a separate system, while V, represents a l l the interactions which involve at least one excitation of the dominant type and at least one other type of (48) excitation. In actual calculations, V, la usually treated as a perturbation on Ho . Since the dominant excitations may be interacting among themselves, in general, we s t i l l have the fundamental problem of how the interaction energy is to be "shared" among these excitations. Fortunately, in the impor-tant case of an ordinary metal, the dominant excitations are usually electrons and these can be assumed to be noninteracting to a good f i r s t approximation. A"reasonable" choice of the electronic energy flow operator i n this example is J ~' » m\ rrr\ J where "H^ , pj. and <™ are the energy operator, momentum operator and mass of the 3 electron. A similar sort of expression i s often used for the phonon energy flow operator (see, for example, Carruthers ( 1 9 6 1 ) ) . feny authors believe that the concept of an energy flow i s basic-a l l y a macroscopic concept and hence the corresponding quantum mechanical observable may not be necessarily an operator which describes the behaviour of the system in microscopic detail. More precisely, It may be that the energy observed by macroscopic flow measurements i s carried by a large number of interacting excitations, and may not be the sum of the energies carried by individual excitations. As an example of this type of treatment, we refer to Mori ( 1 9 5 8 ) . Here the whole system i s s p l i t into many subregions each of which i s macroscopically small but microscopically large. In other words, each subregion has a large number of excitations but the dimension i s smaller than the experimental accuracy 0 To a sufficient approximation, we may now neglect the interactions between the subregions as surface effects and assign " th a definite energy E-and velocity operator Vi ^° * n e <• subregion. In terms of this picture, the energy operator i s often taken as (4.4) • Q = £ p L y i where we sum over a l l the subregions. This type of coarse-graining has been used i n many theories of irreversible processes and in the derivation of 0*9 > hydrodynamical equations. From both an experimental and intuitive point of view, i t is good as a f i r s t approximation. The major weakness of this type of approach is that i t is very difficult to present It rigourously or precisely. In addition, i t takes our attention from one of the basic problems of theoretical physics, the precise definition., of an energy flow. In prin-ciple, there does not seem to be any reason why the energy current cannot be given by some precisely defined operator, although the discussion in the last few pages seems to indicate that a reformulation of the problem may be neces-sary before any further progress is made. As we mentioned before, the electric current operator for a ordinary metal may be taken as (3.33) J = £ e p. where Q is the charge on the electron. Here the choice is unique, even in the presence of interactions among the electrons and other excitations. This is because the charge carried by each electron has a definite meaning to a very good degree of approximation. Another problem arises when we consider the appropriate definition of local densities. When we are discussing the classical N-partiele problem, the conventional definitions of the local number density n(M and the local energy density £ ( /0are in terms of delta functions , f run) = J~[ <$(n _ •) such that n(/v) dn, - N and j " j=, ~ ~ 3 J-n- ~ ~ ^ = Z! i such that f £(n) d/i- = E In this- equation, .n. and E are the total volume and energy} respectively, of the system; N is the total number of particles, a l l of which are identical; /t is "fell the position vector; n,^ is the position vector of the ^ particle; and Ej th is the kinetic energy plus the potential energy-of the j particle, assuming that the potential energy is equally shared among the interacting particles. (50) . The other two quantities of interest, the lo c a l electric current density /A. ' V and the local energy current density are usually defined in terms of nM and £6v) i n such a way that the equation of continuity (4.6) e dm£) + v A .£c&) =o d t and the equation of conservation of energy (4.7) ^ + V f t - ^ ) = Q dt are sati s f i e d . Of course, the equations (4.6) and (4.7) do not uniquely specify and c^Mj . To extend this c l a s s i c a l treatment to a quantum mechanical N-particle system, we may make use of Weyl's correspondence rule. The la t t e r enables us to transform clas s i c a l phase space functions of the coordinates and momenta of the particles into quantum mechanical operators by means of a consistent procedure. By means of this, Me find l o c a l density operators such as ^ (4.8) n(/L) - £ £(&-&i) for the local number density, where , now denotes the position operator of the # * n particle. I t must be emphasized that quantum mechanical operators such as (4.8) are essentially c l a s s i c a l i n content. This may be seen from the method of finding these local density operators from the classi c a l l o c a l density functions. Another nay of seeing this i s by noting that the operator A r\(/i) has a physical meaning only when i t i s used i n a very restricted way. To make, the la s t point more precise, we shall calculate the matrix elements of H(/t) with respect to the N-particle state functions £ ^ K ^ I ^ I , • • • it) j (4.9) < ^ | n ( A ) | ^ , > E L "(*) $u>(4j,-,lL*;t) cU.-.. eU» " x dsi, ...~OIA.J.-I dbi $.+<••• CIA* = W J A ^ ^ . A L R . . , ^ ; < ) ^ ) ( A ; A l r . ^ - t ) ^ L . . . ^ A » ; (51) where i n the la s t step, use was made of the fact that a l l the particles are assumed to be identical. I f we put cy* =. u' , the last expression turns out to be, by definition, the diagonal element (in the 5p<* (-Hr*, \i t • - ^ «/;^) representation) of the single particle reduced density matrix. I t has a definite physical meaning as the probability density of finding a particle at /t at time t , i f the state functions 6£>, ...,Av;-r) are the eigenf unctions of the t o t a l hamiltonian. In contrast, the off-diagonal elements < ^ ifo<'/* $ where , are not equivalent to the off-diagonal elements of the single particle reduced density matrix. Indeed, these off-diagonal elements of YM/J) have no direct physical meaning. For this reason, i n dealing with the operator /1(A) , we have always to keep i n mind that only the diagonal elements of this operator may be used i n a calculation. Although several authors use this operator as the definition of the number density operator, the above point i s an essential drawback of such a definition and hence we w i l l not use this operator i n the following argument. In this thesis, we shall define our operators using a quantized theory, i n contrast to unquantized theory used up to now. In particular the various local densities w i l l be assumed to be expressed i n second quantized form. The density of the number of conserved particles M ( A ) i s given by the usual formula, (4.10) h(/t) = Y U ) with the to t a l number operator N = n (A.) dA- and where ^ ( A . ) denotes the quantized particle f i e l d function and Y V A ) the Hermitian conjugate of ^(^L) . For the pertinent example of electrons i n a metal, we have (52) ( l u l l ) A . ) — II OLKV$KV(A) (4oi2) n where (X^v and are the usual creation and annihilation operators, respectively, of the conduction electrons with the reduced wave vector k and i n electronic energy band V . We have ignored the spin of the elec-trons. ^PKUCA.)} i s some complete orthogonal set of single particle states, properly normalized i n the volume Sl of the whole system. Typically, (PKVCA.) i s the Bloch wave function describing an electron of reduced wave vector k i n the y * n band. The usual fermion anti-commutation rules hold for and fl^v . For the purpose of argument, we shall assume that the local density operators are known so that we can concentrate on the next problem, namely to evaluate &P± . In particular, we assume that the l o c a l density operator to ( A ) corresponding to the energy flow i s known. The next section requires only the existence of such an operator. In chapter 5, where the problem of evaluating the formal expressions discussed i n chapter 3 and k i s tackled, we must be more ex p l i c i t . As an example, we could take Q as the operator defined i n (U.3), with u)(A) defined i n the usual manner, **** S\r If -H- i s the t o t a l Hamiltonian of the isolated system, then (U.1U) ^ r L where i s the local energy density. If we apply the general results of chapter 3, we may now easily find the energy flow due to an electric f i e l d §[tt) s £/• € ~ t u / ^ . Using (3»U8) and (3*1*9), the "response" energy flow i s given by (53) We may introduce the tensor 0~u)(LO) SO that (4,16) <$\= <= rt) • < T u > ( c o ) where (4.1?) cr">cu>) = $?dT z'^JfcUi T A [ ^ Ctt i«» 3^ T+a>} In component form, we have as usual (4.18) < ( ^ > t = X ry^t^EUt) y-i and (4.19) ^ ( c p ) . r J ^ t e 1 " ^ f?d\TA.fpea JyMQ^Ct+O)] SECTION 4.2 : DERIVATION OF A KUBO-TYPE FORMULA. FOR THERMAL CONDUCTIVITY. Although a thermal gradient i s physically very different from an external electric field, i t may be hoped that we can derive a formal expression for thermal conductivity similar in form to that obtained for electrical con-ductivity. The derivation outlined in this section i s based on the approach used in the papers of Mori (1956, 1958, 1959) and Nakajima (1958, I960). Although formally different in development, both these authors make use of local distribution functions and study transport processes involving open systems in terms of the regression of an appropriately chosen fluctuation in the isolated system. Further comments on the literature, which is fairly extensive, will be given in section 4*3* Our f i r s t object is to find the equation of motion of the density matrix of the many-particle system subject to a "sufficiently small" tempera-— ture gradient ^^.TitC) (see section 2.1) which constrains the system from reaching thermodynamic equilibrium. It is assumed that i f the system has been <5U) isolated long enough ("aged system"), or equivalently, i f i t has been i n contact with a constant temperature heat bath, i t w i l l be i n thermal equi-librium. As mentioned before, the equilibrium density matrix pe^ i s taken for a canonical ensemble and therefore Is (U.20) pe<k =. e P*-*3* with the normalization constant chosen so TA(p^) = / • To even define macroscppically a temperature gradient, we must assume the existence of a "local temperature," as was noted i n section 2 .1 . The specimen can then be effectively s p l i t up into small volume elements each with an approximately constant temperature and each being i n thermal e q u i l i -brium. The validity and meaning of this "local equilibrium assumption" has been discussed extensively by Kirkwood and hie co-workers, for the case of classi c a l mechanics, from 1°U6 onward. Mori (1958) puts considerable emphasis on the importance of the existence of macroscopically small mass elements i n effective thermal equilibrium. I f the "interactions" between these small elements are assumed to vanish, then each volume element i s assumed to attain i t s own thermal equilibrium i n a time t0 , the latter time being quite small since i t i s a result of the internal microscopic processes. The other relaxa-tion process i s the hydrodynamical process of attaining spatial uniformity throughout the whole system, the relaxation time Te. being much larger than Much of Mori's analysis (1958) i s b u i l t upon the qualitative distinction between these two relaxation mechanisms and their coupling but, as one would expect, the discussion relies on intuitive arguments i n many places. In our opinion, It i s d i f f i c u l t enough to deal with a macroscopic temperature gradient within the framework of a quantum s t a t i s t i c a l theory without bringing i n additional macroscopic concepts and arguments. Mori, i n trying to give some physical (55) insight into transport and relaxation processes, succeeds i n complicating the theoretical analysis. As we shall see. certain intuitive assumptions are necessary but by stating them as simply as possible and only when needed, we never forget that these are assumptions and'may be eliminated i n future work. An open system, which i s kept under a constant }small temperature gradient by means of heat reservoirs, w i l l approach to a steady state after a sufficiently long time. We shall denote by p the density matrix which describes this steady state. The quantity which we want to calculate is the s t a t i s t i c a l average of the energy flow with respect to the density matrix p • However, the direct evaluation of this quantity i s not easy for the following reasons. F i r s t , the explicit form of the density matrix i s not known; i t i s obviously different from the local equilibrium density matrix since this l a t t e r density matrix neglects the interactions between the different mass elements which give rise to the dlssipative heat flow (without these interactions, the hydrodynamical processes w i l l follow the ideal f l u i d equations). Secondly* the time evolution of the density matrix i s d i f f i c u l t to describe because i t depends on the interaction between the system and the heat reservoirs. Hence, instead of studying the open system directly, many authors consider an auxiliary isolated system which at the i n i t i a l moment i s i n the same macroscopic state as the real open system, and then try to identify the physical situation taking place i n this auxiliary system with that i n the real system under certain conditions. We shall denote the density matrix of the auxiliary isolated system by p t • I f i t i s to describe the same macroscopic state as p at the i n i t i a l moment, the following condition must be satisfied (56) (4.21) T / u ( p t = 0 i(&)= T^CpciA)) A where i s the local energy density operator defined by (4»14). Equation (4 .21) , however, does not determine Pt*-o uniquely and the choice i s dictated by convenience. Mori has pointed out that the choice of Pt^o > with the restriction (a.21), i s rather immaterial i f we assume the distinct separability of the two relaxation mechanisms discussed above (for more details, see the discussion after (4*51))• Accepting Mori's argument, we choose as the i n i t i a l density matrix the "local equilibrium" density matrix defined by (4.22) pL ^ e P o * - S ^ (3(A) £ U) cU with the auxiliary condition (4.21) and the normalization condition (4.23) T/u(/9 L) = f which determines $• In (4 .22) , T i s the average temperature defined by (4.24) T = - ^ - ' | — ^ ] , Using the Fourier transform defined as follows, (4.25) g ( A ) r; ± £ 2« e , , £ ' * i we can rewrite the argument of the exponential i n (4*22) as follows (57) where in the last equation we have used the relation (4.27) i. = J A lu) c U - n In the Fourier expansion of » (4.28) / 3<A) = J-0. + i ^ ' / j , e £ * * - i t . -it- K ' ~ the coordinate-independent f i r s t term can be interpreted, for the case of no temperature gradient, as /S - 7=. » where T is the temperature of the system. If we let the temperature gradient approach zero, this is nequivalent n to assuming that the values of K present in the summation in (4.28). approach zero. In other words, for a small temperature gradient, only the /3* with small values of K. (say K. < *rc ) are nonnegligible. As a result of this analysis, we may rewrite (4.26) as (4.29) 0(A) CC/i)cU = fi*L ±1'fi<£->c To the f i r s t order in the BtcC^-o) , which are assumed to be small since the temperature gradient i s small, we find from (4.22) (4.30) pL=-pev-pj™ f^dxle™ r where £ -K is in the Heisenberg representation generalized to imaginary time variables. More explicitly, we have (58) Thus at t=o , the deviation of the density matrix from p r 4 i s ( p L - p e Q ) * or more ex p l i c i t l y , where A p L = p L - p E Q As the isolated system "relaxes to equilibrium" f o r t > o , i t i s governed solely by the internal Hamiltonian. We are dealing with the "regres-sion of a fluctuation which had reached i t s peak at t = O **. We are not con-cerned with an open system under an external temperature gradient any more but with a fluctuation i n i t i a l l y specified so that, at t" = O , the resulting statis-t i c a l average of the energy density operator i s equal to the macroscopic energy variation found i n the open system. As before, we must solve the usual equation of motion for an isolated dynamical system d t -where p t = pe<s-*- and with the i n i t i a l condition (MY) /^p0-/\pL The formal solution of ( 4 . 3 3 ) and ( 4 . 3 4 ) i s ( 4 . 3 5 ) ^ p ^ e - ^ ^ e 4 ^ * - t > o . I t i s convenient to introduce the Liouvllle operator L defined by the relation (M6) L A - LA,ii] where A i s any operator. Using (4.J6) and the series expansion of an exponen-t i a l , we can derive the useful relation ( 4 . 3 7 ) e L A = e "A e ~ * quite easily (for example, see Kubo (1957a)). Using ( 4 . 3 7 ) , we may rewrite ( 4 . 3 5 ) more compactly as (59) From (4.37) we see t h a t the L i o u v i l l e operator i s , i n 'the quantum mechanical case, simply another way of denoting the Heisenberg r e p r e s e n t a t i o n . A s i m i l a r s e l f - a d j o i n t operator i s o f t e n used i n c l a s s i c a l mechanics t o de s c r i b e the. dynamical e v o l u t i o n of an i s o l a t e d system, composed of an enormous number of degrees of freedom. We s h a l l now make the r i g h t hand s i d e of (4.37) more e x p l i c i t as f o l l o w s (4.38) Ap± = 4 p u + C ^ p t - 4 p J ' . 1 J° di' = A p L + ( - i ) { y e " £ i : t , L * A p t " ft where i n the l a s t two equations we have i n s e r t e d the e x p l i c i t expression f o r Ap L given i n (4.32-) and have made use of C Pets > "H ~\ = O . Using the energy c o n s e r v a t i o n equation w i t h the operators i n the Heisenberg representa-t i o n ( t h a t i s , the dynamical e v o l u t i o n of the system i s i n c o r p o r a t e d d i r e c t l y i n t o the ope r a t o r s ) (4..*0) c) V^.d)(A,t) =Q • w^ere 6 = fuD(/t)c6t i t follows t h a t the F o u r i e r components s a t i s f y (Ml ) - lZi-At)iii]- ix-A-Klt) =o (60) As a f i n a l remark on (4.41), £(%) and are assumed to have no e x p l i c i t time dependence. Now i t follows that (4.42) [ L ^ C - i A ) , * ] =• - ittz-tZ-Kt-iA)) = ^ ^ - K C - C X ) and therefore, from (h»3$) (4.43) Ap± or (4.44) Ap± where (4*37) was used i n the l a s t step. We have succeeded i n doing what we wanted to do, that i s , we have found an e x p l i c i t expression f o r the density matrix describing the i s o l a t e d system whose density matrix i n i t i a l l y was given by pL. i n (4.22), the l a t t e r being chosen so that (4.21) holds. Using (4.44), we s h a l l now c a l c u l a t e the ensemble averages < ffi>± and < T)T. These are the currents a r i s i n g as the Isolated system evolves from i t s i n i t i a l non equilibrium s t a t e . We s h a l l only work out the d e t a i l s for < , since the c a l c u l a t i o n o f <J/. follows i n an analogous manner. A More p r e c i s e l y , we f i r s t f i n d v.^(£)'t , which i s given by (4.U5*) <.uJ(/L)y±- T/L Upt UJIA,)) Rewriting the f i r s t term more e x p l i c i t l y by means of (4.32), (4.46) TA,C*PU = tegd\TA{pe*i-« and performing the operation o f time r e v e r s a l , we see that i t vanishes. A Assuming there i s no magnetic f i e l d s present, £ K and fiK are i n v a r i a n t A i n sign while t O t & ) switches i n s i g n . As a r e s u l t , the l e f t hand sid e of (4.46) must be i d e n t i c a l l y zero since i t i s i n v a r i a n t . We have thus shown that the l o c a l d i s t r i b u t i o n density matrix PL{~ ^ I S + ^ P L ) does not contribute A p u + 2T f*ott'L pea u>-£ ' J « - It Q (61) directly tc the response energy current* Expanding OJ(^) i n K -space we may rewrite (iuU5) as (U.U7) J -n < ^ A e 1 ' - = J. iT'E eitK*[*vU,[fidxT*\(>»&iw& or (4.48) <. £> * . > t = ^ • ^ J 6 t c 6 t ' | / 3 c < A T A f/Prs £ - * < r ' A - t ' ) As i n chapter 3, we introduce the response function (4.4?) (+) = ffdxTA. {peo &-*Co) u>tc'C*'+C\)} which, by using the cyclic invariance property of the trace and the defini-tion of pg-$ , i s equal to the factor enclosed i n the brackets C...7 i n (4 .48) . As before and for similar reasons (see chapter 3,where mechanical external eft vJ^'ft \ disturbances are considered), i t shall be assumed that T C r y dies off within some definite time T« (as a result of the inherently dlssipative character of the many-particle system under consideration)* On this assumption, equation (4.48) can be replaced for (U.5Da) t » with the implicit restriction that only small values of f£ and *r' are present* Now, the question arises under what conditions equation (4*£l) which we have derived for an isolated system with the i n i t i a l condition p0 - f>u can be used to describe the physical situation i n the case of an open system under a temperature gradient* To answer this question, l e t us examine the physical situation i n the case of an isolated system (62) i n more d e t a i l * We have assumed above that the response f u n c t i o n (4.49) vanishes f o r X s a t i s f y i n g (4«S>0a). This implies that the F o u r i e r transform of the energy flow given by (4-U8) a t t a i n s a steady state value a f t e r a time long compared with T& • On the other hand r we assume, as usual, that i f the system i s l e f t to I t s e l f the energy flow w i l l disappear a f t e r a c e r t a i n r e l a x a t i o n time T& and the system w i l l come to equilibrium. We may state t h i s assumption more c l e a r l y as f o l l o w s : we assume that there e x i s t two d i s t i n c t processes, one microscopic and the other macroscopic, which are characterized by two d i f f e r e n t r e l a x a t i o n times, the shorter one being TK and the longer one T / / , s a t i s f y i n g the c o n d i t i o n The "decay" of the response f u n c t i o n (U.49) i s due to a microscopic process. On the other hand, the f a c t that the energy flow given i n (4.48) dies o f f i s due to a macroscopic process which i s assumed to be extremely slow f o r an i n f i n i t e l y large system. I f we accept t h i s assumption, then the macroscopic process taking place i n the time region s a t i s f y i n g (14.50b) « ± « T i i s approximately a steady state process under a temperature gradient which i s approximately constant. Let us now consider an open system i n contact with heat r e s e r v o i r s which keep the system at a constant temperature gradient. ( 6 3 ) I f the system i s sufficiently large, then the interaction between the system and the reservoir may be regarded as a boundary condition and be omitted i n studying the process taking place i n the interior of the system. Then, for the time satisfying (U.50b), we may assume that the macroscopic physical situation i n the interior of an open system i s the same as that taking place i n an isolated system with the same, but "approxi-mate l y n constant, temperature gradient, provided that t>he i n i t i a l conditions of the two systems are the same. There s t i l l remains a difference i n the i n i t i a l conditions since we have assumed that the i n i t i a l condition of the isolated system i s given by p0 - PL > while the i n i t i a l density matrix of the real open system i s different from p L • The latter i s the density matrix which describes the steady state under the influence of heat reservoirs. However, since both i n i t i a l density matrices satisfy the same boundary condition (U.21), they describe the same macroscopic state with respect to the energy distribution. Their difference i s of microscopic scale and w i l l change rapidly i n a time of order •£<> due to the internal interactions of the system. Since, on the other hand, the macroscopic energy distribu-tion varies slowly i n time ( i n time of order TR ), the isolated system which started from the i n i t i a l condition given in (U.3U) w i l l attain the same steady state as the open system i n time of order T 0 . Hence the effect of the difference between the two i n i t i a l conditions w i l l die off i n time T 0 • Thus one may disregard the difference of the i n i t i a l conditions for time sufficiently larger than To • This whole argument, which follows MorljWill give a qualitative answer to our original (64) question, although i t s validity must be checked for any particular case. Returning to the discussion (4.51)> consider the operator uJ * when K-0 j from (4 .20) , we have (4.52) o - J^^C^dA = VJ(0) where W (o) i s the net constant energy flow operator independent of the spatial coordinates. Thus we may rewrite (4.51), for the case £' ~0 as (4.53) < $<o)> t= J - £ t £ j 3 « J" 0°°oU: ,J ifclxTA.{p« «5-!S^) W(t4c>)} The last step makes use of the following transformation, similar to that given i n ( 4 . 2 6 ) , (4.54) - i - H U'0*&-Kto) - 1 1 ^ f i ^ / 3 * e 1 ' * * . A«<o>e^"*ck Sl tc St K. ST. Js\ ~ ~ ~-V/v Bin) cU (65) where ^^cLk-Sl* The l e f t hand side i s independent of the spatial coordinates and; i s the response energy current due to the presence of a weak external temperature gradient* We assume that the tempera-ture gradient i s constant or equivalently, the local temperature varies linearly with A as given in ( 2 . 1 0 ) . If the system i s homogeneous and of regular cross section, a constant temperature gradient w i l l lead to a constant energy current. Thus i t i s i n the steady state, that i s , after the i n i t i a l "transients," present when the external disturbance i s f i r s t applied, are "damped" out. Assuming that T(A ) — T c ••• a • A with (U.55) then (U.56) S- V,v ftU) = T c Vn\ V , , TCn) ' L T c ' c ~ where the system i s "small" and therefore A i s not too large. From —+ (U.56), i f V/\ TCA) l a independent of A then, to a good approximation, so Is Vn (3<4) * Coming back to (1**53)» 0 ft (66) where the last relation defines the tensor f j ^ . In terms of components, we have (iu*8> < vCU)>t = - T cr^ djW with/*, V - 1, 2, 3» or x, y, z. If we make no restriction that the total electric current must be zero in the definition of the thermal conductivity (see section 2.1), then (T 1 4^ is the thermal conductivity tensor. By an exactly similar discussion, we can show that the electric current produced by the temperature gradient is (4.60) < £ ( 0 ) > t = - T ( A ) . cr<3> with the tensor C T (^ given by (4.61) 0-<» ^ $™ol* J , f ih T ^ { p e a W ro) J ( t + t A ) } and the net electric current T(o) is (U.62) J C O ) = | The elements of the CT(3^tensor are (4.63) a-^l = J^d? J* Tr, {pe* WV(O)J^CUL\)] If we consider our system under both a constant electric field Ec and a constant temperature gradient V*. TCA.) F w e c a n VT±te (67) down the resultant electric and energy current, linear in the external disturbances, by combining (J+.60), (4.57)» (3.U3) and (4.16) (4.64) <Jfo)> += E c . o - ( , ) ( o ) — % TU) * c r < 3 ) (4.65) < W ( o » + - f c • CT»>^o) - V * r c * ) . < T w ) where CT ( 0(o) , <TU )(0) ; <T t 3 ) and < T K ) are explicitly given in (3.42), (4.17), (4.63) and (4.58), respectively.- Remembering the usual definition of the thermal conductivity tensor, (4.66) C W ( 0 ) > + = - %TU). IC w i t h < J C 0 ) ) + = O we can easily show that (4.67) x = S - u-^co) <rL3)] or, in component form, where £ Cr l , )(o)]~' = CT">(o) We have thus "derived" a formally exact expression for the thermal conductivity tensor which is analogous to the expression for the electrical conductivity tensor, although slightly more complex since i t involves four tensors, each given by a Kubo-type formula. (68) SECTION U.3 : CRITICAL REVIEW OF OTHER METHODS The essential criterion used i n finding the change due to a small temperature gradient was that a certain response function involving a time-dependent correlation function must be negligible after a time r £ while the fluctuation should have hardly changed, i.e . , i t s relaxation time „ i it . i t i s In » where 7 R » T k . Thus when the time t i s such that Tn<z ~t <•<• TR. , we have a "steady flow." This argument i s modeled after Nakajima's work (1958, I960), though he i s specifically interested i n finding a Kubo-type formula for the diffusion constant. As mentioned before, Mori (1956, 1958, 1959) has also given an analysis of irreversible processes caused by thermal disturbances which i s basically the same as the one given i n section U.2* The discussion given by Kubo, lokota and Nakajima (1957b) i s also based on the study of the time evolution of an isolated system which was i n i t i a l l y i n a non equilibrium state. The system i s considered as macro-(U . 2 ) . In addition, Onsager's hypothesis (1931) i s assumed to be valid -a p r i o r i . We refer to the assumption that the average regression of a f l u c -tuation follows the usual phenomenological laws. Although formal expressions for the kinetic coefficients involved i n electron transport phenomena are e x p l i c i t l y written down by Kubo et a l i i , their derivation can only be described as phenomenological. The analysis makes essential use of the concepts and usual results of irreversible thermodynamics. Montroll has not yet published anything on thermal conductivity, but recently he has derived Kubo-type expressions for both the diffusion constant and viscosity. Both of these coefficients are related to thermal disturbances, as defined by Kubo, but Montroll (1959a) managed to apply Kubo's method of dealing with mechanical disturbances. What he did was to scopically defined by a set of variables (69) find an appropriate boundary oondition which enabled him to express the inter-action operator v t e x p l i c i t l y . His discussion of viscosity i s particularly interesting. Starting with an isolated system In the shape of a cube with a known tot a l Hamiltonian tl , Montroll a r t i f i c i a l l y introduced a time-dependent alteration of the volume. Without going into any of the details, we state that he was able to show that the open system, consisting of the system of interest and the boundary condition i n so far as i t effected the system of interest, could be described by means of a precisely given perturbation V^. on the Hamiltonian. Montroll then applied Kubo's formalism, as outlined i n chapter 3, i n a straightforward manner. Although there i s considerable algebraic manipulation involving several canonical transformations of the original coordinates and momenta of the many-particle system, Montroll's discussion i s perfectly rigourous. I t would be very interesting to see i f this type of analysis, involving a "trick" boundary condition leading to an ex p l i c i t form for the perturbation operator , could be extended to the case of a temperature gradient and thermal conductivity. Another interesting approach follows from the work of Lebowitz and others (1957, 1958: further references are given in these papers) who have set themselves the task of e x p l i c i t l y constructing the stationary Oibbsian ensembles which describe open systems i n the steady state. In order to do this, Lebowitz makes certain assumptions as to the microscopic constituents of the driving systems (heat reservoirs, particle reservoirs, etc.). Using these, he is able to find the interaction between the many-particle system of interest and the driving systems. Although most of the detailed discussions so f a r are for classical systems, the method can be generalized to deal with quantum mechanical systems. A very important result of the detailed calculations given so far i s that the stationary non equilibrium properties of the system do not depend c r i t i c a l l y on the particular mechanism by which we assume the driving (70) systems interact with the system of interest. For concreteness, we take the example of R heat reservoirs, the th -r reservoir being at the constant temperature ' A. Lebowitz assumes that reservoir consists of an i n f i n i t e number of noninteracting components. Prior to the interaction with the many-particle system of interest (henceforward to be denoted by S), the components of the / t * n reservoir are assumed to have a Maxwell-Boltzmann velocity distribution corresponding to the temperature TA. Lastly, each component interacts at most once with S and the interaction i s purely impulsively. We shall denote by yU(x/t) the class i c a l phase space distribution function which describes an ensemble of S systems, *. standing for a l l the degrees of freedom of the ensemble. In the absence of heat reservoirs and other external driving systems, JULlx^-t) would be given by Liouville's equation (4.69) d/A£*l±) = { H C x ^ c ^ f ) } where {...} represents the Poisson bracket and HOx) i s the t o t a l Hamiltonian of S. Lebowitz has "shown" that, with the R reservoirs mentioned above, the new equation of motion i s given by the stochastic modification of (U.69) In (U.70) the function Kn($,X%8 assumed to be a differentiable function of TA. but independent of T/i';A^/t. I t i s defined i n such a way that lOiCjf.vOdxett: i s the conditional probability that S at the phase space point *' w i l l interact with the reservoir i n the time interval di,andas a result, end at the point x., the change being given by c(.K . Recently, Lebowitz and Shimony (1961) have used Kubo's response method i n order to find a Kubo-type formula for thermal conductivity. This approach has the merit of being (71) direct in that i t explicitly evaluates the interaction between S and the driving systems, but i t has the disadvantage that this evaluation involves many ad-hoc assumptions. Of course, Lebowltz's whole analysis is built upon the hypothesis that, in studying open systems in the steady state, the Inter-action between the driving systems and S need not be specified In complete detail. In other words, several different interaction mechanisms may lead to the same stationary non equilibrium ensemble, at least to a f i r s t appro-ximation. The last theory which we should like to discuss briefly is the one given by Kllnger (196&C*) In his series of papers on Kubo-type expressions for kinetic coefficients and their evaluation. The generalized "statistical forces", —» such as V/y. fi(A) } which Kllnger deals with are those used In irreversible , thermodynamics. According to Kllnger, a l l attempts made so far In trying to find formal expressions for the kinetic coefficients related to these statis-t i c a l forces lack generality. That Is,the results are only valid for "a special type of relaxation process and time dependence of the response function". In contrast, Klinger's method is supposed to be valid in a manner Independent of the frequency behaviour of the relaxation process. The great disadvantage of his analysis is that i t Is based on the hypothesis that the statistical forces affect the density matrix p^of the system of Interest in exactly the same way as the mechanical disturbances do. For mechanical distur-bances or forces, wa know the Hamiltonian of the system in the mechanical force field. As a result, we need only solve for pt In the, linear approxima-tion. Statistical forces cannot be described by a definite Hamiltonian. To' do so, we would have to deal with a larger system including the driving systems. Since this has not been successfully done so far (see the remarks on Lebowltz's work), Kllnger makes a "guess" as to the correct solution. These remarks should become clearer as Klinger's method is outlined. We do (72) not discuss the problem In the same generality as Kllnger does. The density matrix p t of the perturbed system of Interest Is given by f (U.71) ^«-=-f. + ?; where po Is the equilibrium density matrix^ 7/v f>» - ' ) , (U.72) p0= e ^ ' 4 ^ - ^ e /3JV + \ J n ( A ) o ^ _ (i J £ ( A ) o U We are working with a grand canonical ensemble and ju. - Sj^ Is the chemical potential of the system i n equilibrium. How we consider the special case of an electric f i e l d , (4.73) E(A,t) = - % $(A,t) where A and t stand for functional dependence on the position and time. It i s assumed that (4.74) #(4,-0 = $• + with l<5$| sufficiently small and <$D a constant. The interaction V, between the system and the external driving system i s known i n this case, (4.75) V T = j ^ e "U) 6§<A/OC6I where €> , J i and nd) are the charge of the electron, the t o t a l volume of the system of interest and the lo c a l number density operator, respectively. By a calculation similar to that given i n (4.22), can be expressed i n K*space as (4.76) v t = J- £en*8$-K(t) In the Heisenberg representation (with respect to the Internal Hamiltonian ft), we have (H.77) v tw = i L e n" K(t)^a). " Now the equation of motion which we must solve i s , i n the Heisenberg represen-tation, , 0.78) t£k*)a iCPk(t),V,W] (73) The linear approximation of (4.78) Is (4.79) ^ t ) = t T p „ V t t t ) ] dt . . ^ ~ ' If we use n«(.t)= i CK, n^Ct)j , ( ^ . 7 7 ) , and Kubo's identity as given in (3.18), ve can rewrite (4.79) as (4.80) <¥±M = - £ f E L ^ e ^ ( t - i A ) ( 5 i , W . dt * We shall not go into details, but i f we solve (4.80) in a straightforward manner, we can use the solution to get the usual Kubo-type formula for elec-trical conductivity. It Is important to note that the result (4.80) depends on the knowledge of V* given in (4.76). We now make a small digression. We assume that the electric f i e l d E(Aft) Is given by (4.81) B(A,t) = - % <$U,t) - £ where yU(A,i:) and yUe(A,+) are known as the generalized chemical potential and electrochemical potential, respectively. In addition, we assume that (4.82) jU.*(A.tt) = yU* + SjU,e(A,t) whereyClf is a constant and / is sufficiently small. KLinger then introduces the operator po by the following definition (4.83) p6 = e / S A ' r ^ J c i ^ j o f A ^ / / t ^ A . t - ) We note that (4.84) J fy*(*,f) in(A)oU_= ^ Z ri* o > £ f t ) Now using (4.84) and substituting (4.82) into (4.83), we may expand with respect to the Fourier components S^U-K ft) hy means of the usual Schwinger formula for an ordered exponential. The result is , in the linear approximation Po - po +• $p } where (7»0 In the Heisenberg representation, we have , (4.86) o>(+) = £ 5 1 e n \ ( t - a ) ^ ( t ) . Now, differentiating (4.86) with respect to time and keeping c£a-*(t) cons-tant, (4.87) \# *P(lA The right hand sidesof (4.87) and (4.80) are the same except for the signs, or ' (4.88) ( * 6 r C t ) ) , m In the case of a s t a t i s t i c a l force {3(A) , we assume (fc.89) (3(Hjt) =(3+ tipC*,*) where p ) T i s the equilibrium temperature, and I I i s sufficiently small. We can easily show that i f (1 i n p 0 la replaced by ( 3 ( * , T ) a n d ^ Is kept constant, we have (4.90) Spit) = -_g ^L^X &<*-i*> <5/3^ «)(^ ) Now Kllnger makes the fundamental assumption that (4.88) again holds, that Is (4.91) (dtpW) = _ Using (4.91) and (4.90), Kllnger was able to f i n d a Kubo-type formula for thermal conductivity. (75) CHAPTER 5 EVALUATION OF KUBO-TYPE EXPRESSIONS. SECTION 5.1: RELATION TO VAN HOVE'S MASTER EQUATIONS AND THE REDUCTION TO' SINGLE PARTICLE EXPRESSIONS. Consider a many-particle system under the influence of several weak external disturbances; experimentally, i t i s well known that various trans-port processes w i l l arise. Some of the disturbances may involve "mechanical'* forces and, as we have seen i n chapter 3, there Is no fundamental d i f f i c u l t y i n finding the linear "response" currents due to this "type of force i n terms of transport coefficients given by Kubo-type expressions. Others may be "thermal" or " s t a t i s t i c a l " disturbances and for these, the above problem i s not so easy. In chapter 4, we discussed the effect of a constant temperature gradient i n some detail, and derived a Kubo-type formula for thermal conduc-t i v i t y . Although there are s t i l l some d i f f i c u l t i e s In the proof, this and other discussions i n the literature seem to give a good Indication that we can express thermal conductivity i n the form of a Kubo-type expression. In this chapter, we shall leave a l l doubts aside and assume that the result given i n (4.68), OM), (4.1.9), (4.63) and (4.58) i s correct. Speaking generally, the results of the work over the l a s t few years seem to Indicate the following: I f a weak external "force",, Pit) ~ Fc e"^*" , acts upon an ensemble of systems characterized by the temperature T, then the. linear "response" of current operator 8 i s given by (5.1) < | \ = %-tU ^ 8 ^ ) where the transport coefficient i s formally expressed as (5.2) c r C 8 M = f f e ^ f ^ x T ^ f e - ^ ceo ejoe^^} ( 76 ) * = Tolt eiu)ir J ? ^ A ^ £ C o ) |(-t + a)^<a The operator £ depends on the particular process we have in mind but in general i t is the observable corresponding to the macroscopic flow most directly related to the force Rf)(for example, i f FJ-+>= , then c tr 7 i i f R*> = (itnjt) , then C = § ) . The next problem is to evaluate the right hand side of (5.2), the basic element being the correlation function < C(o) B(<-+<A» r ( S . This part of the subject is at present in a rapid stage of development and only a few remarks will be made in this section. In general, the transport coefficients given by (5.2) (or "kinetic" coefficients i f the generalized "forces" r(t) have been chosen so as to lead to Onsager's reciprocity'relations) are complex quantities. Following KUnger (I961), we may show that (5.3) ReaKCTcoM) = e >o where 7l,-r\*-= E-iio # y e refer to the end of chapter 3 for the significance - £ + of the convergence factor G .It lends generality to our discussion, but i t is somewhat a r t i f i c i a l and in certain cases, redundant. It is sufficient to find Real ((Tce(UJ)) , since we can easily derive the following disper-sion relation r 0 0 (5.4) R e « T c , < u > ) ) = - L j O^U)' ( J ^ (cTca^))} (77) where we have taken the principal part of the Integral. K e CO"colw))ls often referred to as the dissipative part of the transport coefficient OCBCU)) since (5.5) Jbsvrx Re C<rca(u>)) while (5.6) A^rt I/^((T C 8(u))) = 0 I(rvi( °ca(u))) * 8 o f t e n referred to as the dispersive "part of the trans-port coefficient. As we mentioned i n chapter 3$ i t i s often convenient to introduce a quantity £ca (•<*>) such that (5.7) C<rC8tu>)) = - w * e U c e C ^ ) ) If we had external magnetic f i e l d s , i t would also be advantageous to s p l i t the transport coefficient, as given i n (5.2), into .symmetric and antisymme-t r i c parts (with respect to a reversal of the direction.of the magnetic field}. We refer' to the ar t i c l e s by Kubo (1957a) and KLinger (196l) for further formal relations between the r e a l and Imaginary, symmetric and anti-symmetric, components of the formal Kubo-type expressions. It may be mention-ed that Onsager's reciprocity relations can be derived directly. The e x p l i c i t evaluation of the right hand side of (5.2) involves a trace over the product of an equilibrium density matrix and two operators, and a time integral. As we mentioned before, Van Hove's master equations provide one of the most useful methods of evaluation, especially for the case of weak scattering. It should be emphasized that Kubo-type formulas are, i n principle, v a l i d for strong as well as weak scattering of the "current carriers". Indeed, this i s one of their main advantages over the standard transport equation method which i s , s t r i c t l y speaking, only v a l i d for weakly interacting excitations. (78) We shall restrict ourselves to the type of system which Van Hove worked with (see chapter 1 for details). For convenience we use the notation in Van Hove's f i r s t paper (1955). Our basic representation is denoted by lev*> where (5.8) ft, = ElEuy and we choose the normalization (5.9) <Fo(/ek'> = &LE-E ^ f o c - o C ) As (5.9) implies, we assume that E andoC act as continuous variables in the limit of a large system ( N -*>oo> si^oo such that the density E is constant). A l l our operators are written in the second quantized representation. In the long time, weak coupling approximation, the diagonal singularity can be written as (5.10) vAvlewy = 6(e-E')j(o(-ee) wA(eoe) •+ <E* I X* IE'°<'> . where A is any diagonal operator in the IE»-representation and (£o( | vf f o(> r O . Lastly, we assume (5.11) <^EU I NlEUy = N ( E°0 6(E-E')f(ol-iX') < EU l cje'oiy = c(rot) 6(E- e')S(ot-of) where C(eor) and Btew)are "smooth" functions of E , the unperturbed energy. This last condition on the eigenvalues is essential i f we are to use Van Hove's mathematical technique based on the use of the diagonal singularity. Now, in the weak coupling, long time approximation, Van Hove has shown that, (5.12) ' C(t)/Fi6= <F^|UC-i) C J ( t ) / £F '<*• > = §c(E.ot,) <e«Iu(--t)iE\O{,><E,O<,Iu(VIE'u'>d°t,ctE> (79) where P+ ( £" Eoi) ±8 the solution of the "master equation**, (5.13) ± ^ £ ^ ) = 2 ^ , L [ w ( £ ^ f o / i ) p < e*; oft" with the i n i t i a l condition P0(Eot,; Eoi) = 6(o(,-ot) . i n (5.13), W Y E c ^ F o / i ) = w r E ^ j F o i , ) ^ l<E°f-1 v l F o O l 1 , the .second - t equality following from the Hermitian character of V . A similar result holds for ^ l B ( t ) l £ W ' > I f we expand Re C °cs/u>)) given i n (5.5) in the /e<>0-repre-sentation and make use of (5*15), we have (5.1*0 £e ( O-c0 <u>» • * [ 8(£V) c f(f ' - e>o rW-o / ) J c t « " C ( ^ ^ " ) PtL£oi"-Eu) + cce«c <)0:cer'-£)d"c ,^-rt)Joto<" gee*-) P t ( ^ " , E o < ) j HEp(u) X j B(E<X) Jclo(»C(E*") P+tBd';?<*) -f C f E * ) J " G ^ » 6 ^ " ) P±(EOC;EOL) j (80) where E f l C u O = ^ <J^rJa and Z = T/i { e ft/U". I t should be noted that ve are using a grand partition function here... ' ; - J : Klinger (1961) formally simplifies the right hand side of '(5.1*0 by introducing a resolvent opera-tor j the resulting expression i s very compact and quite useful i n making calculations f o r specific models. As ve noted before, the dispersive part of 0~c eCvo) can be found from the dlssipative part by means of the disper-sion relation (5.15) Lm((TcB<uO) = ~{°° KeCCTcsCu;)) . " '-co co-to' We return to the specific case of thermoelectric transport c o e f f i -cients. In chapters 3 and k, ve have derived the following Kubo-type formulas ( s t a t i c ca.se,u~>=o) J7 dt^dX TA{PEQ Jy(o) w^u + n)} { o*ot± db. Trv [ p£Q VV„(o) J^iUi A)j ["ctt^dX TrulpBQ W v f o ) W A (t+tA ) l As we mentioned before, a l l of these involve a double integration. Verboven (1961) has shown that can be reduced so as to involve a single time integration by making use of the dlssipative properties of the many-particle system. His proof can be easily generalized to 0^/y° j 0^l'y a n d cr^i^ As a result, we may rewrite (5.8) - (5.H) i n the following form, (5.16) ' <rj» = (3 j ^ c t f TA { pESl Jy f o ) 5^ f t ) } (5.17) ' ( 7 ^ = ft T A { fa Sv(o) VV^ f-O} (5.18) ' Jo°°ca T A { ^ V\Uo)£ft>} (5.16) cr^> = (5.17) a-JV -(5.18) ^ = (5.19) 0>iN (81) (5.19) ' cj>^ }= J"ca T/tfpec Wy(o)vv>(:t)} A l l of these formal expressions are many-particle formulas in T -space". As Chester and Thellung (1959) have pointed out and Van Hove (see Verboven A (196l) ) has proved, i f the total Hamiltonian # i s the sum of single particle Hamlltonians Wj (where <y refers to t h e ^ 1 particle), then (5.20) -where H Is the single particle Hamiltonian, £ (+•) is the single particle electric current operator In the Heisenberg representation (with respect to H) (H) is the statistical distribution function for the particles when they are in equilibrium. In the theory of metals, we are usually dealing with electrons which, to a sufficient approximation, are Independent. Electrons obey Ferml-Dlrac statistics and consequently (5.21) J(fj) = ' -where// is the chemical potential. As a last remark, " stands for the trace over any complete set of one particle wave functions. In section 5.2, we assume that the energy is carried only by the electrons, and take as our energy current operator W(o) (5.22) w f 0 ) - £ CPiJVjv) where /vn , /y>i* , and are the mass, "effective" mass and the momen-•th turn operator of the ^  electron. With this definition of W, we can reduce ; CT>ty :J) and (TMV^ ^° t n e following single particle formulas (5.23) V* = - M ^ J > ' 4 l ^ (5.24) 0 > » > = _ t \ { ' 4 L ^ ( 0 | v ( 6 ) + ' ^ ( » j v W ] . (5.25) <£„ w )« -1* \ 6$JP£<& '/zOpit) jy(0) + jMC0) fy(t)l(B. (82) by a calculation similar to that given by Van Hove for G^tv in ( 5 . 2 0 ) . In Klinger (I96I) points out that the reduction to single particle formulas is valid for elastic and inelastic scattering when there is no Fermi degeneracy. If the electron gas Is degenerate, then the reduction is possible only for the case when the electrons are scattered elastically by the phonons, Impurities, etc. SECTION 5.21 EXPLICIT EVALUATION OF THERMAL CORDICTIVITY FOR ELECTRONS  SCATTERED BY IMPURITIES. The papers of Chester and Thellung (1959 ) and Verboven ( I 961 ) have evaluated ( f j j ^ given in ( 5 . 2 0 ) for the simple model of electrons elastically scattered by a large number of randomly distributed static Impuri-ties. The single particle Hamiltonian with which they deal is given by ( 5 . 2 6 ) H = W « + A V where Wo is the Hamiltonian of the conduction electrons moving in the period-ic potential of the lattice and. V is the scattering potential for one elec-tron due to the static impurities. The basic representation is denoted by Ut > where < is the wave vector of the electron. £ is assumed to be diagonal in this representation, i.e., we limit ourselves to a single energy band. The case of impurity scattering is very special in that ve may simple consider one electron interacting with a large number of static impurities which are assumed never to act as "carriers". Thus Van Hove's work in nV -space" imme-diately reduces to equivalent results in "/i -space". In contrast, although we may deal with an electron-phonon system composed of a single electron weakly interacting with a large number of phonons, a complicated summation (83) ' must be made In order that the master equation for the entire system i s reduced to a s t a t i s t i c a l equation involving the degrees of freedom of the single electron. The above authors both found correction terms to the static electrical conductivity tensor evaluated by means of Van Hove's lowest order master equation for a single electron. From the order of magnitude of these corrections, they concluded that the results were correct as long as £ <cr( where t i s some characteristic " c o l l i s i o n time" and Y[ i s of the order of magnitude of the Fermi energy. Chester and Thellung imply that they have managed to verify Landau's conjecture because the single particle Kubo formula (3*20) separates the s t a t i s t i c a l factors such as Jj£ and the dynamic* a l factors such as $ (+) It should also be mentioned tha[t their method of evaluation implicitly uses ,the fact that ' I^(o)jE in (5 .28) <A'ljMfr)lA> = <e'«'ljA(t)\Eot> i s an appropriately slowly varying function of H, Using (5.22) for our energy current operator, we may derive the following lowest order results for spherical symmetric impurities, (5.29) cr^p = _ £ dl t K l f r i K X t c i f a t c y v t E j . K dEK w 0 (5.30) c r ^ c -pZI ^<.KlpK><xijvtlc>T(eK) (5.31) yJZ U «lf\\KXKlfvlK> t(eK) using (5.23), (5 .24) and (5.25). In these formulas , m (5.32) ~ = ¥7T* J*'(l-CoQsO) W(E,a&&) dCcao^e) where (>(E) i a some appropriate density function and W(E, coa-e) i s proportional to the transition probability between the single particle, states (energy E") labelled by the wave vectors £ and K' , with = and 0 the angle between the two vectors. For further details oh the calculation and notation, we refer to page 753 of Chester and Thellung (1959). In this section, we have simply applied the calculations made by Chester and Thellung to the lowest order evaluation of the thermal conductivity tensor. That we may do so i s a result of our particular choice of an energy current operator, i.e., the one defined i n (5.22). . If we chose the energy current operator given i n (4.3), with = & ( H given by (5.26)), then the calculation would not carry over so simply. (85) BIBLIOGRAPHY BERNARD,. W. and CALLEN, H.B., 1959, Rev. Mod. Phys., 31, 1017. CALLEN, H.B. and WELTON, T., 1951, Phys. Rev., 83, 3^. CARRUTHERS, P., I 9 6 I , Revs. Mod. Phys., 33, 92. CAS3MIR, H.B.G., 19^5, Rev. Mod. Phys., 3J, 3^3. CHESTER, G.V. and THELLUNGi A., 1959, Proc. Phys. Soc., 73, 7^5. DE GROOT, S.R., 1952, "Thermodynamics of Irreversible Processes", North-Holland Publ. Comp., Amsterdam. DRESDEN, M., I96I, Revs. Mod. Phys., 33, 265. FARQUHAR, I.E., I 9 6 I , Nature, 190, 17. GREEN, M.S., 1951*, J. Chem. Phys., 22, 398. KARPLUS, R. and SCHWlNGERi.;, J., 19^8, Phys. Rev., 73, 1020. KIRKWOOD, J.G., 19^6, J. Chem. Phys., 14, 180. and FITTS, D., i960, Ibid, 33, 1317. ' KLINGER, V.I., 1960a, Soviet Physics - Solid State, 1, 613. 1960b, Ibid, 1, 782. 1960c, Ibid, 1, 1122. 1960d, Ibid, 1, 12 69. I961, Ibid, 2, 2747. KOHN, W. and LUTTINGER, J.M., 1957, Phys. Rev., 108,.590. KUBO, R., 1956, Can. J. Phys., 3^, 127^. t 1957a, J. Phys. Soc. Japan, 12, 570. , YOKOTA, M. and NAKAJIMA, S., 1957b, Ibid, 32, 1203. LANDAU, L.D. and LIFSHITZ, E.M., "S t a t i s t i c a l Physics", Pergamon Press, London,1958. LAX, M., 1958, Phys. Rev., 109, 1921. LEBOWITZ, J.L. and BERGMANN, P.G., 1957, Ann. Phys., 1, 1. f I959, Phys. Rev., Ilk, 1192. and SHBDNY, , I96I, Preprint. MARTIN, P.C. and SCHWINGER, J., 1959, Phys. Rev., 115, 131*2. (86) MATSUBAIRA, N., I 9 6 I , Prog. Theor. Phys., 25, 153 P MONTROLL, E.W., 1959a, Suppl. to II Nuovo Cimento, "Thermodynamic a del Process! I r r e v e r s i b i l i " , Varenna Summer School (1959) Lectures. and W&KB, J.C., 1959b, Physica, 2 5 , 423. MORI, H., I956, J. Phys. Soc. Japan, Lj., 1029. — , 1958, Phys. Rev., 112, 1829. , 1959, Ibid, 115, 2 9 8 . NAKAJIMA, S., 1956, Proc. Phys. Soc. A, 69, 441. , I958, Prog. Theor. Phys., 2 0 , 948. , 1959a,Ibid, 2 1 , 659. _ f 1959b,Suppl. to II Nuovo Cimento, -"Thermodynamica del Process! I r r e v e r s i b i l i " , Varenna Summer School (1959) Lectures, NAKANO, S., 1956, Prog. Theor. Phys., 15, 77* ONSAGER, L, 1931, Phys. Rev., 37, 405. PRIGOGINE, I., BALESCU, R., RENIN, F. and RESIBOIS, P., i 9 6 0 , Suppl. to Physica, "Proceedings of the International Congress on Many-Particle Problems", Utrecht. VAN HOVE, L., 1955, Physica, 2 1 , 517. , 1957, Ibid, 23, 441. , 1958, "Selected Topics i n the Quantum Statistics of Interacting Particles", Lecture Notes, University of Washington, Seattle. , 1959, Physica, 2 5 , 2 6 8 . , i 9 6 0 , "The Theory of Neutral and Ionized Gases", Les Houehes Summer School ( i 9 6 0 ) Lectures; J. Wiley and Sons, New York. , and VffiffiOVEN, E., 1 9 6 l , Physica, 27, 4 l 8 . VERBOVEN, E., 1961, Physica, 26, 1091. WILSON, A.H., 1953, "The Theory of Metals", 2nd Ed., University Press, Cambridge. (87) ZIMAN, J.M., I960, "Electrons and Phonons", Clarendon Press, Oxford. ZUBAREV, D.N., i960, Soviet Physics - Uspechi, 3, 

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