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Theoretical studies in stochastic processes Blackmore, Robert Sidney 1985

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THEORETICAL STUDIES IN STOCHASTIC PROCESSES by ROBERT SIDNEY BLACKMORE Sc., New Mexico Institute of Mining and Technology, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Chemistry We acc e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d UNIVERSITY OF BRITISH COLUMBIA May, 1985 © Robert Sidney Blackmore, 1985 In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the UNIVERSITY OF BRITISH COLUMBIA, I agree that the Library s h a l l 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 or her representatives. It is understood that copying or publication of thi s thesis for fi n a n c i a l gain shall not be allowed without my written permission. Department of Chemistry UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: May, 1985 Abstract A general method of analysis of a variety of stochastic processes in terms of prob a b i l i t y density functions (PDFs) i s developed and applied to several model as well as physically r e a l i s t i c systems. A model for d i f f u s i o n in a bistable potential is the f i r s t system considered. The time dependence of the PDF for thi s system is described by a Fokker-Planck equation with non-linear c o e f f i c i e n t s . A numerical procedure is developed for finding the solution of thi s class of Fokker-Planck equations. The solution of the Fokker-Planck equation is obtained in terms of an eigenfunction expansion. The numerical procedure provides an e f f i c i e n t method of determining the eigenfunctions and eigenvalues of Fokker-Planck operators. The methods developed in the study of the model system are then applied to the trans-gauche isomerization of n-butane in C C 1 „ . This system i s studied with the use of Kramers equation to describe the time evolution of the PDF. It i s found that at room temperature the isomerization rate obeys a f i r s t order rate law. The rate constant for th i s system is sensitive to the c o l l i s i o n frequency between the the n-butane and CC1„ as has been previously suggested. It i s also found that t r a n s i t i o n state theory underestimates the rate constant at a l l c o l l i s i o n frequencies. However, the acti v a t i o n energy given by t r a n s i t i o n state theory i s consistent with the activation energy obtained in th i s work. i i The problem of the escape of light constituents from planetary atmospheres is also considered. Here, the primary objective i s to construct a c o l l i s i o n a l kinetic theory of planetary exospheres based on a rigorous solution of the Boltzmann equation. It is shown that this problem has many physical and mathematical s i m i l a r i t i e s with the problems previously considered. The temperature and density p r o f i l e s of l i g h t gases in the exosphere as well as their escape fluxes are ca l c u l a t e d . In the present work, only a thermal escape mechanism was considered, although i t is shown how non-thermal escape mechanisms may be included. In addition, these results are compared with various Monte-Carlo calculations of escape fluxes. Table of Contents Abstract i i Table of Contents iv L i s t of Tables v i i L i s t of Figures . . . . . v i i i Acknowledgement x 1. Introduction 1 1.1 Transport Processes 1 1.2 Brownian Motion 5 1.3 Microscopic Descriptions 10 1.4 Model Problems 19 1.5 Problems Considered 21 1.5.1 Chemically Reactive Systems 21 1.5.2 Diffusion in a Double Well Potential 25 1.5.3 Escape of an Atmosphere 27 2. Numerical Methods for Stochastic Equations 32 2.1 Introduction 32 2.2 Solution of the Fokker-Planck Equation 35 2.3 Eigenfunction Expansion 36 2.4 The Discrete Ordinate Method 39 2.5 DO Approximation to Self Adjoint Operators 41 2.6 DO Representation of Non-self Adjoint operators .44 2.7 Summary 47 3. Diffusion in a Bistable Potential 49 3.1 Introduction 49 3.2 Bimode polynomials 50 3.3 Polynomial Representation of Fokker-Planck Operator 51 iv 3.4 Schroedinger Form of the Equation ...52 3.5 Eigenvalue Spectrum 57 3.6 Eigenfunctions ..63 3.7 Time Dependent Solution 67 3.8 Summary 7 3 4. Butane Isomerization 75 4.1 Introduction 75 4.2 Conformations of n-butane 82 4.3 Phenomenological Treatment 85 4.4 Simplified Potential 87 4.5 Molecular Dynamics 91 4.6 Transformation to the Smoluchowski equation 96 4.7 Various Approximations for the Rate Constant ...100 4.8 Present Method of Determining the Rate Constant 108 4.9 Discussion of Numerical Results 112 5. Escape of Atmospheres from Planetary Bodies 129 5.1 Introduction 129 5.2 Plane P a r a l l e l Model of the Exosphere 145 5.3 C o l l i s i o n Operator 150 5.4 Coupled Sets of Equations 153 5.5 Iterative Solution to Boltzmann Equation 155 5.6 Discussion of Numerical Results 162 6 . Summary 186 7 . Appendix A 196 7.1 Introduction 196 7.2 Representation of D i f f e r e n t i a l Operators 198 7.3 Definitions 1 99 v 7.4 D i f f e r e n t i a l Operators 203 7 . 5 Summary 210 8. Appendix B 212 9. Appendix C 217 REFERENCES 188 vi L i s t of Tables Chapter 1 Chapter 2 Chapter 3 3.1 Convergence of Eigenvalues e = 0.1 58 3.2 Convergence of Eigenvalues e = 0.0l 59 Chapter 4 4.1 Potential Parameters in n-Butane Isomerization ..90 4.2 Convergence of Smoluchowski Eigenvalues 113 4.3 Convergence of Kramers Eigenvalues j>=1f0 1 14 4.4 Convergence of Kramers Eigenvalues v=2v0 115 4.5 Convergence of Kramers Eigenvalues P=5I>0 116 4.6 Eigenvalues of Kramers Operator .........117 4.7 Expansion Coefficients for Rate Constant 124 Chapter 5 5.1 Escaping Gas Combinations for Various Planets and S a t e l l i t e s 163 5.2 Convergence of Iteration: Jeans' Ratio 165 5.3 Convergence of Jeans Ratio as a Function of the Number of Points 166 5.4 Comparison of Jeans' Ratio with Monte-Carlo Calculations 172 Appendix A A.1 Numerical Derivatives with the DO Method 208 v i i L i s t of Figures Chapter 1 1.1 Relationship Between Various Kinetic Equations .12 Chapter 2 Chapter 3 3.1 Equilibrium D i s t r i b u t i o n Function 54 3.2 Schroedinger Potential 55 3.3 Comparison of Approximations to X, 61 3.4 Variation of X, with e 62 3.5 Variation of X with e 64 n 3.6 Various eigenf unctions for e = 0.01 65 3.7 Variation of the t h i r d eigenfunction with e 66 3.8 Second Eigenfunction, exact and approximate 68 3.9 F i r s t Eigenfunction: Approach to a Step Function 69 3.10 Time Dependence of PDF 71 Chapter 4 4.1 Conformations of Butane 83 4.2 Intramolecular Potential of n-Butane 84 4.3 Approximate Intramolecular Potential of n-Butane 89 4.4 Rate Constant as a Function of C o l l i s i o n Frequency 118 4.5 Variation of Rate Constant with Temperature ....120 4.6 Transient Rate Constant Calculated from the Smoluchowski Equation 121 4.7 Transient Rate Constant Calculated from the Kramers Equation 122 4.8 X2/X! as a Function of Temperature 126 4.9 F i r s t Eigenfunction of Kramers Operator ..127 v i i i Chapter 5 5.1 Temperature P r o f i l e of the Atmosphere 131 5.2 P a r t i c l e Classes 138 5.3 Plane P a r a l l e l Model 148 5.4 Velocity D i s t r i b u t i o n Function: Escape of Hydrogen From Earth 167 5.5 Ve l o c i t y D i s t r i b u t i o n Function: Escape of Helium from Earth 168 5.6 Velocity D i s t r i b u t i o n Function: Escape of Hydrogen from Mars 169 5.7 Jeans Ratio: Escape of Hydrogen from Earth 173 5.8 Jeans Ratio: Escape of Helium from Earth 174 5.9 Jeans Ratio: Escape of Hydrogen from Mars 175 5.10. Temperature P r o f i l e s : Escape of Hydrogen from Earth 177 5.11 Temperature P r o f i l e s : Escape of Helium from Earth 178 5.12 Temperature P r o f i l e s : Escape of Hydrogen from Mars 179 5.13 Temperature Drop: Escape of Hydrogen from Earth ..180 5.14 Density P r o f i l e s : Escape of Hydrogen from Earth 181 5.15 Density P r o f i l e s : Escape of Helium from Earth 182 5.16 Density P r o f i l e s : Escape of Hydrogen from Mars 183 ix Acknowledgement I g r a t e f u l l y acknowledge the encouragement, guidance and help of my supervisor, Dr. Bernard Shi z g a l . Were i t not for his unselfish support and help, the completion of this thesis would have been impossible. In addi t i o n , I acknowledge the f i n a n c i a l support provide by the Department of Chemistry and the I. W. Killam foundation. Without such support, the preparation of this thesis would have taken much longer. I also would l i k e to thank the many people in the Computer Science Department who cheerf u l l y helped me with my problems. F i n a l l y , I would l i k e to thank Donald Arseneau, who provided me with many tools to create a comfortable computing environment. x 1. INTRODUCTION This thesis deals with the description of the approach to equilibrium of systems far from equilibrium. This i s a very broad subject and there i s an abundance of physical systems which f a l l into t h i s category. The large number of non-equilibrium systems precludes a thorough discussion of them a l l . I w i l l begin by describing several physical systems which are far from equilibrium and their importance in chemistry, physics, biology and applied science. These problems are t y p i c a l of those under active consideration by a large group of researchers and overlap with the work of this t h e s i s . A discussion of the s p e c i f i c problems considered w i l l be presented in section 1.5. 1.1 TRANSPORT PROCESSES It has long been known that mass, energy or momentum transport w i l l take place in macroscopic systems that have non-zero density, temperature and/or v e l o c i t y gradients [R1]. Early researchers were interested in r e l a t i n g these density or temperature gradients with the corresponding mass or energy f l u x . It turned out that these fluxes and gradients are simply related via phenomenological laws such as Fourier's law of heat conduction or Fick's law of d i f f u s i o n . Fick's law states that the mass flux i s proportional to minus the density gradient. S i m i l a r l y Fourier's law of heat conduction states that the heat flux i s proportional to minus the temperature gradient. These 1 2 laws are referred to as macroscopic laws since they relate macroscopically observable quantities (e. g. the mass flux and density gradient) to each other. The proportionality constant r e l a t i n g a flux and a corresponding gradient i s c a l l e d a transport c o e f f i c i e n t . These include the d i f f u s i o n c o e f f i c i e n t and heat conductivity c o e f f i c i e n t in the case of Fick's law and Fourier's law, respectively. It is important to note that the macroscopic laws provide a continuum description of a physical system, in other words, intensive macroscopic quantities such as density and temperature vary continuously. Such a description ignores the microscopic nature of the system. These macroscopic laws by no means provide a complete description of p a r t i c l e transport. Two important li m i t a t i o n s of them are as follows. F i r s t , the transport c o e f f i c i e n t s are not given by the law i t s e l f and must be determined either by experiment or by some other theoretical approach. Another l i m i t a t i o n is that these laws are v a l i d only when the gradients of the number density, temperature or other macroscopic intensive quantity i s small. Much of t r a d i t i o n a l transport theory has been concerned with these considerat ions. In order to understand both the range of v a l i d i t y of the linear macroscopic equations, and to calculate the transport c o e f f i c i e n t s contained within them, i t i s necessary to consider these systems from a microscopic l e v e l . The f i r s t step in th i s direction was taken by 3 Maxwell. Maxwell suggested [R2] that the p a r t i c l e s composing a system have random v e l o c i t i e s , but the d i s t r i b u t i o n of the p a r t i c l e v e l o c i t i e s i s a d e f i n i t e function given by what i s now c a l l e d the Maxwell ve l o c i t y d i s t r i b u t i o n . Later Maxwell showed [R3] that macroscopic q u a n t i t i e s , such as fluxes, are related to moments of ( i . e . integrals over) the d i s t r i b u t i o n function. The next major step was when Boltzmann h e u r i s t i c a l l y derived [R4] an equation for this v e l o c i t y d i s t r i b u t i o n function and showed that i t must, in the absence of external f i e l d s , approach the Maxwell vel o c i t y d i s t r i b u t i o n . This equation i s known as the Boltzmann equation. With respect to macroscopic equations, Chapmann and Cowling [R5] have shown that t r a d i t i o n a l transport equations may be derived from the Boltzmann equation. These equations include Fick's law and Fourier's law previously mentioned. Their derivation of these laws i s important for a number of reasons. The approximations necessary to derive them are c l e a r l y stated and hence the physical conditions for which these equations remain v a l i d i s made c l e a r e r . In ad d i t i o n , the transport c o e f f i c i e n t s , such as the d i f f u s i o n and thermal d i f f u s i o n constants, are expressed in terms of the i n t e r p a r t i c l e potentials and p a r t i c l e masses. Other transport equations which follow from the Boltzmann equation are equations such as the Navier-Stokes equations, employed in many f l u i d dynamical problems. Such equations describe the time and space dependence of the mass density, bulk 4 v e l o c i t y and temperature of a non-equilibrium system. The Boltzmann equation [R6-9], has proven to be extremely useful in the description of many physical systems apart from i t s application to p a r t i c l e transport. It has been applied to the kinetic theory of gases [R6-9], neutron transport [R10], NMR relaxation times [R11], s t e l l a r dynamics [R12], chemically reactive systems [R13-17], dynamics of the upper atmosphere [R18,19] as well as to a host of other problems too numerous to mention. One p a r t i c u l a r application of the Boltzmann equation is to neutron transport. The v e l o c i t y d i s t r i b u t i o n function of neutrons in a nuclear reactor core is an important quantity since the rate of nuclear f i s s i o n is dependent upon i t . This dependence is a consequence of the fact that neutron-nucleus reactive cross sections are s e n s i t i v e l y dependent [R20] on the r e l a t i v e v e l o c i t i e s of the two p a r t i c l e s . In general, this reactive cross section increases with decreasing v e l o c i t y [R20]. The determination of t h i s v e l o c i t y d i s t r i b u t i o n function is complicated by the fact that reactor cores must have a f i n i t e extent. Hence the v e l o c i t y d i s t r i b u t i o n function for the neutrons w i l l be perturbed not only by neutron sources and sinks within the reactor core but also by loss of neutrons at the reactor surface. The s i m p l i f i e d problem of evaporation of neutrons from a reactor surface, assuming a source of neutrons at an i'nfinite distance from the surface, is c a l l e d the Milne 5 problem [R21], It f i r s t arose, not in nuclear engineering, but rather in the description of photon transport from the int e r i o r to the surface of a s t a r . Indeed, photon transport shares many s i m i l a r i t i e s with neutron transport as described by Kourganoff [R22]. In both cases, one is concerned with uncharged p a r t i c l e s , although the photon case is si m p l i f i e d since no energy exchange takes place. It w i l l l a t e r be seen that this Milne problem is also similar to a problem considered in thi s thesis; the escape of light constituents from the upper portions of planetary atmospheres. 1.2 BROWNIAN MOTION Although Brownian motion provides a seemingly very different approach for the description of non-equilibrium systems, i t has been shown that Brownian motion i s closely related to d i f f u s i v e processes [R23]. Einstein [R24] was able to explain Brownian motion [R25], the e r r a t i c motion of macroscopic p a r t i c l e s in a l i q u i d , by assuming that the macroscopic p a r t i c l e suffers many random c o l l i s i o n s with molecules of the l i q u i d . The net result is that the p a r t i c l e i s randomly forced to move in one or another d i r e c t i o n . This motion is that of a random walk. Although the exact trajectory of the p a r t i c l e cannot be predicted, the average distance a p a r t i c l e has moved at any given time can be ca l c u l a t e d . The position of a Brownian p a r t i c l e is an example of what is c a l l e d a stochastic v a r i a b l e . Stochastic variables 6 are variables which do not have a fixed value, but rather they have a range of probable values. Let the position of a Brownian p a r t i c l e be denoted by x and the v e l o c i t y by v. Then the probability that x has a value between x and x + dx and v has a value between v and v + dv is given by some probab i l i t y density function (PDF), P(x,v,t)dxdv. The PDF completely describes the motion of an ensemble of Brownian p a r t i c l e s . If the Brownian p a r t i c l e suffered no c o l l i s i o n s which randomized i t s v e l o c i t y , then i t s motion would follow a deterministic path. Equations for the PDF may be derived [R26-28] from th i s deterministic equation of motion plus some irregular force acting on the p a r t i c l e . These equations are either the Kramers equation or the Fokker-Planck equation. The generic name for equations for PDFs resulting from deterministic equations with random fluctuations is the Fokker-Planck equation. There are numerous physical systems [R26] which may be modeled as the di f f u s i o n of some Brownian p a r t i c l e . The description of the motion of Brownian p a r t i c l e s may be applied to chemically reactive systems. Kramers [R29] was the f i r s t to model a chemically reactive system as a p a r t i c l e moving in a potential with a b a r r i e r . When the p a r t i c l e crossed the potential barrier the system has undergone a reaction. The desired quantity was the rate of reaction determined in terms of the rate of passage of the p a r t i c l e s over the b a r r i e r . Kramers [R29] derived an equation describing the behaviour of the Brownian p a r t i c l e 7 c a l l e d the Kramers equation. The solution to Kramers equation gives the PDF of the p a r t i c l e in the potential well as a function of p a r t i c l e speed, position and time. If the c o l l i s i o n frequency is great enough th i s leads to two time scales for the relaxation of the d i s t r i b u t i o n function. The short time scale is for the relaxation of the velocity d i s t r i b u t i o n to the equilibrium d i s t r i b u t i o n whereas the longer time scale is for the relaxation of the position to the equilibrium d i s t r i b u t i o n . Kramers showed that the approach to equilibrium on the longer time scale is described by the Smoluchowski equation which can be derived from the Kramers equation with an equilibrium d i s t r i b u t i o n in v e l o c i t y . Later, researchers were interested not only in the rate constants of chemical reactions but also in the v a l i d i t y of the macroscopic rate law [R30-34]. In spite of the extreme complexity of such reacting systems, the rates of change of the concentrations of d i f f e r e n t species may usually be described by a simple rate equation. The rate equations are most often proportional to simple powers of the concentration of d i f f e r e n t chemical species. It is important to know under what physical condition these rate laws may be expected to f a i l . Chemically reactive systems are not the only systems which can be described by Kramers equation. Lasers are also systems which are far from equilibrium which have been treated by a number of researchers [R35-38]. B a s i c a l l y , one 8 has a number of atoms or molecules which are excited in such a manner that there i s an inversion in the population of two quantum states. Light, with a frequency corresponding to the energy difference between the two states causes an enhanced emission of l i g h t of the same frequency due to downward t r a n s i t i o n s . The f i n a l intensity of the emitted l i g h t depends on the rate that the upper quantum state is being replenished. One says that the upper state i s being pumped and the replenishment rate depends on a pump parameter. Some la s e r s , referred to as bimode l a s e r s , have the interesting property that they may operate at more than one intensity (of emitted light ) for a given pump parameter. Two questions with regard to bimode lasers naturally a r i s e . F i r s t , one i s interested in knowing at what i n t e n s i t i e s the laser w i l l operate for any given pump parameter. Second, i f the laser can operate at two i n t e n s i t i e s , what is the rate of switching from one intensity to the other? It i s interesting to note that the switching rate in a bimode laser i s analogous to a reaction rate. These questions are of interest not only from a the o r e t i c a l point of view, but also from a p r a c t i c a l point of view since i t has been proposed that bimode lasers may be used as computer switches. These i n t e n s i t i e s and times may be found by the solution of a Fokker-Planck equation for the system as discussed by Blackmore, Shizgal and Weinert [R39] and references therein. The intensity of laser l i g h t i s a stochastic v a r i a b l e . For any given system i t w i l l only have 9 a probable set of values. This i s a consequence of noise in the system. B i o l o g i c a l systems are prime examples of systems far from equilibrium, which can be modeled with stochastic methods. One problem which has been extensively researched is morphogenesis. Morphogenesis is the appearance of macroscopic b i o l o g i c a l structures. It has been explained in terms of competing chemical processes. On one hand there is an activator molecule which promotes the growth of a certain b i o l o g i c a l structure, while on the other hand there i s an inhibitor molecule which i n h i b i t s i t s growth. The r e l a t i v e concentration of these molecules control the rate of growth [R27], A set of coupled d i f f e r e n t i a l equations gives the concentrations of the activator and i n h i b i t o r molecules. These d i f f e r e n t i a l equations are generalized d i f f u s i o n equations and they have a form which i s similar to the Fokker-Planck equation. Using such phenomenological equations as these, i t has been possible to model the growth of leaves for example [R40]. Population dynamics can also be modeled using stochastic processes. Predator and prey populations are crudely modeled by assuming that the rate of change of the predator population increases with prey populations and decreases with predator populations. A similar r e l a t i o n holds for the rate of change of prey population. The rates of changes depend, of course, on many other f a c t o r s . These other factors are modeled by adding a randomly fluctuating 10 'force' to the equations for the populations. The populations then become stochastic v a r i a b l e s . This treatment has been applied to populations of plankton [R41] as well as to other populations [R42], Many non-equilibrium systems can be treated either as stochastic processes and modeled by a d i f f u s i o n type equation such as Kramers equation, or a transport problem based on the Boltzmann equation. In the next section I hope to show that these two approaches are, in the end, s t r i k i n g l y s i m i l a r . 1.3 MICROSCOPIC DESCRIPTIONS In this section, I w i l l point out the precise form of the equations referred to in the last section with other standard equations of s t a t i s t i c a l mechanics, and also discuss the rel a t i o n that they have among each other. Prior to beginning i t is useful to point out that these equations provide neither a complete microsocpic nor macroscopic description of the systems mentioned. A purely macroscopic description of a system would only relate macroscopic variables to each other via some phenomenological law, while a completely microscopic description of a system would provide detailed information on each of the p a r t i c l e s composing a system. To reach the middle ground, as exemplified by the Boltzmann, Kramers and Fokker-Planck equations, one may begin at either the microscopic or macroscopic leve l s and, with suitable approximations, obtain 11 similar equations for velocity d i s t r i b u t i o n functions or PDFs. Figure 1.1 diagramatically shows how to go from an N-body problem to equations of macroscopic v a r i a b l e s . As one moves down the diagram, the complexity of d e s c r i p t i o n , in terms of the number of degrees of freedom, i s reduced. So l i d lines with arrows indicate that, with well defined approximations, the equation with the arrow pointing toward i t can be derived from the other equation. Dashed l i n e s , on the other hand, indicate that two equations have close mathematical s i m i l a r i t y . It i s important to note that there are, broadly speaking, three leve l s of de s c r i p t i o n . The f i r s t l e v e l i s that of N interacting p a r t i c l e s (N being on the order of 1 02 3) . The second le v e l is that of the description of one ty p i c a l p a r t i c l e in terms of i t s PDF. In this level the microscopic nature of the system i s s t i l l recognized but is only approximately treated since, to solve rigorously for the PDF of a t y p i c a l p a r t i c l e , the motions of the rest of the p a r t i c l e s must be taken into account. This i s the l e v e l of approximation that w i l l be used in the description of the physical systems in this t h e s i s . F i n a l l y the lowest le v e l i s that of a purely macroscopic d e s c r i p t i o n , i . e. a phenomenological de s c r i p t i o n . A natural starting point for the description of a macroscopic system i s with the microscopic p a r t i c l e s which compose i t . The f u l l N-body quantum mechanical description of an evolving system is provided by the Von Neumann FIGURE 1.1 N-part icle Von Neumann Equation N-particle Liouvilie Equation Equations for n-particle PDFs Master Equations for One-particle PDFs Boltzmann Equat ion Fokker-Planck Equations Stochastic Equations of Motion Hydrodynamic Equat ions Equations of Motion for a Particle Approximations to go from the N-body problem to equations for single p a r t i c l e PDFs. 1 3 equation [ R 2 5 ] , where the brackets indicate a commutator, H i s the to t a l quantum mechanical Hamiltonian for the system and p is the density operator. This equation is the sta r t i n g point for many time dependent treatments of non-equilibrium systems [ R 4 3 ] . One very useful approximation i s to consider the system c l a s s i c a l l y . This leads to the c l a s s i c a l L i o u v i l l e equat ion [ R 4 3 ] , = {H,PM}, ( 1 . 2 ) at N where the curly brackets indicate a Poisson bracket. H i s the c l a s s i c a l Hamiltonian and P ^ * ! 'X1 '2L2'-2' * ' *-N'-N' FC *D-3 1D-3 1 ' ' 'D^3ND-3N ' T H E N-particle d i s t r i b u t i o n function, which gives the probab i l i t y that p a r t i c l e 1 has a position between x and xl+d3x1 and a velocity between v and v_1+d3v1, and with the remainder of the p a r t i c l e s being s i m i l a r l y d i s t r i b u t e d , is the c l a s s i c a l equivalent of the density operator [ R 4 3 ] . The L i o u v i l l e equation is d i f f i c u l t to solve since i t i s an equation of many v a r i a b l e s . The N-particle d i s t r i b u t i o n function provides far more information than is necessary to give a complete macroscopic description of any system. For this reason i t is convenient to define reduced p a r t i c l e d i s t r i b u t i o n functions, pn^2L'Y-T >x_2 '— 2 '" " "— n '— n ' fc ^ " H e r e 1 integrate out 6(N-n) degrees of freedom leaving a d i s t r i b u t i o n function of 6n + 1 14 vari a b l e s . Thus Pn = ;.../PNd3xN...d3vn+1 ( 1 . 3 ) where the arguments of the functions are l e f t out for convenience. Various schemes for deriving equations for the Pn have been presented [R44-46]. These are important in that they provide equations which are more amenable to analytic and numerical solution than the L i o u v i l l e equation. The f i r s t few Pn are often adequate to provide a complete macroscopic description of a system. P^  , which w i l l hereafter be referred to as P(x,v,t) and c a l l e d a PDF or a velocity d i s t r i b u t i o n function interchangeably, provides s u f f i c i e n t information to calculate the density, bulk ve l o c i t y and other macroscopic quantities for s u f f i c i e n t l y d i l u t e systems. Other Pn are also important in the description of both equilibrium and non-equilibrium systems. For instance, P2 is related to the two p a r t i c l e c o rrelation function and plays an important role in the explanation of neutron d i f f r a c t i o n experiments and the theory of l i q u i d s [R47]. The equation obtained from the L i o u v i l l e equation for P(x,v,t) i s none other than the Boltzmann equation mentioned in the last section in conjunction with transport problems [R20], The Boltzmann equation is given by F £f + v-V P + --VP = JP (1.4) at — —x m —v where J(P) i s the rate of change of the d i s t r i b u t i o n function due to c o l l i s i o n s and F i s some external force. 1 5 The right hand side of Eq.(1.4) describes the rate of change of the d i s t r i b u t i o n , function as a result of p a r t i c l e d r i f t . It should be noted that in going from the L i o u v i l l e equation to the Boltzmann equation some approximations must be made. These include the assumption that two p a r t i c l e correlations die out on a time scale much shorter than the average length of time between c o l l i s i o n s . The solutions to thi s equation are, as a matter of convention denoted by f(x,v,t) instead of P(x,v,t). The precise form of the Boltzmann equation depends on the a p p l i c a t i o n . These applications can be broken into two broad groups consisting of an isolated gas and a d i l u t e l y dispersed gas in a medium. In the l a t t e r case, assuming that the medium i s at thermal equilibrium, the c o l l i s i o n operator given in Eq.(1.4) can be written in the form JP(x,v,t) = fK(v,v')P(x,v',t)d3v' - (1.5) v(v)P(x,v,t) where K(v,v') i s the c o l l i s i o n kernel [R48] and v(v) is the c o l l i s i o n frequency. Eq.(1.4) with the c o l l i s i o n operator defined by Eq.(l.5) i s sometimes c a l l e d the linear Boltzmann equation. This equation in par t i c u l a r has been used to obtain a solution to the Milne problem [R21] mentioned in the last s e c t i o n . The form of the c o l l i s i o n operator for the isolated gas is given by [R48] Jf(x,v,t) = / / ( f ' f ; - f f , ) g a ( g , x ) d2x d3v1, (1.6) where the d i s t r i b u t i o n function f for a p a r t i c l e i s changed 16 by a l l possible c o l l i s i o n s between i t and a tagged p a r t i c l e 1; the primes indicate the values of the d i s t r i b u t i o n functions after a c o l l i s i o n , g i s the r e l a t i v e speed of the p a r t i c l e s , x is the angle which the r e l a t i v e v e l o c i t y vector was rotated after a c o l l i s i o n and a i s the d i f f e r e n t i a l scattering cross section. As previously mentioned, hydrodynamic equations such as the Navier-Stokes equations [R48] follow d i r e c t l y from the Boltzmann equation. These are obtained by multiplying Eq . ( l . 4 ) , with J defined by Eq.(1.6), by 1, v or --krw2 and integrating over v. The integral over the c o l l i s i o n operator w i l l contribute nothing in each of these integrations since the number density, bulk flow and kinetic energy are unchanged with c o l l i s i o n s . Another l o g i c a l starting point for the description of macroscopic systems is with macroscopic equations. Consider the equation of motion for a macroscopic p a r t i c l e moving under the influence of a force F, d2x(t) m--jp— = F. (1.7) Eg.(1.7) i s merely a statement of Newton's second law of motion [R49]. If the i n i t i a l p o s ition and speed of a macroscopic p a r t i c l e is given, then the postion at any subsequent time is uniquely given by the solution to Eq. ( 1 . 7 ) . The molecular nature of a macroscopic system can be modeled by adding a randomly fluctuating force to Eq.(l.7). 1 7 This results in the Langevin equation, dx(t) = F + lit), (1.8) dt where £(t) is the randomly fluctuating force and x is the vel o c i t y of the p a r t i c l e . The position variable in the Langevin equation i s a stochastic variable since, with the randomly fluctuating force, i t cannot take on def i n i t e values. The solution of the Langevin equation gives the d i s t r i b u t i o n of the possible p a r t i c l e positions and v e l o c i t i e s . j[(t) must s a t i s f y the two following properties i f i t is to be physically reasonable. F i r s t , averaged over a long period of time, the fluctuating force yields no net force and, second, the average of the square of the fluctuating force i s a f i n i t e constant. Kramers showed that an equation for the PDF of the vel o c i t y and position of the macroscopic p a r t i c l e may be obtained from Eq.(l.8) [R29]. The force, F, was assumed to have the form given by F(x,x) = -VV(x) - vk, where v is the f r i c t i o n c o e f f i c i e n t and V(x) i s an external p o t e n t i a l . In addition Kramers assumed that the random force was Gaussian white noise [R27], This equation is c a l l e d Kramers equation and, in the one dimensional case, i s given by where v is the f r i c t i o n c o e f f i c i e n t . It is important to point out that t h i s is precisely the one dimensional Boltzmann equation with the Rayleigh form of the hard sphere F cIP m 3v kT 9_ m 3v (1.9) 18 c o l l i s i o n operator [R50] even though the approximations used to obtain these equations are vastly d i f f e r e n t . The Langevin equation, Eq . ( l , 8 ) may be rewritten as mv = F(v) + lit) (1.10) where v=x and the force i s assumed to be position independent. If the PDF i s position independent and there is no external p o t e n t i a l , then the Fokker-Planck equation for the PDF as a function of v alone may be derived [R26], The most general form of the one dimensional Fokker-Planck equation i s given by 9P(v,t) _ 9A(v)P(v,t)^92B(v)P(v,t) , , n, — 3 t dv o^P ' U.1U where A(v) and B(v) are generally referred to as the d r i f t and d i f f u s i o n c o e f f i c i e n t s , respectively. The e x p l i c i t forms of A(v) and B(v) vary considerably and depend on the par t i c u l a r application considered. It should be noted that Fokker-Planck equations may arise from physical considerations other than the ones considered here, and, in other app l i c a t i o n s , the dependent variable may be the position instead of the v e l o c i t y . It should be noted that the Kramers equation, E q . ( l . 9 ) , is often c a l l e d the Fokker-Planck equation, but there is no standard usage for t h i s term. I w i l l reserve t h i s term for Eq.(1.11) as is done by Risken [R26]. A f i n a l equation, the Smoluchowski equation, arises when the c o l l i s i o n frequency, v, in Kramers equation, Eq.(1.9), is large [R29]. The v e l o c i t y portion of the 19 d i s t r i b u t i o n function is almost Maxwellian or w i l l become nearly Maxwellian on a time scale much shorter than the change in the position v a r i a b l e . Thus, the velocity dependence may then be integrated out to leave a Fokker-Planck equation with B equal to a di f f u s i o n constant and A(x) equal to the derivative of the external p o t e n t i a l . This equation has the form 9P(x,t) _ ^92P(x,t) , D dV (x)P(x,t) ,, ,,* ~ T E ~ ° 9 x^ + kT oH  ( K 1 2 ) where D is a d i f f u s i o n constant and V i s a potential energy function. In t h i s section, I have t r i e d to give the precise forms of the equations mentioned in the last section and to explain their r e l a t i o n s h i p . These equations arise from two sources, on one hand, the N-body quantum system may be reduced to an equation for a one p a r t i c l e d i s t r i b u t i o n function by removing many degrees of freedom while, on the other hand, macroscopic equations of motion may be modified so that the molecular nature of systems is approximated. In each case an equation for a PDF i s the r e s u l t . Furthermore these equations have a s t r i k i n g mathematical s i m i l a r i t y . 1.4 MODEL PROBLEMS The three equations, the Boltzmann equation, Eq.(1.4), Kramers equation, Eq.(1.9), and the Fokker-Planck equation, Eq.(1.11), are thus intimately related and, together, provide the description of many physical processes and 20 phenomena. These phenomena include a l l of the examples of the non-equilibrium systems introduced in the beginning of this introduction. These equations are, in general, d i f f i c u l t to solve. The numerical d i f f i c u l t i e s can be overcome in one of two ways. The f i r s t way is to solve problems which share some of the desired c h a r a c t e r i s t i c s of the problem one i s interested i n . The other way is to develop specialized methods for dealing with these problems [R51-56]. I w i l l use both options in t h i s t h e s i s . Physical systems are usually too complex to treat in a completely rigorous manner as has been suggested. It i s possible to understand a complex macroscopic system by solving one of the previous equations for the PDF for the system. However, even for these s i m p l i f i e d systems, i t is often very d i f f i c u l t to solve for the PDF of a real system. In such a case i t is often very useful to consider model systems. Model systems are idealized systems. For example, a chemical reaction may be idealized as a p a r t i c l e moving in a double well p o t e n t i a l . This potential may be further idealized as some very simple analytic function. The usefulness of model systems i s that they may often be understood when real systems present numerical d i f f i c u l t i e s in their treatment [R57-59]. In addition, models are often chosen so that they have a simple dependence on some set of parameters. In the chemical kinetic model a parameter may change the height of a potential b a r r i e r , for example. This dependence may be both 21 simple enough to understand and yet s u f f i c i e n t l y r e a l i s t i c to give some insight into the behaviour of physical systems. 1.5 PROBLEMS CONSIDERED I w i l l consider three applications of the equations discussed. One each for the Boltzmann equation, Kramers equation and the Fokker-Planck equation. The broad outline of the thesis i s to f i r s t develop specialized methods for dealing with these equations. Later, these methods w i l l be applied to a model Fokker-Planck problem, a chemical kinetic problem and, f i n a l l y , to a transport problem. 1.5.1 CHEMICALLY REACTIVE SYSTEMS Although chemical systems are complicated by many degrees of freedom, i t is often possible to model a system using only a subset of these degrees of freedom. For example, t r a n s i t i o n state theory [R60] follows a reaction along what is c a l l e d the reaction coordinate. The reaction coordinate is some combination of molecular coordinates such that this variable changes as the reaction proceeds. This coordinate is also usually defined such that i t follows a minimum in the potential energy funtion. An example is the difference between the carbon-bromine distance and the carbon-chlorine distance in a chloride displacement by bromide in methyl c h l o r i d e . This i s very useful since i t allows the reaction to be modeled as a function of one coordinate 22 only. A chemical reaction may be followed in d e t a i l by following the PDF for the system as a function of time and the reaction coordinate. If the reaction coordinate has a p a r t i c u l a r range of values, then the reactant molecule i s present, otherwise the product molecule is present. In the example mentioned previously, i f the reaction coordinate i s p o s i t i v e then we have methyl bromide and i f i t is negative then methyl chloride is present. The PDF, P(x,t)dx, gives the pr o b a b i l i t y that the reaction coordinate, x, w i l l take on a value between x and x + dx. Integrals over appropriate ranges give the f r a c t i o n of molecules which are reactant or product molecules. I w i l l treat the simpler transformation between the trans and gauche conformations of n-butane in this t h e s i s . In the case of n-butane, the reaction coordinate i s the angle of rotation about the central bond. The PDF for t h i s system is obtained from the solution of the Kramers equation or the Smoluchowski equation in the high c o l l i s i o n frequency l i m i t . This problem has been studied by many researchers [R61-65], primarily since the state of n-butane depends only on two degrees of freedom, the angle of rotation about the c e n t r a l bond and the angular ve l o c i t y about th i s bond. Hence, the analysis of the chemical system i s s i m p l i f i e d . 23 The methods used to solve Kramers equation have included orthonormal function expansions of the velocity portion of the PDF as done, for example, by Risken [R26]. These functions are normally Hermite polynomials for one dimensional problems since, in t h i s case, the c o l l i s i o n portion of the operator, the right hand side of E q . ( l . 9 ) , is diagonal in these functions. This expansion then leads to a set of coupled equations for the expansion c o e f f i c i e n t s of the d i s t r i b u t i o n function, which may be solved by continued fractions [R27,66]. The solution may also be found as an expansion in terms of the eigenfunctions of the operator on the right hand side of Eq.(1.J2). This leads to the problem of-determining these eigenfunctions. This is an especially useful approach since the corresponding eigenvalues describe the decay rates of fundamental modes of the system. The solution to the Kramers equation for n-butane isomerization i s complicated by the fact that the force term i s not a simple function of the angle of r o t a t i o n . Although t h i s leads to d i f f i c u l t i e s when standard methods of solution are used, numerical methods are developed in this thesis which are applied to the Kramers equation in a natural and e f f i c i e n t manner. The results of the present determination of the rate constant w i l l be compared with the results of two recent publications. Marechal and Moreau (MM) [R30] 24 have studied the kinetics of this system in terms of the f i r s t few eigenvalues of the Smoluchowski operator, that i s , the high c o l l i s i o n l i m i t of Kramers operator. They transformed the Smoluchowski equation into the form of a Schroedinger equation and used the WKB approximation to find the corresponding eigenfunctions and eigenvalues. Montgomery, Berne and Chandler (MCB) [R31] considered this system from a s l i g h t l y d i f f e r e n t point of view. They studied this problem by gathering s t a t i s t i c s on the t r a j e c t o r i e s of many p a r t i c l e s moving under the influence of an approximate intramolecular p o t e n t i a l . The compiled s t a t i s t i c s were used to compute the derivative of a time autocorrelation function. In a previous paper Chandler [R67] showed how this c o r r e l a t i o n function can be related to the relaxation time and the rate c o e f f i c i e n t . In Chapter 4, I w i l l solve this problem in both the high c o l l i s i o n l i m i t and with the Kramers equation. It w i l l be shown that the f i r s t non-zero eigenvalue of the Kramers operator and the Fokker-Planck operator may be i d e n t i f i e d with the reciprocal of the relaxation time for t h i s isomerization process. The relaxation time w i l l be calculated as a function of c o l l i s i o n frequency and temperature. It w i l l also be shown under what condition the f i r s t order phenomenological law for isomerization may be considered to be v a l i d . This phenomenological law takes the form of a f i r s t order 25 rate law (1.13) where C is the difference between the concentration of the gauche isomer and i t s equilibrium concentration at time t . F i n a l l y the present results w i l l be compared with those obtained by MM and MCB 1.5.2 DIFFUSION IN A DOUBLE WELL POTENTIAL The chemical isomerization problem i s treated using Kramers or the Smoluchowski equation with a double well p o t e n t i a l . As was previously noted, the applications of these equations with double well potentials has wide a p p l i c a b i l i t y . These applications include the bimode laser [R39], and population dynamics [R42]. It is useful to consider the solution of a Fokker-Planck equation with a model double well p o t e n t i a l . One of these model Fokker-Planck equations is defined with the c o e f f i c i e n t s , where the independent variable has been changed from v to x and for which A(x) is nonlinear in such a way that the equilibrium solution may possess two states, that i s , P(x,°°) i s bimodal. For such systems, there is the p o s s i b i l i t y for the time dependent solution to exhibit a A(x) = gx3-ax, (1 .14a) B(x) = e, (1.14b) 26 bifurcation and the solution w i l l in general depend on the i n i t i a l condition. As mentioned previously, Fokker-Planck equations of the general form given by Eq.(1.11) are employed to model the behaviour of such systems. It is important to note that an analysis based on the FPE is one of many methods employed in the study of such systems and, for some cases, i t may not be the appropriate d e s c r i p t i o n . This FPE has been considered by many authors in a study of the role of fluctuations in systems far from equilibrium and the subsequent evolution of such systems [R61-65]. The main interest in thi s thesis is to obtain numerical solutions for t h i s system for a wide range of values of the parameters in Eq.(1.14). It is useful to note that in these studies, 1/e is i d e n t i f i e d as the system size parameter and e is then a measure of the fluctuations in the system. If e = 0, a nonequi1ibriurn state w i l l relax d e t e r m i n i s t i c a l l y to i t s equilibrium state, that i s , i f macroscopic variables i n i t i a l l y have a determinate value, then at any subsequent time they w i l l have determinate values. As e becomes larger, the fluctuations tend to dominate and the variance of the PDF, P(x,t), becomes large. Several workers have sought numerical and semianalytical solutions to Eq.(1.11) with A(x) and B(x) defined by Eq.(l.14). Among the methods used are scaling theory, [R68] which i s based on a nonlinear 27 transformation of the FPE and i s v a l i d for e — > 0 . Gaussian decoupling [R69] i s another method that applies in this l i m i t and employs a gaussian approximation of the solution at each time. Recently, Indira et a l [R64] have employed a f i n i t e element method and a Monte Carlo simulation to solve this FPE. They compared their e s s e n t i a l l y exact results with these approximate methods and suggested that the scaling theory i s the most accurate method in the e—>0 l i m i t . 1.5.3 ESCAPE OF AN ATMOSPHERE The Boltzmann equation can also be used to calculate the rates of chemical reactions [R70], Instead of treating a chemical reaction as a d i f f u s i o n process i t i s treated as a scattering problem. The number of reactions is related to the frequency of c o l l i s i o n s and the fraction of these c o l l i s i o n s leading to a reaction. The precise r e l a t i o n for the time dependent rate, R ( t ) , is given by R(t) = N / / f1( v , t ) f ( y / , t ) a ( v - v ' ) | v - v ' | d3v d3v ' (1. where a is the t o t a l reactive cross section [R5]. Eq.(1.l5) merely sums up, for a l l possible c o l l i s i o n s , the probability that a reaction w i l l occur times the frequency of such c o l l i s i o n s . A rigorous treatment of such systems would involve solution of the Boltzmann equation for the velocity d i s t r i b u t i o n function. 28 In the present work, the Boltzmann equation w i l l not be applied to chemical reactions, but rather to the escape of l i g h t contituents from planetary atmospheres. Although there i s no reaction involved, this problem i s similar to that of a chemically reactive system. In some chemically reactive systems, the reactive cross section i s zero below some value of the r e l a t i v e velocity c a l l e d the threshold v e l o c i t y . The problem of the escape of a li g h t constituent from an atmosphere is similar in that i t might escape only i f i t has velocity above the escape v e l o c i t y . In the upper atmospheres of Earth, Mars, and various s a t e l l i t e s [R19], a portion of the hydrogen atoms have v e l o c i t i e s in excess of the escape v e l o c i t y . This hydrogen w i l l escape i f i t is traveling away from the planet and i t suffers no further c o l l i s i o n s with other atmospheric p a r t i c l e s . Helium w i l l also escape but the escape flux is much smaller since i t s average velocity is much smaller as a consequence of having a greater mass. A useful model of the upper atmosphere i s to consider i t as a two component [R18] system (on earth H and 0 atoms) with the v a r i a t i o n of the density of the heavy component given by the barometric formula. The barometric formula gives the density of each component as an exponentially decreasing function. The reciprocal of the decay constant is c a l l e d the scale height. Each component of the atmosphere has a different scale 29 height, which increases with decreasing mass. Phy s i c a l l y , this just says that the heavy components of the atmosphere sink while the li g h t e r ones r i s e . In addition, the heavy component i s assumed to be in thermal equilibrium and hence i t s vel o c i t y d i s t r i b u t i o n function is given by the Maxwell-Boltzmann d i s t r i b u t i o n . The vel o c i t y d i s t r i b u t i o n function of the l i g h t component i s then calculated from the Boltzmann equation. A standard model of the [R18] upper atmosphere i s based on the complete lack of c o l l i s i o n s between atoms above a certain a l t i t u d e , c r i t i c a l l e v e l , r , while below t h i s l e v e l the atmosphere i s supposed to be c o l l i s i o n dominated. This model is rather a r t i f i c i a l since i t assumes discontinuous behaviour of the atmosphere. In a real atmosphere, the t r a n s i t i o n from the c o l l i s i o n dominated case to the c o l l i s i o n l e s s case is gradual. This standard model of the upper atmosphere leads to an approximation of the escape flux as the Jeans flux [R71]. This flux i s obtained by assuming that at some exobase the ve l o c i t y d i s t r i b u t i o n function i s Maxwellian and p a r t i c l e s moving upwards with speeds greater than the escape speed w i l l suffer no further c o l l i s i o n s . This overestimates the escape flux since the velocity d i s t r i b u t i o n function is perturbed by the escaping p a r t i c l e s and hence there should be fewer high speed p a r t i c l e s than i s given by the Maxwellian d i s t r i b u t i o n . 30 There have been many calculations of the correction to the Jeans flux [R72-75], most of which have been based on Monte-Carlo c a l c u l a t i o n s . The Jeans escape flux i s solely a result of thermal escape. There has also been interest in cal c u l a t i n g the escape flux due to non-thermal mechanisms [R21,76]. The PDF of the li g h t atmospheric component in the tra n s i t i o n region may be calculated with the Boltzmann equation. This w i l l y i e l d P ( X , V , M ) where x i s the a l t i t u d e , v is the speed and u i s the cosine of the angle between the velo c i t y and zenith. The number density is eas i l y calculated by integrating over the veloc i t y and the escape flux i s determined by integration of the d i s t r i b u t i o n function over v e l o c i t i e s greater that the escape v e l o c i t y . These quantities w i l l be calculated in Chapter 5 and compared with the values contained in the primitive model. Many non-equilibrium processes may be studied using one of three very similar equations. One objective of this thesis i s the development of methods by which these problems may be solved numerically. The numerical and analytic techniques which are used to solve the foregoing problems are developed in Chapter 2. The model Fokker-Planck equation is considered in Chapter 3. Chapter 4 is concerned with the butane isomerization problem in CC1*,. The escape problem is dealt with in Chapter 5. F i n a l l y , there are three Appendices. A 31 general numerical method for solving d i f f e r e n t i a l equations and eigenvalue problems is developed in the f i r s t one. This method i s applied to the Fokker-Planck, Kramers and Boltzmann equations in Chapter 2. The recurrence relations for two sets of non-classical polynomials are developed in the other Appendices. 2. NUMERICAL METHODS FOR STOCHASTIC EQUATIONS In the f i r s t chapter, i t was shown that many problems of non-equilibrium s t a t i s t i c a l mechanics may be dealt with using either the Boltzmann, Kramers, or Fokker-Planck equation. In thi s chapter, I wish to develop some of the numerical techniques which w i l l be used to solve these equations. These techniques are particular applications of the discrete ordinate method (DO) of solving d i f f e r e n t i a l and/or integral equations, and eigenvalue problems developed recently by Shizgal and Blackmore [R77]. The DO method, developed s p e c i f i c a l l y for the problems in t h i s t h e s i s , although applicable to many other problem, is discussed in d e t a i l in Appendix A. 2.1 INTRODUCTION There is no general way to solve Eqs.(1.4), (1.9) and (1.11). Some of the techniques for solving these equations were given in the introduction. These methods f a l l - i n t o a number of broad categories. The f i r s t i s Monte-Carlo type calculations where the t r a j e c t o r i e s of a large number of p a r t i c l e s are followed. A crude PDF is calculated from the ra t i o of p a r t i c l e s with various ranges of position and vel o c i t y to the t o t a l number of p a r t i c l e s . It should be noted that when Monte-Carlo techniques are used, the forms of the equation to be solved are different than the ones mentioned before. For example, the Fokker-Planck equation i s not solved, rather the corresponding Langevin equation i s 32 33 solved. These techniques have the advantage that p r a c t i c a l l y any equation that may be written down may be solved, provided that enough t r a j e c t o r i e s are followed. Their great disadvantage i s that these are very costly calculations since i t i s necessary to follow many p a r t i c l e s (on the order of hundreds of thousands) to obtain a precise d i s t r i b u t i o n . The second class are semi-analytic approximations of the sol u t i o n , including, for example, scaling theory. Scaling theory [R68] i s used to find approximate solutions of the Fokker-Planck equation. This method e n t a i l s finding some non-linear variable change which w i l l approximately transform the Fokker-Planck equation into the ' l i n e a r ' Fokker-Planck equation, i . e. the c o e f f i c i e n t A(v) in Eq.(1.11) i s l i n e a r . The lin e a r equation i s solved and the inverse variable transformation is made. . These approximations have the disadvantage that i t is d i f f i c u l t to estimate the magnitude of error they introduce. Another commonly used method for solving these equations is to represent the solution as a f i n i t e sum of orthogonal functions. This r e s u l t s in a set of linear ( d i f f e r e n t i a l ) equations which are then solved by standard methods such as Runge-Kutta integration. Other methods include f i n i t e difference approximation of the derivatives [R64]. This also leads to a set of coupled equations which are often s t i f f , and hence numerically unstable. The Kramers, Fokker-Planck and Boltzmann equations may a l l be written in the form ; 34 (2.1 ) or, for steady state s i t u a t i o n s , LP - 0 , (2.2) plus the associated boundary and/or i n i t i a l conditions. In Eqs.(2.1) and (2.2), L i s the sum of the d r i f t and c o l l i s i o n operators of the Boltzmann equation, and the d r i f t and d i f f u s i o n operators of the Fokker-Planck equation. Eq.(2.1) describes the time evolution of a PDF, given an i n i t i a l condition while the solution to Eq.(2.2) gives the steady state PDF of some system. The outline of the method used to solve Eqs.(2.1) and (2.2) i s as follows. The discrete ordinate method, developed in Appendix A, w i l l be applied to the solution of these equations. The basic idea i s to develop a matrix representation of the derivative operator. This i s used to construct an approximate representation of the d i f f e r e n t i a l operator L in Eq.(2.2). The approximate d i f f e r e n t i a l operator matrix is multiplied with a vector, whose elements are proportional to a function evaluated at a set of points. The r e s u l t i n g vector i s approximately the result of the d i f f e r e n t i a l operator applied to the function evaluated at the same set of points. In the case of Eq.(2.l), t h i s gives a set of coupled f i r s t order linear d i f f e r e n t i a l equations while in the case of Eq.(2.2), t h i s gives a set of coupled algebraic equations. Standard techniques for solving such sets of equations are used to solve them. The DO 35 representation is useful in that the order of the matrix approximation to L i s small enough that the resu l t i n g algebraic and f i r s t order d i f f e r e n t i a l equations may be solved in a reasonable amount of computer time and are numerically stable. This present method of solving these equations is a hybrid method. On one hand, i t approximates the d i f f e r e n t i a l equation by approximating the derivative operator as is done with f i n i t e difference c a l c u l a t i o n s . On the other hand, i t i s c l o s e l y related to the representation of the d i f f e r e n t i a l equation in some basis of orthonormal functions. In both of these cases one is led to a set of coupled equations. I w i l l begin with the case for which L i s a sel f - a d j o i n t operator. In general, neither the Kramers or the Boltzmann operator (the c o l l i s i o n plus d r i f t operators) is s e l f - a d j o i n t , however the Fokker-Planck equation given by E q . ( 1 . 1 1 ) i s s e l f - a d j o i n t provided the scalar product is properly defined. This i s a nice starting point because the matrix approximation to L i s symmetric and there are many standard methods for solving the resulting sets of f i r s t order algebraic equations. 2.2 SOLUTION OF THE FOKKER-PLANCK EQUATION I begin the development with the standard eigenfunction expansion of P(x,t). Later, I develop a technique to determine numerically these eigenfunctions and corresponding 36 eigenvalues. In p a r t i c u l a r , FP equations with bimodal stationary solutions are considered, although, the present method i s by no means limited to such equations. I seek solutions to Eq.(l.1l) which s a t i s f y the boundary condition P(x,t)—>0 as x — ^±» , for a l l of the models considered in th i s t h e s i s . Eq.(1.11) may be rewritten as 9 P* * 'U = LP(x,t), (2.3) where the v.variable has been replaced by x and where ~ _ 6A(x)P a2B ( x ) P ( . is the FP operator operating on a PDF. The stationary solution of Eq.(1.11) is assumed to exist and i s given by P0(x) = Nexp(-/|i^4-dx'-ln(B(x) ) ) , (2.5) 0 v ; where N i s a normalization constant such that /P0(x)dx=1. — oo P0(x) plays an important role in the.standard eigenfunction expansion since the eigenfunctions are orthonormal with the weight function P51(x). The formal solution of Eq.(2.3) i s given by P(x,t) = eL tP(x,0), (2.6) where P(x,0) i s the i n i t i a l p r obability density function. 2.3 EIGENFUNCTION EXPANSION The formal soluti o n , Eq.(2.6) may be evaluated by expanding the i n i t i a l d i s t r i b u t i o n function in eigenfunctions of L and then applying eL t term by term. 37 Thus, i f Pn(x) are the eigenfunctions of L, defined by LP (x) = -A P (x), (2.7) n n n I have the expansion OO P(x,0) = L anPn( x ) , (2.8) where the expansion c o e f f i c i e n t s , an, are given by oo an = ; 0 ( x ) P ( x , 0 ) . (2.9) — 00 The functions 0 in Eq.(2.9) are related to P by n ^ n J P_(x) * n( x ) = P T T i T ' ( 2-l 0 ) and with Eqs.(2.7) and (2.10), I get from Eq.(2.6) the eigenfunction expansion' 00 P(x,t) = I aexp(-X t ) P „ ( x ) . (2.11) A n c n n n = 0 Eq.(2.11) is the standard eigenfunction expansion as discussed often in the l i t e r a t u r e . It is a useful description in that the reciprocals of the eigenvalues are the fundamental relaxation times of the system and the nature of the eigenvalue spectrum governs the approach of the system to equilibrium [R78,79]. It i s useful to note that since ^n>0, n>0 and Xo = 0, then P(x,t)—s>-P0(x) as t — z - ° = . The task, then, i s to find the eigenfunctions and corresponding eigenvalues of Eq.(2.7). It i s convenient to consider the transform of L of the form S"1(x)LS(x) where S(x) i s some positive d e f i n i t e function. The eigenfunctions of the transformed operator are 38 P (x)/S(x). Two of these transformations have been widely used. The f i r s t i s defined by S=P0/ giving the operator defined by L = p-0 1LP0 . (2.12) The eigenfunctions of L, 0n( x ) , are given by Eq.(2.l0) and the eigenvalue equation, L0n=-Xn0n, is given e x p l i c i t l y by d0 (x) d20 (x) -A ( x )- W -+ B ( x )~ d ^ - = "Vn( x )' { 2'1 3 ) where Eq.(2.12) has been used. With the d e f i n i t i o n Eq.(2.12), or the e x p l i c i t form Eq.(2.13), i t i s clear that L i s self - a d j o i n t with the scalar product defined with P0(x) as the weight function. From a computational standpoint, i t is more convenient to consider Eq.(2.13) than Eq.(2.7) since 0n(x) are more slowly varying than ?n(x) and more eas i l y evaluated (compare 0O = 1 and P0( x ) , Eq.(2.5)). In addition, i t i s easier to obtain a symmetric representation of L than L, since in the former case a basis set orthonormal over P0(x) i s required while in the l a t t e r case a set of functions orthonormal over P01 i s needed, which is much more d i f f i c u l t to generate. The second widely used transformation i s defined by L = P ;1 / 2L P J/ 2. (2.14) This form of the operator is self-adjoint with the scalar product defined with unit weight function. In p a r t i c u l a r , for systems for which B(x)=e, the eigenvalue problem may be cast into the form of a Schroedinger equation. The 39 eigenf unctions, <$> n' sat i sfy e0n-[V(x)-XnUn = 0, (2.15) where the p o t e n t i a l , V(x), i s given by V(x) = 2 2e dx (2.16) This i s especially u s e f u l , since many techniques have been developed in quantum mechanics to f i n d the eigenvalues and eigenfunctions of Eq.(2.l5). Conversely, the techniques developed in this thesis may be applied to problems of the form Eq.(2.15). These are not the only transformations of the operator which are use f u l . In the next section i t w i l l be shown how to find a transformation, S(x), for which a se l f - a d j o i n t operator i s defined with respect to an arbitrary weight function. With such a transformation, the DO method developed in Section 2.4, becomes very f l e x i b l e . 2.4 THE DISCRETE ORDINATE METHOD The discrete ordinate method [R77], developed in Appendix A, i s employed to determine the eigenfunctions and eigenvalues of the Fokker-Planck operator. This method consists of representing functions, in par t i c u l a r the eigenfunctions and d i s t r i b u t i o n functions, as column vectors, t_, whose N elements are f.=/w.f(x.). The points, 40 {x^}, are the quadrature points and {w^}, are the corresponding quadrature weights of some integration r u l e . This quadrature rule is defined by oo N-1 / w(x)f(x)dx * I w.f(x.), (2.17) i = 0 1 1 where f(x) i s some function defined on the i n t e r v a l (-00, 0 0) , and w(x) i s a suitable weight function. If f(x) i s a polynomial of degree 2N-1 or l e s s , then Eq.(2.17) i s exact [R80]. The points, {x^}, are the zeros of the Nth order polynomial, R^, of the set of polynomials orthonormal with respect to the weight function w(x), that is 00 f w(x)R R dx = 6 . (2.18) J n m nm — oo It was also shown [R77], provided f(x) is a polynomial of degree less than N, that the representation of the function f(x) described above is equivalent to representing i t as the vector of i t s expansion c o e f f i c i e n t s , f ^ , in the Rn polynomial basis. The N components of fP are defined by 00 fP = / w(x)R (x)f(x)dx. (2.19) n ' n — 00 The equivalence of these representations is given by unitary transformation between the two b a s i s , that i s , f = TT- fp, (2.20) where the elements of the matrix T are T . = R (x . )i/w . . (2.21) nD n J J 41 The basis of the DO method is the representation of the derivative operator, d/dx, in the discrete space defined by the points in the quadrature r u l e , Eq.(2.17). This is eas i l y done by using the transformation, T, to transform the derivative operator from the polynomial basis to the DO basis, thus P = TT'DP-T, (2.22) P . where D is the polynomial representation of the derivative operator and D is the DO representation of t h i s operator. The operator p so constructed s a t i s f i e s [R77] f ' = D-f, (2.23) which represents a high order algorithm for numerical d i f f e r e n t i a t i o n based on quadrature weights and points. 2.5 DO APPROXIMATION TO SELF ADJOINT OPERATORS The matrix representative of d i f f e r e n t i a l operators can be written in a simple fashion by replacing derivatives with D, and functions with their values at the set of points {x^}. Although one can proceed in thi s way with the solution of the eigenvalue problem, Eq.(2.13), i t i s useful to introduce a second set of functions, defined by 1/2 Q (x) = ^n w (x) Rn(x), (2.24) Po (x) which are orthonormal with respect to P0, that i s , 42 / P0Q Q dx = 6 . — CO The matrix elements of L in the Qn basis are given by (2.25) L P = f P0Q LQ dx. nm  J °^T\ m^ (2.26) With an integration by parts, Eq.(2.26) becomes P _ _ nm ; p0BQnQ;dx, — 00 (2.27) It is clear that LPm i s symmetric in th i s representation and the superscript p denotes a representation in the polynomial basis. In terms of the polynomial set, Rn and the weight function w(x), upon which the quadrature rule is based, I have, substituting Eq.(2.24) into Eq.(2.27), that Jnm •J B(x)w(x) d + w M x ) . P 0 ' ( x ) dx 2w(x) 2P0(x) (2.28) d_+wJ_00_Po_l(x) dx 2w(x ) 2P0(x) R dx, m It is important to note that Eq.(2.28) i s not the representation of L in the Rn polynomial basis, defined by Eq.(2.12), which would be nonsymmetric, but rather i t i s the representation of the operator _ i / 2 ~ t/2 [P0(x)w(x)] ' L[P0(x)w(x)] / . This operator i s symmetric in the Rn basis. The re l a t i o n Eq.(2.28) i s a general rel a t i o n independent of choice of basis and equilibrium weight and has a form which i s c l e a r l y appropriate for the quadrature r u l e , Eq.(2.l7), thus 43 Lnm " \ \ B ( xk) wk[ Rm( xk) +9( xk) Rm( xk) ] ( 2'2 9 ) k = 0 [ Rn( xk ) + 5( xk) Rn( xk) ]' where n(x) - (x)_P0'(x) , n, g U ) " 2wTxT 2P0(x) * ( 2 > 3 0 ) The derivative of Rn may be evaluated with Eq.(2.23) and I have that N-1 N-1 Lnm ~ ~ j = o l / WDRm( xj)i f 0 ^ i W ( 2'3 1 ) \lQ B ( xk) [ Dk i+g( xk) 5i k] [ Dk j+9( xk) 6j k]' where the Kronecker 5-function has been introduced and the summations have been rearranged. The representation of the operator L in the DO basis may be written down by noting that the transformation between the two basis sets i s given by L = TT-LP-T. (2.32) With Eq.(2.31) and the u n i t a r i t y condition, TT-T=I,the unit matrix, I have that Li j * \l0 B ( xk) [ Dk i+^xk) 6i k] [ Dk j+g( xk) 6j k]- ( 2-3 3 ) The great advantage of the DO method is that the matrix representation of the FP operator is easily written down and evaluated for arb i t r a r y c o e f f i c i e n t s A(x) and B(x). Although any convenient set of polynomials Rn could be employed, i t i s expected that the convergence of Eq.(2.1l) 44 would be rapid for w(x)=P0(x). For thi s case g(x)=0, and L ^ j , given by Eq.(2.33), is s i m p l i f i e d . However, the polynomial basis set for this choice of weight function may be d i f f i c u l t to construct. Hence alternate choices of basis sets need to be made and Eq.(2.33) provides a symmetric DO representation for such basis sets. 2.6 DO REPRESENTATION OF NON-SELF ADJOINT OPERATORS The Kramers operator F 9 9 ^ 9 9x m 9v 9v . kT 9 V + -r— m 9v (2.34) and l i n e a r i z e d Boltzmann operator F L = -v-Vx - --Vv + J , (2.35) are not s e l f - a d j o i n t as is the Fokker-Planck operator defined by Eq.(2.4). This is a consequence of the fact that these operators are a sum of a d r i f t and c o l l i s i o n operator, the d r i f t operator has purely imaginary eigenvalues while the c o l l i s i o n operator has real eigenvalues. Thus, the operators may not be approximated by symmetric matrices. These two operators are further complicated by the fact that they may not be reduced to operators of a single v a r i a b l e , but rather are dependent on both position and v e l o c i t y . In the case of two var i a b l e s , the DO representation of the function, f(x,y) is again a column vector f_. The elements are ordered such that the f i r s t N elements of f_ are f(x,y) evaluated at the f i r s t N x^ quadrature points and at 45 y0, the next N elements are the function evaluated at x^ and at y i and so on. Thus components of J_ are given by fn = » / V^jf ( xi 'yj) n=M(j-D + i (2.36) where {x^,i=1...N} are a set of quadrature points with the corresponding set of weights {w^ } and s i m i l a r l y {yj,j=1,M} and { V j } are another set of quadrature points and weights. It should be noted that t h i s ordering of elements i s somewhat a r b i t r a r y . It i s easy to see how this d e f i n i t i o n may be extended to an a r b i t r a r y number of v a r i a b l e s . In multidimensional problems, an e f f i c i e n t method of taking p a r t i a l derivatives i s needed so a to reduce the computer memory requirements. The p a r t i a l derivative operators are just the corresponding derivative operators based on the {x^} and {yj} quadratures. Thus, l e t Dx be the p a r t i a l derivative operator with respect to x'and Dy be the p a r t i a l derivative operator with respect to y. Hence I have that Dmn = Di j m=M(k-1)+i;n=M(k-1)+j (2.37) where k=1,2,...M, s i m i l a r l y Dy = D.. m=M(i-1)+k;n=M(j-1)+k (2.38) mn I j J where k=1,2,...N. It i s i m p l i c i t in the d e f i n i t i o n s of the p a r t i a l derivative operator that the appropriate derivative operators have been used on the right hand sides of Eqs.(2.37) and (2.38). Any m u l t i p l i c a t i o n operator H(x,y) is represented by a diagonal matrix whose elements are 46 H(x^ / Y j ) in t n e appropriate order. With the use of these d e f i n i t i o n s of the p a r t i a l derivative operators and m u l t i p l i c a t i o n operator, an arbitrary linear operator of two independent variables may be e a s i l y represented as a matrix. The DO method for the solution of the Boltzmann equation i s p a r t i c u l a r l y useful since d i f f e r e n t i a l and integral operators are treated equivalently. In the DO approximation of d i f f e r e n t i a l operators, the points at which the function is evaluated are precisely those which are needed for the numerical evaluation of i n t e g r a l s . The pa r t i c u l a r form of the c o l l i s i o n operator w i l l be considered in f u l l d e t a i l in Chapter 5 in connection with the escape of l i g h t constituents of an atmosphere. The important point here i s that the integral and d i f f e r e n t i a l portion of the Boltzmann operator may be treated by a similar approximation. Thus, the c o l l i s i o n operator is approximated by one matrix and the d r i f t operator is approximated by another matrix and the two approximations may be added together to give an approximation to the entire Boltzmann operator. Once the approximations for the Kramers and Boltzmann operators have been found one is l e f t with a set of coupled equations to solve. These coupled equations may be solved in various ways including eigenfunction expansion, Runge-Kutta integration and i t e r a t i o n . When the equation to be solved has the form Eq.(2.l), the procedure given by Eqs.(2.6) - (2.11) may be followed d i r e c t l y , the only 47 difference being that the eigenfunctions are functions of two variables and not one. The approximate eigenfunctions are found by diagonalizing the matrix approximation of the operator. Unfortunately, i t is often d i f f i c u l t to accurately calculate the eigenfunctions and eigenvalues since the order of the matrix i s MN where M and N are both on the order of 10-50. However, once the approximation for L in Eq.(2.l) i s obtained t h i s equation takes on the form of M-N coupled f i r s t order d i f f e r e n t i a l equations for the f ^ . This set of equations i s e a s i l y solved by stepwise integrat ion. When the steady state problem, Eq.(2.2), is considered the problem reduces to a set of linear equations, subject to boundary conditions. Although these equations may be solved d i r e c t l y , due to the large size of matrices which result i t is computationally more e f f i c i e n t to solve them by i t e r a t i o n . This is done by guessing the solution to the equation. This crude approximation is used as an i n i t i a l condition for the corresponding time dependent problem. The time-dependent problem i s then numerically integrated u n t i l the solution vector remains constant. 2.7 SUMMARY The techniques which w i l l be used to solve the Kramers, Boltzmann and Fokker-Planck equations have been presented. In later chapters, these methods w i l l be applied to a model problem and two physical systems. The basic method i s to 48 approximate these p a r t i a l d i f f e r e n t i a l and p a r t i a l i n t e g r o d i f f e r e n t i a l equations as sets of ordinary linear f i r s t order d i f f e r e n t i a l equations and linear algebraic equations. Although this has been done previously, the par t i c u l a r DO method of repesenting d i f f e r e n t i a l and integral operators i s novel and a highly e f f i c i e n t technique. It should be noted that di f f e r e n t DO approximations result from d i f f e r e n t choices of quadrature points and weights. The appropriate quadrature rule to be used to represent a d i f f e r e n t i a l operator depends on the par t i c u l a r problem. The DO method is f l e x i b l e in that a DO representation of an operator, based on non-classical polynomials, may be obtained e a s i l y . 3. DIFFUSION IN A BISTABLE POTENTIAL There has been considerable interest in the solution of Fokker-Planck equations with bistable equilibrium d i s t r i b u t i o n s . These are useful in the description of a wide variety of physical phenomena such as the bimode lasers [R39] mentioned in the introduction. Other examples include chemically reactive systems where the Smoluchowski equation has been used to calculate the angular d i s t r i b u t i o n function of the methyl group in n-butane [R30]. The angle of rotation of the methyl group is more probable at some values than others due to s t e r i c repulsion. Often these Fokker-Planck equations are d i f f i c u l t to solve since the potential term, A(x) in Eq.(1.11), i s not suited to available numerical methods. Consequently there has been a great deal of interest in model Fokker-Planck equations with bistable steady state d i s t r i b u t i o n s . In this chapter, a model Fokker-Planck equation w i l l be considered. The potential term in the equation i s bimodal. This equation is given by Eq.(1.11) with the A(x) and B(x) given in Eq.(1.13) . 3. 1 INTRODUCTION This model Fokker-Planck equation i s defined by 9P(x,t) _ 9(gx3-ax)P(x,t) ^ 52P(x,t) ,, 9t 3x e olT1 ' It has received considerable attention as a model for d i f f u s i o n in a double well p o t e n t i a l . Van Kampen [R26,61] 4 9 50 and Dekker [R62] and van Kampen have determined a few eigenvalues of this FPE. Suzuki [R55,68] has employed t h i s model in an application of scaling theory. A preliminary study, based on the DO method, was also carried out by Shizgal and Blackmore [R81] for the special case g=a=e=1. Recently Indira et a l . [R64] have obtained numerical solutions based on a f i n i t e element method as well as with a Monte Carlo simulation. Brand et a l . [R63] have applied v a r i a t i o n a l methods in the calcu l a t i o n of upper and lower bounds to the lowest two eigenvalues. In t h i s chapter the numerical method developed in the last chapter w i l l be applied to this Fokker-Planck equation. However prior to doing so I w i l l f i r s t introduce a basis set which is appropriate for this problem. 3.2 BIMODE POLYNOMIALS There are no standard polynomial sets orthonormal over bimode weight functions, therefore I have generated sets of polynomials, {R (x),n = 0,1 . . .} , orthonormal with respect to the weight functions defined by w(a,7;x) = N e(-7xV2 + ax*)f { 3 > 2 ) where N is a normalization constant and a and 7 are two parameters. This weight function is bimodal i f both a, and 7 are p o s i t i v e . The peaks are found at ±^ and the width of the peaks are inversely related to the size of 7 . The weight function, w(a,7;x) with a=a/2e and 7=g/2e is the equilibrium d i s t r i b u t i o n of the Fokker-Planck equation given 51 by Eq.(3.l) and discussed at length l a t e r in the chapter. The corresponding polynomials were generated s p e c i f i c a l l y for the solution of this problem, although they are more widely app l i c a b l e . They w i l l be used in the next chapter for the problem of n-butane isomerization. In Appendix B, I discuss the calculation of these polynomials, the calculation of the points and weights of the corresponding quadrature r u l e , and the construction of the derivative operator defined by Eq.{2.23). 3.3 POLYNOMIAL REPRESENTATION OF FOKKER-PLANCK OPERATOR. The equilibrium solution i s given by P0(x) = V(^-,2-;x), (3.3) and coincides with the weight function Eq.(3.2) with a=a/2e and 7=g/2e. An immediate consequence of t h i s is that the DO representation of the Fokker-Planck operator defined by Eq.(2.33) and the polynomial representation defined by Eq.(2.28) are equivalent. It i s , however, more convenient to work in the polynomial basis since these matrix elements are simply related to the recurrence c o e f f i c i e n t s , 0n, that define the polynomials as discussed in Appendix B. If a and g are r e s t r i c t e d to take on positive values one can set a=g=1 since a l l that i s required is a r e d e f i n i t i o n of t , x and e. Thus the FP equation may be written as 9P(x,t) 9(x3-x)P(x,t) ^ 92P(x,t) ,, . \ —dt = ~te € 9x 2 . (3.4) where e i s the only parameter and remains as a measure of 52 either the system size or temperature. The matrix elements of L, Eq.(2.27), Lnm = -/ ew(l-,i-;x)R'R'dx, (3.5) nm 2 e 2 e nm — 00 are given e x p l i c i t l y by f n 2 i +iPr,P„-,Pr,-or n= m' (3.6a) nm B eFnpn-1Kn-2 n Lnm = m ' V n-r m = n ' 2 ' ( 3 ' 6 b ) L = n/0 , n=m-2, (3.6c) nm mm-1 Ln m = 0, otherwise, (3.6d) where the relation KU) - 7 l T ;Rn - 1( x ) +7 ^ n ^ - 1 ^ - 2Rn - 3( x )' ( 3'7 ) derived in Appendix B, and the orthonormality of the polynomials have been used. The approximate eigenfunctions, 4>n(x) , in the Rn basis and eigenvalues, X , are found by diagonalizing L given by Eq.(3.6). Since'-Eq. (3.6) defines a pentadiagonal matrix, the convergence of the eigenvalues and eigenfunctions is expected to be very rapid. 3.4 SCHROEDINGER FORM OF THE EQUATION If the d i f f e r e n t i a l operator on the right hand side of this Eq.(3.4) i s transformed to the Schroedinger form, Eq.(2.l5), then the potential Eq.(2.!6) i s , V(x) = ( x 3l ex ) 2- l ( 3 x2- 1 ) . 53 (3.8) The p o t e n t i a l , V(x), and the corresponding equilibrium d i s t r i b u t i o n , P0( x ) , are shown in Figures 3.1 and 3.2, respectively, for various values of e. This potential is characterized by three minima at . o _ = 0, = + | + / l / 9 + 2 e 1/2 (3.9a) (3.9b) The potential barriers between these three minima become larger as e becomes smaller. In the l i m i t of e—>0, approximate eigenvalues may be found by expanding the potential about the minima, keeping only the quadratic terms and neglecting terms of order e. The resulting harmonic potentials given by v*U) = ( X"X") 2- 1 , x^x ,(3.10a) V°(x) = — + -v i ; 4 e+2, x^O,(3. 10b) approximate V(x) near x=x" and x = x ° , respectively. The eigensolutions of Eq.(2.l5) with the potentials defined by Eq.(3.10) are given by *k = Hk [(x-x1)] r-u-x1)2] exp L ^ e J 2e X r-x2i exp 4e (3.11a) (3.11b) and their corresponding eigenvalues are given by 54 F I G U R E 3 . 1 P o t e n t i a l i n t h e S c h r o e d i n g e r e q u a t i o n , E q . ( 3 . 8 ) ; e i s e q u a l t o ( a ) 0 . 0 0 5 , ( b ) 0 . 0 2 5 ( c ) 0 . 0 5 a n d ( d ) 0 . 1 . FIGURE 3.2 55 Equilibrium d i s t r i b u t i o n function, Eq.(3.3 ) ;e i s equal to (a) 0.005, (b) 0.025 (c)0.05 and (d) 0.1. 56 X* = 2k, k=0,1,2...(3.12a) X° = k+1, k=0,1,2...(3.12b) where i s the kth Hermite polynomial. Thus in the l i m i t of very small e the eigenvalues approach integer values. The zero eigenvalue is doubly degenerate and the remaining even eigenvalues are t r i p l y degenerate. In t h i s l i m i t , where there is no coupling between adjacent wells, the eigenf unctions C ^>n} of Eq.(2.15) with V(x) given by Eq.(3.8) are linear combinations of {0^r0£}« 0n e such combination is given by ^n = T^k^k* ' n=4k, (3. 13a) 0n = 72{(t>i~rk]' n=4k+1,(3.13b) <t>n = 0£, n = 2k + 2, k even,(3.13c; 0n = 0 j , n=2k+1, k odd,(3.13d) where the set 0n is obtained by l e t t i n g k take on integer values in accordance with Eq.(3.13). This i s not the only possible set of linear combinations that may be used, although i t must be chosen such that 0n comes in even and odd p a i r s , since the potential given by Eq.(3.8) i s even. This analysis is similar to the one given by Larson and Kostin [R82] for a chemical kinetic model. 57 3.5 EIGENVALUE SPECTRUM The eigenvalues of Eq.(2.7) for L defined by the right hand side of Eq.(3.4) are found by diagonalizing the truncated polynomial representation of this operator given by Eq.(3.6). The numerical convergence of Xn for e=0.1 and e=0.01 i s given in Tables 3.1 and 3.2, respectively. It is clear from the tables that.the DO method is an e f f i c i e n t and accurate computational method for determining many excited states in t h i s t r i p l e well p o t e n t i a l . The only other calculations of the eigenvalue spectrum include estimates of the lowest eigenvalues with v a r i a t i o n a l methods [R63], estimates based on a f i n i t e difference calculation [R62] and asymptotic WKB approximations [R65]. None of these calculations come near to the accuracy shown in Tables 3.1 and 3.2. The v a r i a t i o n a l c a l c u l a t i o n was used to obtain upper and lower bounds on the f i r s t two eigenvalues alone. The smallest nonzero eigenvalue, X,, becomes very small as e decreases and as the barrier between adjacent wells (Figure 3.1) increases. This eigenvalue may be approximated by Kramers approximation. Kramers was able to approximate th i s eigenvalue with the assumptions that the potential wells are well represented by a harmonic potential and the peak of the potential barrier also has a parabolic shape. He then calculated [R29] the rate at which p a r t i c l e s escape over the b a r r i e r . His approximation gives X, = j2e~WUe)/ir, (3.14) which is v a l i d for e—*-0. Recently Larson developed a Table 3.1: Convergence of Eigenvalues. Quart lc Potent ia l " ' N V. X.) Vs Vi o V i > V i o Vi • 3 1 . 15(-1) 1 .64 5 5 . 15(-2) 1 . 196 2 .83 8 3 .5016(-2) 1 .031 1 .863 5 .05 10 3 .3962(-2) 0 .9647 1 .7509 4 .34 15 3 ,3574302( -2) 0. .928412 1 . 68828 3. 844 2 14.83 20 3 .3545699( -2) 0 .927440 1 .680430 3. .738081 12.671 35.01 30 3 3545300( -2) 0. 927372 1 .680264 3. .733990 1 1 .7001 23.299 43.33 74, .83 40 3. 3545300( -2) 0. 927372 1 .680264 3. 733985 11.687463 22.64789 36.79 56 .512 50 3 . 733985 11.687442 22.639923 36.04413 51 .9850 60 11.687442 22.639908 36.031815 51 5419 1. 1=0.1. a=g=1 tn 00 Table 3.2: Convergence of Eigenvalues. Quart ic Potent ia l " ' N X > v.. v.* V. ° X. > » X.. o 10 3 37(-7) 1 .866176 1 .867351 3 .52 20 1 92(-10) 1 . .865756 1 .865765 3 .37 7 . 12 13 .94 30 1 . 05(-11 ) 1 .865745 1 .865754 3 .307 5 .23 9 .41 17 .53 29 .37 40 6 . 452(-12) 1 , . 150 1 .865749 2 .301 4 .47 7 .51 12 .901 23 .03 50 6. 1 6 8 K - 1 2 ) 0 .9831 1 .865725 1 .9222 4 . 146 6 56 17 , .112 16 . 33 60 6 . 15498(-12) 0. .96847 1 .865337 1 .86933 3. .9797 6 . 114 9 .543 14 . 18 70 6 , 1546497(-12) 0 .967877 1 .864581 1 .867016 3 .945756 5 .9785 8 .9845 12 .986 80 6. 1546497(-12) 0. ,967865 1 .864542 1 .866975 3. .943588 5 96159 8 81464 12 .433 90 0. 967865 1 .864542 1 .866975 3. 943532 5. 960854 8 . 79404 12 . 284 100 3. 943531 5. 960839 8 . 793163 12 . 2693 DVK 0. 968 1 .862 1 . .867 1. t=0.01, a=g=1 o H Dekker and N. G. van KamDen 60 perturbative expansion for the f i r s t few eigenvalues in e [R82]. The zeroth order term agrees with Kramers result and the f i r s t order term y i e l d s X, = i/2e"l / ( 4 e )(1-3e/2)A, (3.15) A graph of the r a t i o of these approximate eigenvalues to the numerical value of X, as obtained in the present work i s given in Figure 3.3 as a function of -Loge. This indicates the way in which the Kramers approximation and the improved approximation of Larson and Kostin become v a l i d as the potential barrier becomes very large. It i s interesting to note that, especially for the smaller e, that some of the smaller eigenvalues converge more slowly than some larger eigenvalues; see X2 and X3 in Table 3.2. The comparison of the rates of convergence in these two cases indicates the general trend that as e i s decreased the rate of convergence of Xn becomes slower. This may be understood in terms of the approximate eigenfunctions discussed above. The eigenfunctions are approximated by the square root of the equilibrium weight times a sum of the polynomials. When e i s small some of the eigenfunctions are small in the region where the equilibrium weight i s large and vice-versa. Consequently, many basis functions are needed to represent those functions. It i s interesting to show the variation of the eigenvalues and eigenfunctions as a function of e. This is useful because i t gives some indication of the v a l i d i t y .of d i f f e r e n t approximation schemes. Figure 3.4 gives a plot of FIGURE 3.3 Variation of X, with e. 63 logX, versus -loge which i l l u s t r a t e s the very rapid decrease of this smallest nonzero eigenvalue with an increase in the barrier between the minima in the p o t e n t i a l . The reciprocal of this eigenvalue corresponds to the relaxation time between the stable s t a t e s . Figure 3.5 shows the variation of the eigenvalues Xn (n=2-17) for many of the excited states in thi s p o t e n t i a l . The approach of the eigenvalue spectrum to the form in the e—=»-0 l i m i t as given by Eq.(3.12) i s clear from the fig u r e . The asymptotic values of the eigenvalues to the right of the figure are very close to integer values, p a r t i c u l a r l y for the lower states, and the successive s i n g l e t - t r i p l e t pattern i s c l e a r l y seen. 3.6 EIGENFUNCTIONS Figure 3.6 shows several of the eigenfunctions for e=0.01, which include some highly excited states. The features that are clear include the symmetric form of 0n for n even and antisymmetric for n odd, and that the number of nodes i s equal to n. It is interesting to note that 0n for the lower states appear concentrated in the region of the minima of the p o t e n t i a l , V(x). For the excited states (n=7 and 10), 0 i s not concentrated near the minima, n Figure 3.7 shows the var i a t i o n of a p a r t i c u l a r eigenfunction, 03(x) as a function of e. The eigenfunctions become more concentrated in the region of the minima of the potential with decreasing e. It is interesting to note the dramatic change in the form, of the eigenfunction between FIGURE 3.5 64 Variation of Xn with e; n=2 (lowest curve) to n=17 (upper curve) . 65 FIGURE 3.6 Eigenf unctions of (x). e = 0.0l 66 FIGURE 3.7 -0.8 -0.4 0 0.4 0.8 x Eigenfunctions of $3( x ) . e is equal to (a) 1/20, (b) 1/40, (c) 1/60, (d) 1/140 and (e) 1/180. 67 6=1/140 and e =1/180. Figure 3.8 provides a comparison of the calculated 02(x) and the approximate form in terms of Hermite functions given by Eq.(3.13), v a l i d for e—>0. This comparison i s useful since, together with Figure 3.4, i t suggests that the e—z-0 l i m i t i s q u a l i t a t i v e l y attained for e<0.0l, at least for the lowest states. It i s also interesting to look at the f i r s t eigenf unction, ^ ( x ) as a function of x and e. An approximate solution for the f i r s t eigenfunction is given by <p ,-er f (x/\/2e) [R82]. This form of the f i r s t eigenf unction is analogous to the situ a t i o n in quantum mechanics where, for some p o t e n t i a l s , one obtains a degenerate pair of symmetric and anti-symmetric wave functions. A plot of 0,(x) for two values of e is given in Figure 3.9. This c l e a r l y shows that the eigenfunction approaches this form for the small e l i m i t , and as e—>0 the function c l e a r l y approaches a step function. This form of the f i r s t eigenfunction w i l l be useful when the chemical isomerization problem i s discussed. 3.7 TIME DEPENDENT SOLUTION The time dependent solution of the FPE i s completely determined once the expansion c o e f f i c i e n t s of the i n i t i a l d i s t r i b u t i o n are determined. For an i n i t i a l delta function d i s t r i b u t i o n , 5(x-x0), the PDF i s given by CO P(x,t) = I P (x)0 ( x 0 ) e x p ( - X t> . (3.16) n=0 FIGURE 3.8 FIGURE 3.9 Variation o f . ^ ^ x ) . (a) e = 0.1 and (b) e=0.0l. 70 Figure 3.10 shows the time evolution of the d i s t r i b u t i o n function for an i n i t i a l 5 function d i s t r i b u t i o n with xo=0 and e=0.0l25. The s o l i d curves are the numerical results while the dashed curves are obtained with scaling theory, [R68] v a l i d in the l i m i t e—>0. For an i n i t i a l delta function d i s t r i b u t i o n the numerical result with a f i n i t e number of polynomials w i l l deviate from the actual solution for s u f f i c i e n t l y small t . The numerical results shown in Figure 3.10 employed 100 polynomials and are converged to the resolution of the graph except in the wings of the d i s t r i b u t i o n at the smallest times; see Figure 3.10a, x>0.6. The computation time for these calculations is less than the time reported by Indira et a l . [R64] with a f i n i t e element method and considerably less than the time involved with the Monte Carlo simulations. However, one should note that the value of e used by these workers is very much less than I could use with existing algorithms. From the results in Figure 3.10, one can notice that the scaling theory solution approximates the numerical calculations in some intermediate time regime as has been discussed by other authors [R55,65,68]. The separation in the s o l i d and dashed curves near x==0 in Figure 3.10a decreases with increasing t . However, the deviation of the scaling theory result from the numerical solution increases for long times as shown in Figure 3.10b. There is some overlap in the two solutions at intermediate times. Caroli et a l . [R65] have shown that the normal mode expansion, 71 FIGURE 3.10 T 1 : r -0.5 0 0.5 x -0 .5 0 0.5 x Time va r i a t i o n of the pr o b a b i l i t y density function for e= 0.0125. present r e s u l t , scaling theory r e s u l t . t is equal to (a) 0.7, (b) 0.8, (c) 1.0, (d) 1.2, (e) 1.4, (f) 1.6, (g) 1.8 and (h) 2.0. 72 Eq.(3.16) yiel d s Suzuki's scaling theory result i f the eigenvalues are approximated by their harmonic values, Eq.(3.12), and the eigenfunctions are approximated by Weber functions. They then are able to perform approximately the sum in Eq.(3.16) and derive Suzuki's r e s u l t . In terms of the present numerical r e s u l t s , i t appears that Suzuki's result i s v a l i d at intermediate times for the following reasoning. For small, but f i n i t e e, the largest eigenvalues w i l l depart from the harmonic approximation and Suzuki's result w i l l depart from the exact solution at short times, times for which the solution i s dominated by the largest eigenvalues. With an increase in time the lower order eigenvalues contribute most and i f these are close to the harmonic values then Suzuki's solution w i l l be close to the true s o l u t i o n . It should be noted, though, that scaling theory f a i l s during a time regime for which the harmonic approximation of the eigenvalues is v a l i d . From the present study i t appears that there is r e a l l y only two rather than three d i s t i n c t time domains as defined by 1/X, and the reciprocal of some average eigenvalue 1/Xn, n>1. However i t i s always possible to define different time regimes for which a group of eigenvalues make a dominant contribution to the s o l u t i o n , but a comparable separation as occurs between A, and X2 does not occur elsewhere in the spectrum. 73 3.8 SUMMARY A complete study of a model Fokker-Planck equation has been presented in the present chapter. The method of solution of the equation was by eigenfunction expansion as developed in Section 2.3. This problem has been extensively studied by various researchers [R61-65] and a number of approximate methods have been used to solve i t . It was found that scaling theory could not accurately treat t h i s problem unless the value of the parameter e was very small. The e—>0 l i m i t i s useful for the study of low noise systems [R68]. In this l i m i t the PDF usually follows a deterministic path except at i n s t a b i l i t y points [R68]. In a bistable system, in the e—^0 l i m i t , the f i r s t eigenvalue, X1f is e s s e n t i a l l y zero and hence both states are stable. Often the Fokker-Planck equation is used to describe the rate at which one state goes over to the other state. In t h i s case, the exact value of X, becomes very important. It was shown that Kramers approximation becomes v a l i d as the value e becomes very small. This i s equivalent to saying that the barrier height in the potential between two stable states must become very large and the rate very small before Kramers approximation may be used. In addition, i t was shown that the f i r s t eigenfunction is approximately a step function times the zeroth eigenfunction. This is important since i t means that most of the PDF may be l o c a l i z e d in the f i r s t or second well of the potential with appropriate combinations of the f i r s t two eigenfunctions. This result w i l l be used in the next chapter on the chemical kinetics of n-butane isomerization. 4. BUTANE ISOMERIZATION The numerical methods for solving the Fokker-Planck and other d i f f e r e n t i a l equations, developed in previous chapters, are used to solve the Kramers equation as well as the Smoluchowski equation. These equations are used to describe the time dependent trans-gauche isomerization of n-butane in carbon t e t r a c h l o r i d e . The isomerization rate constant i s calculated from these equations and compared to the value given by t r a n s i t i o n state theory and by other researchers [R30,31]. The range of v a l i d i t y of the phenomenological rate law is also examined. The methods of solution of the Smoluchowski and Kramers equations should prove useful in a variety of other physical situations [R26]. 4.1 INTRODUCTION Chemical kinetics has long been a problem considered by researchers in s t a t i s t i c a l mechanics [R33,34,82-86]. Owing to the complexity of most chemically reactive systems, a th e o r e t i c a l description from a molecular point of view necessarily involves several approximations, p a r t i c u l a r l y for systems far from equilibrium. These approximations may be introduced in a number of ways which depend on the method of approach. For reactions in the d i l u t e gas region, for which one may assume that only two p a r t i c l e interactions are important, the reaction rate can be expressed in terms of the reactive scattering cross section from the reactants to 75 76 products. In p r i n c i p l e , the calculation of cross sections must be done quantum mechanically [R87], although semi-classical and c l a s s i c a l methods can give useful results for some systems. The macroscopic rate c o e f f i c i e n t i s given in terms of a s t a t i s t i c a l average of the reactive cross section with the appropriate d i s t r i b u t i o n function [R5]. The d i s t r i b u t i o n function may be determined from the Boltzmann equation [R5] as discussed by Ross and Mazur [R88] and as used by Neilsen and Bak [R70] to describe the dissociation of diatomic molecules. However, the Boltzmann equation i s not appropriate to the study of reaction rates in l i q u i d s since i t i s not v a l i d in dense systems. The Boltzmann equation is employed in Chapter 5 to study the planetary atmospheric escape problem. In cases where the Boltzmann equation is not v a l i d , as in l i q u i d s , detailed information with respect to the coupling of the reactive system and the host medium is generally not av a i l a b l e . E x p l i c i t treatment of the c o l l i s i o n a l dynamics of the system cannot be done except perhaps with costly computer simulations. Hence, one needs to employ some model of the effects of c o l l i s i o n s . For example, t r a n s i t i o n state theory [R60] estimates the reaction rate from the flux of p a r t i c l e s crossing the maximum of the potential energy as a function of the reaction coordinate, discussed in the introduction. The molecular configuration at the maximum of the potential 77 energy is c a l l e d the t r a n s i t i o n state and hence the name tran s i t i o n state theory [R60], Transition state theory is a rather crude approximation, ignoring the c o l l i s i o n nature of reaction dynamics in i t s simplest form. A somewhat more sophisticated approximation i s to model the c o l l i s i o n process [ R 8 9 - 9 2 ] . This allows the e f f e c t s of the c o l l i s i o n s to be included yet avoids the mathematical complexity of treating the c o l l i s i o n dynamics. For l i q u i d s , c o l l i s i o n s may be modeled by a stochastic, that i s random, force acting on a reactant molecule. Thus, the internal degrees of freedom of a molecule are randomly changed at random times. A random force acting on a p a r t i c l e was f i r s t used to describe the motion of a Brownian p a r t i c l e . The Brownian p a r t i c l e i s influenced by a - f r i c t i o n a l force, proportional to velocity of the Brownian p a r t i c l e with the proportionality constant c a l l e d the f r i c t i o n c o e f f i c i e n t , and by a random force. These forces are related by the fluctuation d i s s i p a t i o n theorem [ R 2 7 ] . Such modeling has been successfully used in the description of NMR relaxation [ R 9 3 - 9 5 ] . A chemically reactive system may be modeled using the same formalism as used to describe motion of a Brownian p a r t i c l e . Although I w i l l not consider a space dependent f r i c t i o n c o e f f i c i e n t , this is an area of active research [ R 9 6 - 9 8 ] . The need for a space dependent f r i c t i o n c o e f f i c i e n t may arise when, for example, d i f f e r e n t conformations of a molecule are coupled d i f f e r e n t l y to the 78 heat bath. The potential energy of a system as a function of the reaction coordinate has two l o c a l minima corresponding to the stable states of the system. At any point in time, the state of a chemically reactive system is given by the value of the reaction coordinate and the rate of change of the reaction coordinate. An isomorphic description is to speak of a point p a r t i c l e moving in an external potential described by the potential energy of the reactive system. Thus, the position of the p a r t i c l e corresponds to reaction coordinate and i t s v e l o c i t y corresponds to the rate of change of the reaction coordinate. The evolution of the chemically reactive system is modeled by the Brownian motion of t h i s p a r t i c l e with molecular c o l l i s i o n s modeled as a stochastic force on the Brownian p a r t i c l e . Kramers [R29] was the f i r s t to treat chemical kinetics in t h i s manner and his analysis i s based on the Kramers equation, which gives the rate of change of the probability density function (PDF) in terms of d r i f t and c o l l i s i o n operators. These two operators are often refered to as the reversible and i r r e v e r s i b l e portions of the Kramers operator, respectively. The r e l a t i v e magnitude of these two terms is controlled by a f r i c t i o n constant, which is a measure of the strength of coupling between the heat bath and the reactive system. Kramers showed that the Kramers operator i s equivalent to the Smoluchowski operator in the l i m i t of a large f r i c t i o n c o e f f i c i e n t . In addition, he 79 interpreted the reaction rate as the rate of passage of pa r t i c l e s over the potential b a r r i e r . This rate was shown to be proportional to the f i r s t eigenvalue of the Smoluchowski operator for which he derived an estimate. The connection between the smallest eigenvalue of a linear master equation operator and the rate c o e f f i c i e n t is well known in chemical k i n e t i c s . For example, Widom [R32] i d e n t i f i e d the f i r s t eigenvalue of a kinetic operator for a system of o s c i l l a t o r s as the f i r s t order rate constant for a chemical reaction. He also asserted that the v a l i d i t y of a f i r s t order rate law depends on the separation of the two lowest non-zero eigenvalues. The eigenvalue spectrum of a kinetic operator i s important since the reciprocals of the eigenvalues are the decay rates of the fundamental modes of the system. In t h i s chapter, I w i l l describe the isomerization of n-butane in CC1„ in terms of the solution of Kramers equation. The reaction coordinate for this reaction is the angle of rotation about the central bond of the n-butane. This problem was chosen for a number of reasons. It i s perhaps the simplest chemical kinetic problem which may give physical insight into more complicated chemical kinetic problems. This molecule has also been the object of rather extensive research [R30,31,67,99-103], allowing for comparison with the present r e s u l t s . In p a r t i c u l a r , the calcul a t i o n of the rate constant for the isomerization of n-butane in CC1« has been the object of two recent papers. 80 The f i r s t of these [R30], by Montgomery, Chandler and Berne (MCB), calculates the rate constant by following the t r a j e c t o r i e s of a large number of p a r t i c l e s moving in a model potential chosen to approximate the actual intramolecular potential energy of n-butane as a function of the angle of rotation about the central bond. The t r a j e c t o r i e s were used to calculate the derivative of a time autocorrelation function. In a previous paper [R67], Chandler showed the way in which the f i r s t order rate constant for the isomerization reaction is related to this autocorrelation function. In a second paper, Marechal and Moreau (MM) [R31], provide a method of c a l c u l a t i o n of the correlation function mentioned above in terms of the eigenfunctions of the Smoluchowski operator. The eigenfunctions of this operator were found by transforming i t to the form of a Schroedinger operator whose eigenfunctions were approximated by a WKB approximation [R30]. The objective of t h i s chapter i s to calculate the rate of isomerization for butane i f the molecules start in predominately the gauche conformation. As Chandler [R67] correctly points out, a f i r s t order rate law cannot be derived but can be shown to be consistant with microscopic dynamics. The f i r s t order rate constant w i l l be estimated from the eigenvalues of the Kramers and Smoluchowski operators calculated with the methods introduced in Chapters 2 and 3. 81 Montgomery et a l . [R31] raised two objections to the c a l c u l a t i o n of the rate constant in this way. The f i r s t objection was that there is no guarantee that the bulk of the reactants w i l l follow the f i r s t order rate law even i f the gap in the eigenvalue spectrum e x i s t s . The second objection was that i t i s very d i f f i c u l t to calculate the eigenvalue spectrum of a kinetic operator for a physically reasonable system. I w i l l demonstrate that the DO method permits a study of the eigenvalue spectrum of both the Kramers and Smoluchowski operators. In addition, i t w i l l be shown that, under certain conditions, the bulk of the reactants must follow the f i r s t order rate law. It should be noted that the rate constant depends on the f r i c t i o n c o e f f i c i e n t which is related to the c o l l i s i o n frequency between the n-butane and C C la. In the next section, I consider the conformations of n-butane. This is followed by a phenomenological treatment of the isomerization process. In Section 4 . 5 , the derivation of the Kramers equation is discussed and the following section shows how i t i s reduced to a Smoluchowski equation. Eigenvalues of these equations are related to the rate c o e f f i c i e n t in Section 4 . 7 . The method of solution of these equations are presented in section 4 . 8 and a discussion of the numerical results is given in the last s e c t i o n . 82 4.2 CONFORMATIONS OF N-BUTANE Butane can exist in more than one conformation. The two important conformations are the trans and gauche conformations. These are shown in Figure 4.1. This system has been extensively studied owing to i t s r e l a t i v e s i m p l i c i t y . The analysis of the isomerization process is further s i m p l i f i e d by assuming that the energy of the molecule depends almost e n t i r e l y on only one internal degree of freedom, that is the angle of rotation about the central bond. The potential energy of n-butane as a function of angle of r o t a t i o n , <f>, about the central bond is given in Figure 4.2. The intramolecular potential energy is a minimum at 0=0. As 0 is increased from 0 there are repulsive methyl-hydrogen interactions and the potential energy increases u n t i l i t reaches a l o c a l maximum at TT/3. A lo c a l minimum in the potential energy w i l l again be reached at 0=27r/3 when the hydrogen-methyl s t e r i c interaction i s again at a minimum. This minimum w i l l not be as low as the minimum at 0=0 since the terminal methyl groups are closer together. F i n a l l y , at 0=7r a maximum in the potential energy, w i l l be reached since the methyl groups are closest together. The difference between the energy of the two conformations i s around 0.6 Kcal/mole or kT at room temperature. The barrier height between the two conformations is about 4kT at room temperature. This potential energy has been experimentally determined by microwave absorption spectra [R104]. FIGURE 4.1 83 Gauche and trans conformations of n-butane. FIGURE 4.2 Intramolecular potential of n-butane as a function of V(0) in units of Kcal/mole. 85 4.3 PHENOMENOLOGICAL TREATMENT Suppose I have N n-butane molecules d i l u t e l y dissolved in carbon t e t r a c h l o r i d e . The number of molecules in the gauche conformation is denoted by N^ and the number of molecules in the trans conformation is denoted by N0. Since the number of molecules of n-butane must remain constant, NA+Ng=N. Chemical systems such as this one are eas i l y treated in a phenomenological manner as done, for example, by Skinner and Wolynes [R33]. The rate of change in the concentration of molecules in the gauche conformation is given by dN = -k._N. + kn„ Nn. (4.1) dt AB A BA B At equilibrium the l e f t hand side of Eq.(4.l) must be equal to 0, hence ^ B A B ^ N ^ / N ^ • Eq.(4.1) may be cast into the form of a simple f i r s t order rate equation by setting r"1=kA B+kB A and 6NA(t)=NA(t)-N^q. Thus d5N - g ^ = - T - 1 6NA. (4.2) As mentioned before, my objective i s to calculate r ~1 as a function of temperature and frequency of c o l l i s i o n s between the n-butane and C C 1 „ . It i s also interesting to know i f T "1 obeys the Arrhenius equation, i.e r"1(T) = Aexp(-AEA/kT), (4.3) where AEA is the ac t i v a t i o n energy. The accuracy of the foregoing macroscopic description is contingent on a number of things. F i r s t of a l l , i t i s 86 necessary that the time scale for rotation about the central bond be much larger than the time scale for the relaxation of other internal degrees of freedom. This follows from the assumption that the potential energy along the reaction coordinate i s equal to the average internal energy of the n-butane as a function of the angle of rotation. The rate constant, T "1, must go to a constant value i f the phenomenological rate law, Eq.(4.1), i s to be physically reasonable. F i n a l l y , the n-butane molecules must be independent of each other for the f i r s t order rate law to hold. The v a l i d i t y of the f i r s t two assumptions depends on the depth of the potential wells, as w i l l be seen l a t e r , and i f the butane concentration is low enough, the molecules w i l l be independent so that f i r s t order kinetics i s applicable. The number of each isomer may be easily calculated from the PDF, P(0,cj,t), for t h i s system. P( 0,w, t )d<pdu> gives the probability that a molecule w i l l have an angle of rotation around the central bond between 0 and 0+d0, and the methyl group rotates with an angular v e l o c i t y between CJ and CJ + da>. Thus --7T /3 7T NA( t ) = NT ; P(0.,cj,t)d0 + / P(0,w,t)d0 TT 7r/3 dw, (4.4) and TT / 3 J -co-TT / 3 and hence the PDF, P(0,o),t), is a l l that is required to Nf i(t) = NJ" f P(0,w,t)d0dw, (4.5) 87 calculate macroscopic quantities and test the v a l i d i t y of Eqs.(4.2) and (4.3) 4.4 SIMPLIFIED POTENTIAL Prior to proceeding, i t i s useful to make some approximations of the molecular system. The problem I am concerned with i s the cal c u l a t i o n of the average rate which the PDF is reduced for |0|>7r/3. Since the potential is comparatively large at -ir and n, the PDF should be small at those points. Consequently, the potential is modeled so that i t tends to ±°° at the endpoints. Furthermore, only the flux across one barrier is desired and hence the t r i p l e well potential may be approximated by a double well p o t e n t i a l . It is anticipated that these approximations introduce a small e r r o r . If the domain of the position variable is ( -=> f +co) and the PDF vanishes at the endpoints, then the use of the bimode polynomials, developed in Appendix B, is appropriate for the application of the DO method to this problem. This may be done by changing the position variable from the angle of rotation to the length, x, that the methyl group moves as i t i s rotated as was done by Chandler [R67]. The bond length between the two end carbon atoms is approximately 1.57A while the angle between the three carbon atoms is approximately 1 0 7 ° . Thus, the variable x may be defined by x = (1.500-1.57)A, (4.6) where the 1.57A1 s h i f t s the middle of the central barrier in 88 the potential to x = 0 . A similar change of variable may be used to go from the angular v e l o c i t y , u>, to the velo c i t y of the methyl group, v. While t h i s change of variables i s , s t r i c t l y speaking, v a l i d only for a small range of <f>, i t is convenient to modify the potential energy of the system such that i t becomes large as |x| becomes large. The potential may then be approximated by a polynomial in x. Figure 4.3 shows several approximate f i t s of the potential energy function, U(x), where x=x/x0, x 0 =1.57A. The f i r s t f i t is an accurate f i t of the potential in two of the wells, the second is a piecewise harmonic f i t of the potential and the last is a polynomial f i t of the piecewise harmonic p o t e n t i a l . The harmonic potential i s given by /imu2 (x+x^ ) 2 x<-a (4.7) £ 3 3 U(x) = 2 v 2 U --~mu>ix a 2 -a<x<b Ub + Imu2(x-xb)2 x>b where a= U x , b=U,x, , C J 2 = U C J 2 / ( 1 - U ) and C J 2 = U , C J2/ ( 1 - U , ) with 3 3 D D 3 3 3 D O O U = 2 U /(ma>2x=) and U , = 2 ( U - U , )/(mcj2x, ) . The quantities a, a a a o a D O b, k>a and C J ^ are chosen such that the potential and f i r s t derivative are continuous. The other parameters used are given in Table 4 . 1 and are chosen to agree with previous researchers [ R 3 0 , 3 1 ] . 89 FIGURE 4.3 0.06-50.04-0.02-x Intramolecular potential of n-butane as a function of x; U(x) and x in units of m^2x2 and x respectively. (a) Harmonic potential (b) polynomial f i t of harmonic p o t e n t i a l , polynomial f i t of the true p o t e n t i a l . 90 Table 4.1: Potential parameters in n-butane Isomerization 2.93 KJ mole"1 u = a 12.34 KJ mole"1 *o = 1 .57 A &>,= 1.06X1013 sec"1 »o = 3.0X1012 sec" 1 T= 3.00 K m= 1.85X10"23 g Although these potentials are considerably d i f f e r e n t from the true periodic p o t e n t i a l , t h i s should not introduce much error since the potential in the region of the barrier is well f i t t e d and the size of the potential barrier should be the controling factor in the reaction rate. It should also be noted that the reaction rate is not changed s i g n i f i c a n t l y with the use of thi s approximate potential since in the t r i p l e well case the reactive flux over the larger barrier must be much smaller than the flux over the smaller b a r r i e r . This i s to say that once molecules are in the gauche conformation, the methyl group may rotate in one of two d i r e c t i o n s . Rotation in one dire c t i o n w i l l leave the molecule in a dif f e r e n t gauche conformation, while rotation in the other direction w i l l lead to a tr a n s i t i o n to the trans conformation. However the barrier height for the gauche-gauche t r a n s i t i o n i s much higher than the barrier 91 height for the gauche-trans t r a n s i t i o n . This l a t t e r rate is the major component of the over a l l rate constant r "1. With the use of the approximations in the previous paragraphs, the PDF becomes a function of velocity of the methyl group, i t s re l a t i v e position and time. The d e f i n i t i o n of the number of the gauche conformers now becomes 00 00 N.(t) = N/ fP(x,v,t)dxdv. ( 4 . 8 ) A - » 0 A similar d e f i n i t i o n defines the number of molecules in the trans conformation. 4 . 5 MOLECULAR DYNAMICS The macroscopic state of n-butane is completely described by various integrals over the PDF, in particular the number of molecules in each conformation is given by E q ( 4 . 8 ) . Thus i t suffices to calculate this PDF. The purpose of this section is to introduce equations which w i l l give the probability density function. This may be done by considering the system to be a stochastic system. That i s , both the position and velocity of the p a r t i c l e s are considered to be random variables which are altered by a random force. I w i l l speak of the motion of a Brownian p a r t i c l e which models the motion of the methyl group. The random force is meant to model the c o l l i s i o n s and the Brownian p a r t i c l e follows a trajectory defined by and 92 mv = F(x)" - 7V + *( t ) , (4.9) v = x, (4.10) where and U i s the intramolecular p o t e n t i a l , y i s the f r i c t i o n c o e f f i c i e n t and * is a random force. If *(t)=0 and 7=0, then there are no c o l l i s i o n s and the molecule obeys Newton's equations of motion. It is convenient to model the random force as a series of random impulses, that is ¥(t) = c l 6(t-t .)(±1) .. (4.12) j 1 3 where {tj} i s some random sequence of times, the (±1)J indicates that the jth impulse is random in a p o s i t i v e or negative direction in the one-dimensional model, and c is a measure of the strength of the c o l l i s i o n s . The stochastic force should not result in any net force on the p a r t i c l e , that is <*(t)> =0, (4.13) where the angular brackets, <....>, indicate an ensemble average. Furthermore, i f the t j are a Poisson process, i t may be shown [R27] that the correlation function, <*(t)*(t')> = c2^ 6 ( t - t ' ) , (4.14) where v i s the reciprocal of the average length of time -between the successive t j or, equivalently, the c o l l i s i o n 93 frequency. Einstein showed [R24] that the autocorrelation function for the fluctuating force i s proportional to the f r i c t i o n c o e f f i c i e n t , that is <*(t)*(t')> = ^ | ^ 6 ( t - t ' ) . (4.15) I may relate the c o l l i s i o n frequency, v, to the f r i c t i o n c o e f f i c i e n t , 7 , i f i t i s assumed that the constant c in Eq.(4.12) is given by c = v/2mkT, (4.16) where the right hand side of Eq.(4.16) is the most probable momentum of a p a r t i c l e . This i s equivalent to saying that the momentum of the p a r t i c l e changes by an amount equal to the most probable momentum after each c o l l i s i o n . Comparing Eq.(4.14) and Eq.(4.l5) and using the d e f i n i t i o n of c, Eq.(4.l6), I find that the c o l l i s i o n frequency and the f r i c t i o n c o e f f i c i e n t are related to each other by v = y/m. (4.17) Kramers showed [R29] that the Langevin equation, Eq.(4.9), may be rewritten in the form of an equation for the PDF. This equation is given by, 3P 3 3v . kT 3 V + -r— m 3v - — - i L m 3v "3x P, (4.18) where F i s given by Eq.(4.1l). It i s useful to write Eq.(4.!8) as, 94 fa = LcP + LdP' ( 4'1 9 ) where 3 L = v-~— c 9v kT 9 m 9v (4.20) is the c o l l i s i o n operator, and is the d r i f t operator. The c o l l i s i o n operator is the i r r e v e r s i b l e portion of the Kramers operator and describes the coupling of the system with the heat bath. The d r i f t operator gives the rate of change of the PDF as a result of the motion of the methyl group in the intramolecular p o t e n t i a l . It i s interesting to note that t h i s equation has exactly the same form as the Boltzmann equation. It is assumed that the n-butane is di l u t e enough that the only c o l l i s i o n s are those between i t and the solvent. The c o l l i s i o n frequency in a l i q u i d is given by [R105] v = |v/kT/MffT?(2-T})/( 1-ij) 3 (4.22) where M i s the mass of the solvent p a r t i c l e , a i s the hard sphere diameter of the solvent and 77 is the packing fraction of the l i q u i d . The c o l l i s i o n frequency calculated by Eq.(4.22) i s only an approximate r e s u l t , hence the effect of varying the c o l l i s i o n frequency w i l l be examined. In this problem not a l l of the c o l l i s i o n s couple with the reactive degree of freedom, in fact only about 1/3 of the c o l l i s i o n s are so coupled [R31]. This to say that c o l l i s i o n s between 95 the CC1„ and n-butane only lead to rotation of the methyl group i f the geometry of the c o l l i s i o n allows i t . Here I assume that only a t h i r d of the c o l l i s i o n s have a geometry that allows rotation of the methyl group. The f i n a l approximation i s then to use 1/3 the value of the c o l l i s i o n frequency given by Eq.(4.22). A set of reference parameters has been collected in Table 4.1. This table also includes some of the parameters for the piecewise harmonic f i t of the p o t e n t i a l , GJ, is a parameter which describes the curvature of the potential at x=0. The values w i l l be discussed in further d e t a i l when the work of MCB and MM is discussed. Eq.(4.!8) may be put into dimensionless form by making the substitutions x=x/x0, t=tv/2 and y=/m/2kTv, thus where and 3P 3t 3_ 3y P - -7 2 H b " * ( x ) ! y -P (4.23) 7 = /mxjs pV2kT (4.24) *(x) = U'(x)x0/kT (4.25) The PDF for t h i s system may be calculated with Eq.(4.23) at any time given some i n i t i a l PDF. In the next section, I w i l l show that the Kramers equation reduces to the Smoluchowski equation in the l i m i t of high c o l l i s i o n frequency. This equation i s also used to determine a PDF. 96 The main o b j e c t i v e i s to use these PDFs to c a l c u l a t e the r e l a x a t i o n time, T . These r e s u l t s w i l l be compared to each other and with those obtained by Montgomery et a l . and Marechal and Moreau. It i s a l s o i n t e r e s t i n g to study the c o l l i s i o n frequency and temperature dependence of r . Montgomery et a l . s t u d i e d t h i s system by f o l l o w i n g the t r a j e c t o r i e s of a l a r g e number of p a r t i c l e s moving under the i n f l u e n c e of the piecewise harmonic approximation to the p o t e n t i a l given i n F i g u r e 4.3. They a l s o used an even simpler square w e l l p o t e n t i a l with a v a r i e t y of b a r r i e r widths and h e i g h t s . Marechal and Moreau, on the other hand, only c o n s i d e r e d t h i s system i n the high c o l l i s i o n frequency l i m i t . In t h i s l i m i t , the v e l o c i t y p o r t i o n of the PDF re l a x e s to i t s e q u i l i b r i u m d i s t r i b u t i o n on a time s c a l e much f a s t e r than the time s c a l e f o r the r e l a x a t i o n of the s p a t i a l p o r t i o n of the PDF. 4.6 TRANSFORMATION TO THE SMOLUCHOWSKI EQUATION Eq.(4.23) may be r e w r i t t e n as |f = LkP ( x , y , t ) (4.26) where i s the Kramers o p e r a t o r . As d i s c u s s e d i n the i n t r o d u c t i o n , the lower order eigenvalues of t h i s operator are important s i n c e they are r e c i p r o c a l s of r e l a x a t i o n times. These 9y a y - + 2 y (4.27) 97 eigenvalues are often d i f f i c u l t to c a l c u l a t e , especially i f the potential is a complicated function. As was previously mentioned, the Kramers operator i s the sum of a streaming and c o l l i s i o n operators. The streaming operator has an imaginary eigenvalue spectrum while the c o l l i s i o n operator has a real spectrum. Consequently, the Kramers operator i s not s e l f - a d j o i n t and, in general, has a complex spectrum. In this section, I w i l l show that in the high c o l l i s i o n frequency l i m i t the small eigenvalues of thi s operator are approximated by the eigenvalues of a simpler operator, the Smoluchowski operator. This s i m p l i f i c a t i o n comes about since there are two relaxation time scales, as discussed below. It should be mentioned that there are a number of ways of looking at the relationship between these two operators, for which Risken [R26] provides a good di scussion. The basic idea of this approximation is that when the c o l l i s i o n frequency becomes large the PDF must become Maxwellian in velo c i t y on a time scale much shorter than the time scale for r e d i s t r i b u t i o n of methyl group po s i t i o n s . The PDF may then be set equal to a Maxwellian in velocity and the v e l o c i t y dependence integrated out. Unfortunately, as the equation stands, the velo c i t y dependence may not be integrated out d i r e c t l y since each term on the right hand side of Eq(4.26) is an odd function of ve l o c i t y and with a Maxwell-Boltzmann velocity d i s t r i b u t i o n a l l these terms w i l l vanish when the velo c i t y integration is performed. The 98 t r i c k is to rewrite the operator in such a way that, once the ve l o c i t y integration is done, a l l terms w i l l vanish except for one well defined term. The operator may be written as Lk = y 9 _ 9y 1 i _ 7 3x "2Z + 1 3 + * + 1^ 9 7 7 oy 7 7 ox (4.28) 33T*(x) + 3x^ Consider the variable changes defined by x = x + y / 7 (4.29) y = x - y 7 (4.30) If the f r i c t i o n c o e f f i c i e n t , 7 , i s s u f f i c i e n t l y large then, for small values of y and x, y - ~ 7 y and x^x. The Kramers operator may now be rewritten using t h i s variable change to give With the use of Eq.(4.30) I may integrate Eq.(4.26) with respect to y. This w i l l remove the v e l o c i t y dependence and leave only the position dependence of the PDF. Thus the terms with p a r t i a l derivatives with respect to y may be evaluated d i r e c t l y . They must vanish because of the 2(x+y) 7Z + 1 1' 72-1 •J. 9 9y 2 7: F x (4.31) 99 boundary conditions imposed on P(x,y,t), i . e . P(x,y,t)—> 0 as y—>-±o». The remaining integral i s given by / L, P = / — J k J 7 — oo —oo ' P(x,y,t)dy (4.32) The only y dependence the integrand has is in <l>(x), however in the region that P(x,y,t) i s not vanishingly small this i s approximately independent of y i f 7 i s s u f f i c i e n t l y large. Thus Eq.(4.26) may be rewritten as ItX P(2>y,t)dy = 1, — 00 ' / P(x,yrt)dy. (4.33; Also, since x=^ x in t h i s l i m i t , I may make the replacement 00 x=x. Define the reduced PDF f(x,t)=/ P(x,y,t)dy which —00 s a t i s f i e s the equation, 9 f - r f " LSf' (4.34) where 7' 32 W7 f(x,t) (4.35) Equation (5.34) is the Smoluchowski equation and i s v a l i d provided 7 > > 1 . It was previously mentioned that, in the high c o l l i s i o n frequency l i m i t , the relaxation of the non-Maxwellian modes in v e l o c i t y i s rapid. This implies that the eigenvalues governing their relaxation must have large real parts in comparison with their imaginary parts. 100 4.7 VARIOUS APPROXIMATIONS FOR THE RATE CONSTANT Chandler showed [R67] that T ~1 may be related to the derivative of the equilibrium time co r r e l a t i o n function by r c M t ) = - ( N x ^ ) - (4.36) where C(t) = <NA(0)NA(t)>, (4.37) and xA and xg are the equilibrium mole fractions of the molecules in the gauche and trans conformation respectively. One may also calculate r ~1 d i r e c t l y by rewriting Eq.(4.2) as 1 d5N (t) T _ , ( t ) = " m^tT — ( 4 - 3 8 ) and substitution of the d e f i n i t i o n of NA, Eq.(4.8), to obtain N/ /^fdxdy -CO (J V1( t ) = . (4.39) OO CO NJ ;P(x,y,t)dxdy --coQ Eq.(4.26) has the solution given by P(x,y,t) = I aexp(-X t)P (x,y), (4.40) n=0 n n n where Pn are the eigenfunctions of defined by L1 ,P„ = - x^p„ ' (4.41 ) k n n n and X =Xr + iX1 are the associated complex eigenvalues. It n n n c should be noted that the f i r s t eigenfunction, P0(x,y) is 101 important since i t is the equilibrium d i s t r i b u t i o n function and the eigenfunctions are orthonormal with respect to PS1. The expansion c o e f f i c i e n t s , an, in Eq.(4.40) are determined from the i n i t i a l condition by 00 00 an = / J P(x,y,0)P*(x,y)/Po(x,y)dxdy, (4.42) — 00— oo * where Pn(x,y) is the complex conjugate of P (x,y). The expression for T "1, Eq.(4.39) may be evaluated by substituting the expansion for P(x,v,t) into i t and taking the derivative term by term. This yi e l d s I anexp(-Xnt)Xn/ ;Pn(x,y)dxdy n=1 -ooO (t) = (4.43) 00 oo oo L a exp(-X t ) J /Pn(x,y)dxdy n=1 -ooQ If the expansion c o e f f i c i e n t s , a , are a l l of the same order of magnitude and X,<<X2 then for t>>1/X2, Eq.(4.43) reduces to T "1 = X,. (4.44) It i s interesting to show that the d e f i n i t i o n s of T _ 1 given by Eq.(4.43) and 1 given by Eq.(4.36) lead to the same asymptotic value. The value of N^(t) is given by N N.(t) = I T ? . ( xn( t ) )f (4.45) A n=1 A n. where xn( t ) i s the position of the methyl group and *?A(t) is 1 for x>0 and zero otherwise. At d i l u t e concentrations the 1 0 2 p a r t i c l e s are independent hence <NA(0)NA(t)> = N<r?A(x(0) )7?A(x(t) )>. (4.46) The correlation function on the RHS of Eq.(4.46) may be evaluated by noting that i t is just the t o t a l probability of finding a molecule in the gauche state at time t , given that i t was in the gauche state at t = 0 , times the probability that i t was in the gauche state at t = 0 . This probability may be found by solving for the i n i t i a l value problem for P(x,y,t) with the i n i t i a l conditions given by P(x,y,0) = P0(x,y) x>0, (4.47) P(x,y,0) = 0 x<0. Thus, the correlation function, Eq.(4.46), is equal to Ooo <NA(0)NA(t)> = N J 7 P(x,y,t)dydx, (4.48) Q-co provided P(x,y,0) is defined by Eq.(4.47). It should be noted that P(x,y,t), so defined, is normalized to the equilibrium probability of finding a molecule in the gauche state. The rate constant r~ 1 , defined by Eq.(4.36), i s evaluated with the substitution of the correlation function, Eq.(4.45) to give oo oo r M t ) = -(xAx B)- 1J / d P U ^ y , U 0 ( x ) d y d x , ( 4 > 4 g ) — oo—co where the step function 0(x) has been introduced; It i s 103 interesting to note that as t approaches 0, r "1 may be evaluated a n a l y t i c a l l y . Thus, at t = e«\/v CD 00 rc1( e ) = - ( XAXB) ~1/ / 0(x)LkPoU,y)t9(x)dydx, (4.50) — 00—00 where Kramers equation has been used. Since L^Po(x,y)=0, Eq.(4.50) may be written as 00 oo rc1( e ) = - ( XAXB) "1/ ; t9(x)Po(x,y)Lk0(x)dydx. (4.51) — 00—00 With the d e f i n i t i o n of Lfc, Eq.(4.27), I find that oo o T'c = 2 ( 7 XAxB)"1-f $ e(x)P0(x,y)y6(x)dydx, (4.52) — oo —oo where only the term in Eq.(4.27) contributes to the y integration performed on [-°°,0]. The l i m i t s on the y integration arise in the following manner. At times much shorter than the reciprocal of the c o l l i s i o n frequency the PDF is changed only by p a r t i c l e d r i f t . Thus, in order for there to be p a r t i c l e s at x=0 and y>0 at t=e, these p a r t i c l e s must be at x=-ey at t=0. However, the i n i t i a l condition i s such that P(x<0,y,0)=0 and consequently, i f y>0, then P(-ey,y,0)=0. The two integrals in Eq.(4.52) are evaluated e a s i l y with the substitution of the d e f i n i t i o n s y=i/mxg i>z/2kT, y=v/m/2kTv and x = x/x0, noting that is in units of v/2 and P0(x,y) i s given by x P0(x,y) = N exp(-(y2 + J *(x')dx'), (4.53) where N is a normalization constant and $ i s defined by Eq.(4.25). This yie l d s after some algebra 104 = (XAXB)-1S(0)<|v|>/2, (4.54) where -U(x)/kT CO U(x)/kT S(x) /S e dx, (4.55) — CO which i s the t r a n s i t i o n state theory approximation to the rate constant [R60], This result was o r i g i n a l l y derived by considering the rate at which p a r t i c l e s cross the potential b a r r i e r , assuming that the methyl groups have a Maxwell-Boltzmann velocity d i s t r i b u t i o n . This rate is found by multiplying the average v e l o c i t y of the p a r t i c l e times the p r o b a b i l i t y per unit length that the p a r t i c l e w i l l be found at the t r a n s i t i o n s t a t e . Thus, the value of < | v | >=j/2kT/m7r i s fixed by the temperature of the solvent and S(0) depends only on the potential and temperature. Consequently, the value of the t r a n s i t i o n state theory approximation to the rate constant is independent of the c o l l i s i o n frequency. It should be noted that this equivalence between the i n i t i a l value of the rate constant and the t r a n s i t i o n state theory value of the rate constant was previously noted by Chandler [R67]. For a r b i t r a r y t , i t i s necessary to evaluate the PDF, P(x,y,t), as a solution, to the Kramers equation. With the eigenfunction expansion, Eq.(4.40), in the expression for rc1, Eq.(4.49) , I find that, n=0 (4.56) where the fact that the eigenvalues come in complex 1 05 conjugate pairs has been used. The An c o e f f i c i e n t s are given by An = < W ' / fP(x,y)dxdy -oo(J .00 oo / /P (x,y)dxdy -ooQ (4.57) where the f i r s t bracket term arises from the integration in Eq..(4.49) and the second term that arises i s a , Eq.(4.42), for the i n i t i a l condition given by Eq.(4.47). If the eigenvalues are well separated, then, for a time 1/X,>>t>>1/X2, Eq.(4.56) reduces to rc1 = X,A,. (4.58) However, A, on the right hand side of Eq.(4.58) may be shown to be approximately equal to one provided that the barrier between the potential wells is great enough. It was seen in the last chapter that Fokker-Planck operators characterized by a double well potential have a f i r s t eigenfunction that tends toward a step function as the barrier between the wells increased. I make the assumption that the f i r s t eigenfUnction for the Kramers equation w i l l also tend toward a step function. This is a reasonable assumption since the f i r s t eigenvalue tends toward 0 and thus, at least where the eigenfunction i s large, i t w i l l approximately s a t i s f y the eigenvalue equation. Hence I choose for the f i r s t eigenfunction the form P,(x,y) * P0(x,y)(B0(x)+C0(-x)) (4.59) where 0(x) i s the unit step function, B=/Xr>/x» and C=-v/x77xZ a A A D are constants chosen such that P, i s orthonormal to P0. 1 06 With these approximations and noting that the equilibrium mole fraction of isomer A i s given by CO oo XA = // P0(x,y)dydx, (4.60) 0-co i t follows that J O O _ 2 ( xAxB) - ;/ P,(x,y)dydx •0-o° (4.61) with the immediate consequence that A,=*1. It should also be noted that A0 and A, are much larger than the other An i f the approximation, Eq.(4.59), i s v a l i d since the i n i t i a l condition may be approximately represented by a sum of the f i r s t two eigenfunctions. This guarantees that most of the reactants w i l l have to follow the rate law given by Eq.(4.2). There exists a gap in the eigenvalue spectrum only in the case of a bimode p o t e n t i a l . Hence, as long as there is a separation in the eigenvalue spectrum, the conditions for r"1=X, are s a t i s f i e d . In the high c o l l i s i o n frequency l i m i t , exactly the same arguments may be followed to give an approximation of the rate constant to be the f i r s t eigenvalue of the Smoluchowski equation, Eq.(4.34). If the potential barrier i s large enough, then the form of f i ( x ) approaches a step function times the equilibrium PDF. This form of the f i r s t eigenfunction may be c l e a r l y seen in Figure 3.9. A set of An are given by A n " ' y B ) ' 1 ^ J f n ( x ) d x LQ n (4.62) 107 which i s analogous to Eq.(4.58). These c o e f f i c i e n t s are used to evaluate the r a t e c o e f f i c i e n t i n the high c o l l i s i o n frequency l i m i t s t a r t i n g from the c o r r e l a t i o n f u n c t i o n f o r m a l i s m , thus OO T ;1= Z A*Jexp(-X t) (4.63) S n=0 n n There have been a number of recent d i s c u s s i o n s on the r e l a t i o n between the Kramers and Smoluchowski equations [R26,33]. In g e n e r a l , one cannot expect that the time dependence of the r a t e constant given by the Smoluchowski equation to match that given by Kramers equation s i n c e the former approximation depends on an e q u i l i b r i u m v e l o c i t y d i s t r i b u t i o n f u n c t i o n . Furthermore, the e i g e n f u n c t i o n of the Smoluchowski operator are c o l l i s i o n frequency independent while the ei g e n v a l u e s have an i n v e r s e c o l l i s i o n frequency dependence. Hence, the time dependent rate constant given by Eq.(4.63) has an i n v e r s e c o l l i s i o n frequency dependence at t=0 and cannot match the t r a n s i t i o n s t a t e theory r e s u l t at more than one c o l l i s i o n frequency. In the high c o l l i s i o n frequency l i m i t , the f i r s t e i g e n v a l u e of Lg, the Smoluchowski o p e r a t o r , may be approximated by Kramers formula [R29] TKRM = ( ^ t U" ( x a ) U " ( x 0 ) |exp( |U (xa)-U (x 0 ) | A T ) + (4.64) v/ j U" (xb)U" (x0) |exp( |U(xb)-U(x0) | /kT) )/2irwn where xfi and xb are l o c a l minima of the p o t e n t i a l energy f u n c t i o n . The Kramers approximation i n d i c a t e s t h a t , i n the 1 08 high frequency l i m i t , the rate constant should be proportional to the inverse of the c o l l i s i o n frequency. This is the same dependence that the f i r s t eigenvalue of the Smoluchowski operator w i l l have since this operator only depends on the c o l l i s i o n frequency, v , as a m u l t i p l i c a t i v e f a c t o r . The v a l i d i t y of Kramers approximation w i l l depend on both the c o l l i s i o n frequency and the exact nature of the intramolecular potential as discussed in the introduction. 4.8 PRESENT METHOD OF DETERMINING THE RATE CONSTANT An 11th degree polynomial f i t of a portion of the potential was made. As discussed in Chapter 2, the Smoluchowski operator, Eq.(4.35), which has the form of a Fokker-Planck operator, is conveniently transformed by writing the eigenfunctions in the form of fn( x ) = exp(-J *(x')dx') hn( x ) , (4.65) and with the substitution into Eq.(4.35), yie l d s an eigenvalue equation for hn( x ) , that i s , Lshn(x) = -Xnhn(x), (4.66) where L h (x) = 1; s n 7 9x 9xz hn( x ) . (4.67) This eigenvalue equation is solved with the DO method discussed in Chapter 2. The eigenvalues and eigenfunctions are computed with the numerical diagonalization of the DO representation of 109 Lg as given by Eq.(2.33). It should be noted that the solution vector gives the value of hn(x) exp(-J $(x')dx') w(x) 1/2 evaluated at the quadrature points times the corresponding quadrature weight. The eigenvalues of the Kramers operator were also found by diagonalizing a matrix representation of the operator, as described as follows. With the eigenfunction Pn(x,y) written in the form (4.68) I find with Eq.(4.23) the following equation for \J/ (x,y), L, \L (x,y) = -X \b , k*n '1 n*n' where the operator L, is defined by (4.69) Vn(x,y) - * U ) h (4.70) 9_ a y 2y An obvious choice of basis functions for representing this operator in the speed variable are the Hermite polynomials since the second term in big brackets on the right hand side of Eq.(4.70) i s the Hermite d i f f e r e n t i a l operator. When this is done, I obtain the following tri-diagonal matrix representation of the operator L^, in units of v/2 1 10 0 ydx. 0 ^ < * - 2 - ) 7 9x; -2 .•49 79x 0 0 <^*-|-> 7 9x -4 .•6 3  79x 7 9x .•83 79x The natural choice of basis functions for representing the s p a t i a l portion of the eigenfunctions of are the eigenfunctions, hn( x ) , of Lg. This y i e l d s , after doing one integration by parts, <n'm'|L, |mn> = -2m5m m,6„„ , - • 2 ( m + 1 ) „ - G , , - (4.71) 1 k1 m  nn m+1,m n n • 2m 6 , ,G >, v m-1,m nn ' where the m index refers to the Hermite basis and the n index refers to the Smoluchowski eigenfunction basis. The matrix G > i s defined by 111 °° x dh Gn,n = / exp(-f #(x')dx') h , ^ d x . (4.72) — CO These matrix elements are determined n u m e r i c a l l y as f o l l o w s . The i n t e g r a t i o n i s done n u m e r i c a l l y with the quadrature r u l e d e f i n e d by Eq.(2.17). The e i g e n f u n c t i o n s , hn( x ) , were determined at the quadrature p o i n t s of the bimode polynomials with the numerical d i a g o n a l i z a t i o n of Eq.(2.33) as d e s c r i b e d p r e v i o u s l y . The d e r i v a t i v e i n the integrand of Eq.(4.72) was determined at the same set of quadrature p o i n t s with the DO r e s u l t , Eq.(2.23). In t h i s way, the matrix elements G , are determined a c c u r a t e l y and n n •* e f f i c i e n t l y . T h i s e x e m p l i f i e s the f l e x i b i l i t y of the DO method in t r e a t i n g i n t e g r a t i o n and d i f f e r e n t i o n i n an analogous f a s h i o n . The matrix r e p r e s e n t a t i o n of the Kramers o p e r a t o r , Eq.(4.71), was d i a g o n a l i z e d using M Hermite polynomial b a s i s f u n c t i o n s and N Smoluchowski eigenfunct i o n s . I t should be noted that the matrix approximation to i s not symmetric and can not be symmetrized. T h i s i s a consequence of the f a c t that the Kramers operator i s not s e l f - a d j o i n t . R e c a l l that a s e l f - a d j o i n t operator has a r e a l eigenvalue spectrum, while the Kramers operator has a complex spectrum. The p h y s i c a l reason f o r t h i s i s that the d r i f t term d e s c r i b e s the motion of a p a r t i c l e moving i n an e x t e r n a l p o t e n t i a l . T h i s motion, f o r the case of v=0, i s p e r i o d i c and hence the e i g e n v a l u e s are imaginary. As the c o l l i s i o n frequency i s i n c r e a s e d , these p e r i o d i c modes w i l l be damped out and the eigenvalue spectrum becomes complex. 1 1 2 4.9 DISCUSSION OF NUMERICAL RESULTS Table 4.2 shows the convergence of the eigenvalues of the Smoluchowski operator as a f u n c t i o n of the number of quadrature p o i n t s . The r a t e of convergence i s remarkable with the f i r s t 10 ei g e n v a l u e s c a l c u l a t e d to 8 s i g n i f i c a n t f i g u r e s with 50 quadrature p o i n t s . Since these e i g e n f u n c t i o n s are used to c o n s t r u c t the matrix r e p r e s e n t a t i o n of the Smoluchowski o p e r a t o r , i t i s necessary that they be c a l c u l a t e d w e l l . Since the r e s u l t s of the c a l c u l a t i o n of the r a t e constant with the Smoluchowski equation i s v a l i d only i n the high c o l l i s i o n frequency l i m i t , i t i s u s e f u l to d i s c u s s these r e s u l t s with the r e s u l t s obtained from the Kramers equation to show t h e i r domain of v a l i d i t y . However, i t i s f i r s t necessary to d i s c u s s the convergence of the Kramers e i g e n v a l u e s . Tables 4.3-4.5 give the convergence of the lowest order e i g e n v a l u e s of the Kramers operator f o r v a r i o u s values of v as a f u n c t i o n of N and M. I t i s c l e a r from these t a b l e s that the convergence becomes more r a p i d as the c o l l i s i o n frequency i s i n c r e a s e d . T h i s f o l l o w s from the f a c t that the matrix r e p r e s e n t a t i o n of L^ becomes d i a g o n a l l y dominant as the c o l l i s i o n frequency, 7 , becomes l a r g e with the o f f d i a g o n a l elements being i n v e r s e l y r e l a t e d to 7 . Since the order of the matrix d i a g o n a l i z e d i s NM, which i s on the order of 200, i t i s p o s s i b l e to a c c u r a t e l y o b t a i n only the lowest few eigenvalues of the Kramers o p e r a t o r . However c o n s i d e r i n g the small number of p o i n t s in each dimension, 1 13 TABLE 4.2: Convergence of the Smoluchowski Eigenvalues1 N M X2 x3 x5 x9 1 0 0 .001383516 0. 13105988 0. 20780348 0. 41295570 2. 81987476 20 0 .000804173 0. 12615340 0. 21 121901 0. 39738190 0. 92250019 30 0 .000799649 0. 12615442 0. 2113211 1 0. 39750087 0. 90081948 40 0 .000799619 0. 12615448 0. 21132117 0. 39750009 0. 90043109 50 0 .000799619 0. 12615448 0. 21132117 0. 39749997 0. 90042930 60 0. 39749997 0. 90042930 1. Harmonic potential 2. X in units of 3Xl01 3sec"1 especially in the speed va r i a b l e , the convergence i s rapid. The f i r s t eigenvalue is taken to be the rate c o e f f i c i e n t as discussed and i t i s interesting to note the variation of this and a few other eigenvalues as a function of c o l l i s i o n frequency. These results are shown in Table 4.6 Figure 4.4 shows a comparison of T ~ \ calculated from the Kramers and Smoluchowski operators as well as from Kramers approximation and the t r a n s i t i o n state theory approximation. These results q u a l i t a t i v e l y agree with those obtained by MCB who used a square well intramolecular p o t e n t i a l . As Chandler has previously noted, the t r a n s i t i o n state theory approximation overestimates the rate constant. In this l a t t e r approximation, as well as for the Kramers estimate of the f i r s t eigenvalue of the Smoluchowski operator, the numerical value of th i s rate constant i s c r i t i c a l l y dependent on the height of the potential b a r r i e r . 1 1 4 Table 4.3: Convergence of eigenvalues v=v0 N\M 5 7 9 11 X] 5 0 . 2 1 5 1 ( - 2 ) 0 . 2 2 0 0 ( - 2 ) 0 . 2 2 0 1 ( - 2 ) 0 . 2 2 0 1 ( - 2 ) 1 0 0 . 1 1 8 0 ( - 2 ) 0 . 1 4 6 6 ( - 2 ) 0 . 1 6 2 9 ( - 2 ) 0 . 1 6 8 3 ( - 2 ) 1 5 0 . 1 1 0 8 ( - 2 ) 0 . 1 3 7 5 ( - 2 ) 0 . 1 4 9 3 ( - 2 ) 0 . 1 5 4 6 ( - 2 ) 2 0 0 . 1 1 0 7 ( - 2 ) 0 . 1 3 3 3 ( - 2 ) 0 . 1 4 3 9 ( - 2 ) 0 . 1488C- 2 ) x 6 10 0 . 081 21 0 . 0 9 1 4 4 0 . 0 9 5 1 6 0' . 0 9 6 8 6 1 5 0 . 0 7 7 4 1 0 . 081 9 8 0 . 0 8 3 2 0 0 . 0 8 3 6 5 2 0 0 . 0 7 7 4 1 0 . 0 8 1 9 2 0 . 0 8 3 1 5 0 . 0 8 3 6 0 x f 5 0 . 0 6 4 6 4 0 . 0 6 4 7 3 0 . 0 6 4 7 6 0 . 0 6 4 7 6 1 0 0 . 0 7 6 6 4 0 . 0 7 8 3 8 0 . 0 7 8 1 6 0 . 0 7 7 4 4 1 5 0 . 0 7 3 2 5 0 . 0 7 8 3 3 0 . 0 7 9 0 4 0 . 0 7 9 0 7 2 0 0 . 0 7 3 2 6 0 . 0 7 7 6 8 0 . 0 7 9 0 7 0 . 0 7 8 7 2 x£ 5 0 . 4 1 9 3 3 0 . 4 2 0 6 4 0 . 4 2 0 7 3 0 . 4 2 0 7 3 10 0 . 4 2 4 8 3 0 . 4 2 8 3 0 0 . 4 2 8 8 5 0 . 4 2 8 7 2 1 5 0 . 4 2 7 9 1 0 . 4 2 8 8 0 0 . 4 2 8 5 8 0 . 4 2 8 7 0 2 0 0 . 4 2 8 4 2 0 . 4 2 8 9 5 0 . 4 2 8 3 8 0 . 4 2 8 5 5 1. X i n u n i t s of 3X101 3 s e c '1 1 1 5 Table 4.4: Convergence of eigenvalues v=2v0 N\M 1 1 X] 5 10 15 20 0. 1653(-2) 0. 1459(-2) 0. 1434(-2) 0.1433(-2) 0.1670(-2) 0.1578(-2) 0.1556C-2) 0.1549(-2) 0.1671(-2) 0.1610(-2) 0.1589(-2) 0.1581(-2) 0.1671(-2) 0. 1617(-2) 0.1597(-2) 0.1589(-2) X6 10 1 5 20 0.1548 0. 1 483 0. 1483 0.1692 0.1576 0.1575 0.1725 0.1597 0.1595 0.1617 0.1603 0.1601 X2 5 1 0 1 5 20 0.12003 0.13109 0.13087 0.13087 0.11995 0.13150 0.13189 0.13182 0.11995 0.13141 0.13202 0.13197 0.11995 0.13136 0.13202 0.13199 5 1 0 15 20 0.41630 0.41732 0.41763 0.41767 0.41651 0.41777 0.41749 0.41750 0.41651 0.41779 0.41757 0.41755 0.41651 0.41781 0.41760 0.41759 1. X in units of 3X1013 sec"1 1 1 6 Table 4.5: Convergence of eigenvalues N\M 5 7 9 1 1 X] 5 0. 1 1 301(-2) 0.11304( -2) 0. 1 1 304( -2) 0. 11304( -2) 10 0. 1 1475(-2) 0.11523( -2) 0. 1 1 526( -2) 0. 11526( -2) 15 0. 11474(-2) 0.11526( -2) 0. 11530( -2) 0. 11530( -2) 20 0. 1 1 474(-2) 0.11526( -2) 0. 11530( -2) 0. 11530( -2) X7 10 0. 39278 0.41356 0. 41 488 0. 41 492 1 5 0. 37979 0.39913 0. 401 26 0. 401 50 20 0. 37984 0.39914 0. 40126 0. 401 51 x^  5 0. 27256 0.27256 0. 27256 0. 27256 10 0. 29460 0.29455 0. 29454 0. 29454 1 5 0. 29503 0.29502 0. 29502 0. 29502 20 0. 29504 0.29502 0. 29502 0. 29502 xi 5 0. 36622 0.36623 0. 36623 0. 36623 10 0. 36429 0.36419 0. 3641 8 0. 3641 8 1 5 0. 36432 0.36431 0. 36431 0. 36430 20 0. 36432 0.36431 0. 36431 0. 36431 1 . X in units of 3X1013 sec_ 1 1 1 7 TABLE 4.6: Eigenvalues of Kramers operator 2u0 3v0 5J^O 1 0i>o x! 0.00149 0.00159 0.00144 0.00115 0.000716 x^  0.0787 0. 1 32 0. 186 0.222 0.124 x| -0.428 -0.418 -0.405 0.0 0.0 x^  0.0787 0. 1 32 0.186 0.295 0.232 xj 0.428 0.418 0.405 0.364 0.0 x^  0.0791 0.133 0. 1 88 0.299 0.543 x, 0.612 0.602 0.593 0.566 0.0 x^  0.111 0.203 0.282 0. 402 0.602 x^  0.0 0.0 0.0 0.0 0.418 x^  0. 175 0.313 0.422 0.642 0.7.15 xi 0.412 0.826 0.806 0.736 0.0750 x^ 0.191 0.323 0 .440 0.668 0.935 x,j 0.564 1 . 1 96 1.186 1 . 1 54 0.0 x,f 0. 198 0.356 0.509 0.815 1 .253 x,i 0.844 0.538 0.510 0.473 0.930 1. X in units of 3X1013 s e c- 1 On the other hand, the rate constants estimated from the numerical value of the f i r s t eigenvalue of the Smoluchowski and the Kramers operators are r e l a t i v e l y insensitive to the exact shape of the potential b a r r i e r . The Kramers 118 FIGURE 4.4 T ~1 as a f u n c t i o n of v/v0 c a l c u l a t e d by (a) Kramers e q u a t i o n , (b) Smoluchowski e q u a t i o n , (c) Kramers approximation and (d) MCB [R31]. 119 approximation to the value of the r a t e constant i s a reasonable approximation to the f i r s t eigenvalue of the Smoluchowski e q u a t i o n . The two r e s u l t s agree to w i t h i n about 10%. Apparently the b a r r i e r need not be extremely high f or t h i s approximation to be v a l i d . I t should be r e c a l l e d that the b a r r i e r height i s only about 4kT and t h i s approximation i s v a l i d f o r b a r r i e r h e i g h t s much higher than kT. It i s a l s o apparent that c o l l i s i o n f r e q u e n c i e s must be much gr e a t e r than 1.0ro for the Smoluchowski equation to provide a reasonable approximation of the r a t e c o n s t a n t . Within the context of the approximations used, the c o l l i s i o n f r e q u e n c i e s f o r which the Smoluchowski equation i s v a l i d are l a r g e r than p h y s i c a l l y reasonable ones. F i g u r e 4.5 shows a p l o t of the L o g ( T " 1 ) vs 1/T f o r the v a r i o u s approximations to T "1. T h i s p l o t should be a s t r a i g h t l i n e a c c o r d i n g to both the phenomenological law as w e l l as the t r a n s i t i o n s t a t e theory approximation. The slope of the l i n e g i v e s the a c t i v a t i o n engery AE^. The numerical s o l u t i o n of Kramers equation a l s o y i e l d s the constant A EA« A l l of these curves give an a c t i v a t i o n energy on the order of 9.6KJ/mole. F i g u r e s 4.6 and 4.7 show graphs of the r a t i o of k ( t ) / kTgT versus vt, where k ( t ) i s r e l a t e d to T~ 1 by k ( t ) = XAXBr - \ (4.73) and k ( t ) i s the r e a c t i v e f l u x f o r the system. The t r a n s i e n t 120 FIGURE 4.5 Log ( T~ 1 /I Oi'o) as a function of l/T calculated by (a) Kramers equation v=l0vOr (b) Smoluchowski equation v=\0vo and (c) Kramers approximation »>=10I>O, (d) Kramers equation u=2u0r (e) Smoluchowski equation V=10VO and (f) Kramers approximation v=2vQ. 121 FIGURE 4.6 1000K/T R e a c t i v e f l u x k ( t ) as a f u n c t i o n of time at T=300 c a l c u l a t e d from the Smoluchowski equation with (a) the f i t t e d p o t e n t i a l ^ l O ^ O f (b) the f i t t e d harmonic p o t e n t i a l V = 1 0 V O , (MM) r e s u l t of Marechal and Moreau v=A0vOl and (MCB) r e s u l t of Montgomery et a l . , v=v0. FIGURE 4.7 Rea c t i v e f l u x k ( t ) as a f u n c t i o n of time at T=300 c a l c u l a t e d from the Kramers equation with (a) v=]0vOr (b) v=bv0 and (c) v=2v0. Curves 1, 2, and 3 are c a l c u l a t e d with 20, 30, and 40 terms, r e s p e c t i v e l y , in the expansion f o r k ( t ) . 123 k ( t ) was c a l c u l a t e d with Eq.(4.56) or Eq.(4.62) and Eq.(4.73) depending on whether the Kramers or Smoluchowski equation was used to c a l c u l a t e the time dependence of the r a t e c o n s t a n t . Table 4.7 g i v e s a t a b l e of the expansion c o e f f i c i e n t s A „ c a l c u l a t e d with both the Smoluchowski n equation and the Kramers e q u a t i o n . It i s c l e a r l y seen t h a t , although the p o t e n t i a l energy b a r r i e r i s r e l a t i v e l y low, A0 and A, are much l a r g e r than than the other A . It i s 3 n i n t e r e s t i n g to note that these r e s u l t s seem to be i n q u a l i t a t i v e agreement with Skinner and Wolynes' a s s e r t i o n [R33] that the ln(A ) f o r n>1 are p r o p o r t i o n a l to minus the p o t e n t i a l b a r r i e r h e i g h t . Thus, most of the r e a c t a n t s must f o l l o w the f i r s t order r a t e law s i n c e , by the d e f i n i t i o n of NA and the use of the eigenvalue expansion f o r f ( x , t ) or P ( x , y , t ) , I have that oo NA = N Z A exp(-A t ) xA, (4.74) n = 0 given that a l l of the molecules are i n the gauche s t a t e at t=0. If the An were a l l of the same order of magnitude then a l a r g e p o r t i o n of the r e a c t a n t s would not f o l l o w the r a t e law given by Eq.(4.2), even though a f t e r s u f f i c i e n t time the remainder of the r e a c t a n t s would approach t h i s r a te law. The present r e s u l t s are s i m i l a r to those obtained by MCB, but not those obtained by Marechal and Moreau. In the l a t t e r c a s e , the e i g e n f u n c t i o n s of the Smoluchowski operator were obtained by a WKB approximation. T h i s approximation may not be v a l i d , in t h i s case s i n c e the b a r r i e r height i s so 1 24 Table 4.7: Expansion c o e f f i c i e n t s A1 A2 n A 3 n 1 0.9961 0.9955 2 0.6366(-2) 0.5780(-5) 3 0.70l2(-3) 0.8601(-3) 4 0.4917(-3) 0.2228(-3) 5 0.1993(-3) 0.6432(-4) 6 0.4576(-3) 0.4242(-5) 1. * = 1 0 . J > O 2. Smoulchowki equation 3. Kramers equation small [R102]. It a l s o should be noted that the s t r u c t u r e which i s found in the r e a c t i v e f l u x c a l c u l a t e d by. MCB cannot be found from a s o l u t i o n of the Smoluchowski equation s i n c e the An must be p o s i t i v e and the eigenvalues are r e a l and p o s i t i v e . Since the e i g e n f u n c t i o n and eigenvalues of the Kramers operator are not n e c e s s a r i l y r e a l , the r e a c t i v e f l u x need not be a monotonically d e c r e a s i n g f u n c t i o n . These r e a c t i v e f l u x e s were c a l c u l a t e d with the use of 14 and 18 terms r e s p e c t i v e l y f o r curves a, b i n Fig u r e 4.6. In the case of Kramers e q u a t i o n , i t was shown that the sum Eq.(4.56) should be equal to the t r a n s i t i o n s t a t e theory value at t=0 i n S e c t i o n 4.7. The present r e s u l t s are not 1 25 agreement with t h i s value at t=0. For curves a and c i n F i g u r e 4.7, 20 expansion c o e f f i c i e n t s were used, while 20, 30 and 40 terms were used in curve b. I a t t r i b u t e t h i s disagreement to the numerical problems a s s o c i a t e d with c a l c u l a t i n g a l l of the An and to the d i f f i c u l t y i n f i t t i n g the i n i t i a l c o n d i t i o n . I t should be r e c a l l e d that to o b t a i n these c o e f f i c i e n t s , i t i s necessary not only to n u m e r i c a l l y evaluate the matrix elements of the Kramers o p e r a t o r , but a l s o to n u m e r i c a l l y i n t e g r a t e the e i g e n f u n c t i o n s of t h i s o p e r a t o r . My main o b j e c t i v e , however, was to c a l c u l a t e the long time r a t e c o e f f i c i e n t and to study the c o n d i t i o n s for which i t i s v a l i d . The p o t e n t i a l energy b a r r i e r i s only about 4 times the average thermal energy, consequently t h i s system i s near the l i m i t of the v a l i d i t y of the a n a l y s i s presented i n S e c t i o n 4.7, which depends on a l a r g e p o t e n t i a l b a r r i e r . The v a l i d i t y of choosing the r e a c t i o n rate c o n s t a n t , T "1, to be the r a t e constant may be t e s t e d by l o o k i n g at the r a t i o X2/X1 and the nature of the f i r s t e i g e n f u n c t i o n . F i g u r e 4.8 shows the r a t i o of the f i r s t two non-zero eigenvalues of Kramers equation as a f u n c t i o n of temperature. T h i s c l e a r l y shows that the phenomenological r a t e law must begin to f a i l as T i s i n c r e a s e d . T h i s i s because the e f f e c t i v e b a r r i e r height i s reduced as the temperature i s i n c r e a s e d . T h i s system i s , even at room temperature, near the l i m i t of v a l i d i t y of f i r s t order k i n e t i c s . FIGURE 4.8 126 X2/X, as a function of temperature for v=2v0. 1 27 FIGURE 4.9 # 1 (x ) vs x. 1 28 It i s also interesting to plot the f i r s t eigenfunction of the Kramers operator. Consider the function defined by 00 00 $,(x) = J P1(x,y)dy/J P0(x)dx. (4.75) — oo —oo I had suggested that this should approach a step function as the barrier height increases. Figure 4.9 shows a graph of # at T=500K. It is c l e a r l y seen that t h i s approximates the form suggested by Eq.(4.59). In this chapter I have studied the s t a t i s t i c a l mechanics of conformational changes in n-butane. The f i r s t order rate law and rate constant may be r a t i o n a l i z e d in terms of kinetic operators such as the Kramers and Smoluchowski operators. The rate constant obtained from the f i r s t eigenvalue of these operators is i d e n t i c a l to the one calculated from the co r r e l a t i o n function formalism developed by Chandler and others [R67,106,107] provided the potential barrier is large enough. The numerical methods developed here should be useful in other applications of the Kramers equation. Other applications are discussed in recent books by Risken [R26] and by Gardiner [R25]. 5. ESCAPE OF ATMOSPHERES FROM PLANETARY BODIES I w i l l c o n s i d e r the problem of the escape of l i g h t components of p l a n e t a r y atmospheres i n t h i s c h a p t e r . T h i s problem o v e r l a p s s t r o n g l y with the problems c o n s i d e r e d i n pr e v i o u s c h a p t e r s . The two main areas of o v e r l a p are the methodology of a n a l y s i s and the p h y s i c a l s i m i l a r i t i e s . The formal s i m i l a r i t y of t h i s problem with the p r e v i o u s problems w i l l become c l e a r as the formalism i s developed. I t i s a c l a s s i c problem i n k i n e t i c t h e o r y , and the methodology developed i n the previous c h a p t e r s and in Appendix A can be c o n v e n i e n t l y a p p l i e d to t h i s problem. I w i l l begin with a b r i e f d e s c r i p t i o n of the p h y s i c a l problem, and i t s importance to p l a n e t a r y s c i e n c e and r a r e f i e d gas dynamics. A p a r t i c u l a r model of the upper atmosphere w i l l then be c o n s i d e r e d . F i n a l l y , d e n s i t y and temperature p r o f i l e s along with escape f l u x e s w i l l be c a l c u l a t e d and comparisons with the r e s u l t s of other r e s e a r c h e r s w i l l be d i s c u s s e d . 5.1 INTRODUCTION In t h i s c h a p t e r , a model of p l a n e t a r y exospheres w i l l be developed. P r i o r to dev e l o p i n g t h i s model, i t i s necessary to f i r s t g ive a short background d i s c u s s i o n on r e l e v a n t f e a t u r e s of a p l a n e t a r y atmosphere with the earth's atmosphere as an example. The atmosphere of the e a r t h i s c o n v e n i e n t l y d i v i d e d i n t o three major zones, the homosphere, the heterosphere and the exosphere i n accordance with c h a r a c t e r i s t i c f e a t u r e s of 1 29 1 30 the temperature p r o f i l e as shown i n Figu r e 5.1. The lowest l e v e l , the homosphere or the p o r t i o n of the atmosphere w i t h i n 100 Km of the e a r t h , i s composed of molecular n i t r o g e n and oxygen. At the lower l e v e l s of the homosphere, the number d e n s i t y i s on the order of 101 9 cm- 3. T h i s • de n s i t y i s s u f f i c i e n t to ensure that the mean f r e e path of the molecules i s very s m a l l , about 10"5 cm, assuming a t o t a l e l a s t i c s c a t t e r i n g c r o s s s e c t i o n on the order of 30 square angstroms. A consequence of the high d e n s i t y and high c o l l i s i o n frequency i s that the gas w i l l have, at t h i s l e v e l , almost a Maxwell-Boltzmann v e l o c i t y d i s t r i b u t i o n f u n c t i o n . T h i s , coupled with the f a c t that the g r a d i e n t s of macroscopic i n t e n s i v e q u a n t i t i e s , such as the number d e n s i t y , are very small i n u n i t s of r e c i p r o c a l mean free path, ensures that o r d i n a r y hydrodynamic equations are v a l i d . As the name suggests, the homosphere i s a region where the atmosphere i s w e l l mixed, and hence homogeneous. This r e g i o n i s a l s o s u b d i v i d e d i n t o v a r i o u s subregions, the lowest of which i s the trop o s p h e r e . T h i s r e g i o n i s the subject of a great d e a l of r e s e a r c h s i n c e most of the earth's weather occurs h e r e . Above 100 Km, the number d e n s i t y of p a r t i c l e s i n the atmosphere becomes too small to support c o n v e c t i v e c u r r e n t s , and hence t h i s region of the atmosphere g r a d u a l l y changes from one composed mainly of molecular oxygen and n i t r o g e n to one composed of hydrogen and oxygen atoms. T h i s region i s a l s o c h a r a c t e r i z e d by i n c r e a s i n g temperature and d i m i n i s h i n g FIGURE 5.1 Temperature p r o f i l e of the e a r t h ' s atmosphere. 1 32 number d e n s i t i e s as the a l t i t u d e i n c r e a s e s . The temperature i n c r e a s e i s caused by the a b s o r p t i o n of u l t r a v i o l e t r a d i a t i o n while the change i n the composition of the atmosphere i s a r e s u l t of the f a c t t h a t , i n the absence of mixing, the heavier gases w i l l tend to sink to lower a l t i t u d e s while the l i g h t e r gases r i s e above these heavier gases. These statements may be made mathematically p r e c i s e . The requirement for momentum c o n s e r v a t i o n y i e l d s the equation f o r h y d r o s t a t i c e q u i l i b r i u m , that i s , g f = PF, (5.1) where p i s the p r e s s u r e , r i s the d i s t a n c e from the center of the e a r t h to some atmospheric l e v e l , p i s the number d e n s i t y , and GMm . F = - - p r1, (5.2) i s the g r a v i t a t i o n a l f o r c e a c t i n g r a d i a l l y on the p a r t i c l e s . In Eq.(5.2), G i s the g r a v i t a t i o n a l c o n s t a n t , M i s a pl a n e t a r y mass and I T U i s the mass of the j t h gas p a r t i c l e . T h i s , together with the assumption of i d e a l gas behaviour, P=nkT, immediately leads to the number d e n s i t y of each molecular component being given by r i j ( r ) = r i j ( r0) e x p [ X j ( r ) - X j ( r0) ] , ' (5.3) where, X j ( r ) = GMm_j/kTr, (5.4) 1 33 and r0 i s some re f e r e n c e d i s t a n c e . Eq.(5.3) i s c a l l e d the barometric law. It i s u s e f u l to r e w r i t e Eq.(5.3) as i s the s c a l e h e i g h t . T h i s d e f i n i t i o n of the s c a l e height i s somewhat d i f f e r e n t than that used by other r e s e a r c h e r s , however H^(r) changes slowly i n the region of i n t e r e s t in the atmosphere. Equation (5.5) s t a t e s that each component of the atmosphere has an approximate e x p o n e n t i a l decrease i n number d e n s i t y as a f u n c t i o n of r - r ^ with a decay constant of 1/Hj. T h i s decay constant i s p r o p o r t i o n a l to the mass of the atmospheric p a r t i c l e and t h e r e f o r e the number d e n s i t y of heavy molecules and atoms w i l l decrease more r a p i d l y than that of l i g h t e r p a r t i c l e s . It should be noted that the barometric law cannot be v a l i d f o r a l l l e v e l s of the atmosphere s i n c e , a c c o r d i n g to t h i s law, the number d e n s i t y approaches a constant as the r a d i u s approaches i n f i n i t y . If Eq.(5.3) were v a l i d at a l l a l t i t u d e s , then the atmosphere would have an i n f i n i t e mass. The reason the law f a i l s i s t h a t , at s u f f i c i e n t l y l a r g e a l t i t u d e s , the v e l o c i t y d i s t r i b u t i o n f u n c t i o n d e v i a t e s s i g n i f i c a n t l y from a Maxwell-Boltzmann d i s t r i b u t i o n and hence hydrodynamic equations such as E q . ( 5 . l ) become i n v a l i d . T h i s w i l l be d i s c u s s e d i n more d e t a i l l a t e r . (5.5) where H.(r) = kTrr0/GMm. (5.6) 1 34 It i s convenient to d e f i n e a t r a n s i t i o n r e gion where, because of the small number d e n s i t y of atmospheric gases the mean f r e e path of a p a r t i c l e i s of the same order as the s c a l e height and hydrodynamic equations such as Eq . ( 5 . 1 ) are no longer v a l i d . It i s convenient to d e f i n e an a l t i t u d e c a l l e d the exobase or c r i t i c a l l e v e l , rn, where the mean f r e e path i s equal to the s c a l e h e i g h t . The t h i c k n e s s of the t r a n s i t i o n region i s on the order of s e v e r a l mean free p a t h s . T h i s corresponds to an a l t i t u d e change on the order of a few hundred kilometers from the top to the bottom, and hence the s c a l e height changes by only a few percent i n t h i s r e g i o n . The region j u s t above the t r a n s i t i o n r e g i o n i s the exosphere. The exosphere i s a region where the number d e n s i t y of p a r t i c l e s i s so small that the mean f r e e path i s much gr e a t e r than the s c a l e h e i g h t , H. In t h i s r e g i o n , the atmosphere of the e a r t h i s composed mostly of atomic oxygen with minor amounts of atomic hydrogen and helium. The oxygen and hydrogen atoms w i l l not recombine s i n c e the frequency of three p a r t i c l e c o l l i s i o n s i s too low. The exospheres of other p l a n e t s are s i m i l a r l y composed of a massive background gas as w e l l as a smaller amount of a l i g h t gas. There are s e v e r a l important consequences of the low number d e n s i t y of the upper atmosphere. A p a r t i c l e with v e l o c i t y g r e a t e r than the escape v e l o c i t y may escape from the e a r t h ' s atmosphere without ever c o l l i d i n g with other 1 35 p a r t i c l e s . P a r t i c l e s with v e l o c i t i e s in excess of the escape v e l o c i t y may be generated in a number of ways which w i l l be d i s c u s s e d l a t e r . Owing to the l o s s of e n e r g e t i c p a r t i c l e s , the v e l o c i t y d i s t r i b u t i o n f u n c t i o n w i l l depart from a Maxwell-Boltzmann d i s t r i b u t i o n , and, as s t a t e d b e f o r e , hydrodynamic equations such as E q . ( 5 . l ) w i l l no longer h o l d . The o b j e c t i v e of t h i s chapter i s to c a l c u l a t e the v e l o c i t y d i s t r i b u t i o n f u n c t i o n f o r the l i g h t escaping c o n s t i t u e n t s of a p l a n e t a r y atmosphere. T h i s v e l o c i t y d i s t r i b u t i o n f u n c t i o n i s perturbed from a Maxwell-Boltzmann d i s t r i b u t i o n f u n c t i o n by the escape of l i g h t components of the atmosphere. Owing to the infrequency of c o l l i s i o n s , t h i s p e r t u r b a t i o n to the d i s t r i b u t i o n f u n c t i o n p e r s i s t s s e v e r a l mean f r e e paths i n t o the atmosphere. The standard model of the upper atmosphere [R73,108] has been to c o n s i d e r i t to be c o l l i s i o n dominated below the c r i t i c a l l e v e l and c o l l i s i o n l e s s above t h i s l e v e l . The t r a n s i t i o n from the c o l l i s i o n dominated gas to the c o l l i s i o n l e s s gas in the v i c i n i t y of the exobase i s not c o n s i d e r e d . At the exobase, the atmosphere i s assumed to have a Maxwell-Boltzmann d i s t r i b u t i o n , except f o r the p o r t i o n of the d i s t r i b u t i o n f u n c t i o n r e p r e s e n t i n g p a r t i c l e s with speeds g r e a t e r than the escape speed and which are incoming, that i s , f(v,/i,rQ) f o r v > ve s c a n <3 M<0, where ju=v»r/vr. T h i s p o r t i o n of the d i s t r i b u t i o n f u n c t i o n i s set equal to z e r o . The d i s t r i b u t i o n f u n c t i o n i n the region 1 36 above the exobase may be found as a solution to the c o l l i s i o n l e s s Boltzmann equation, a F-V l| t + v - V r - = -^ } f - 0, (5.7) where F i s the gravitational force acting on the p a r t i c l e , Equation (5.7) is to be solved subject to the appropriate boundary conditions discussed l a t e r . In a spherical geometry, Eq.(5.7) may be rewritten as 9f . 9f GMM 9f . - T T - + V M — - £ — + 9t 9x 9v v2r-GM vr "-"''Is 0 (5.8) Equation (5.8) is a f i r s t order hyperbolic p a r t i a l d i f f e r e n t i a l equation. The standard method of solving such equations i s by the method of c h a r a c t e r i s t i c s . C h a r a c t e r i s t i c s are curves along which the d i s t r i b u t i o n function remains constant. Equation (5.8) is well posed when the boundary values define the value of the function once and only once for each c h a r a c t e r i s t i c . For the c o l l i s i o n l e s s Boltzmann equation, the c h a r a c t e r i s t i c s are the curves defined by the tr a j e c t o r i e s of p a r t i c l e s which conserve both the t o t a l energy, E = mv2/2-GMm/r, (5.9) and the o r b i t a l angular momentum, j 2 = m2r2v2( 1 - M2) . (5.10) These t r a j e c t o r i e s are, of course, those along which a pa r t i c l e t r a v e l s in a central p o t e n t i a l . 1 37 It i s u s e f u l to d i v i d e these c h a r a c t e r i s t i c s i n t o v a r i o u s c l a s s e s depending on the energy and angular momentum of the p a r t i c l e s . For convenience, I w i l l r e f e r to these c l a s s e s of c h a r a c t e r i s t i c s by the corresponding c l a s s e s of p a r t i c l e s . These c l a s s e s are shown i n F i g u r e 5.2. The f i r s t c l a s s corresponds to the t r a j e c t o r i e s f o l l o w e d by b a l l i s t i c p a r t i c l e s . These are p a r t i c l e s which c r o s s the exobase, but have i n s u f f i c i e n t energy to escape from the g r a v i t a t i o n a l f i e l d of the p l a n e t , hence they f o l l o w p a r a b o l i c paths i n t e r s e c t i n g with the exobase. A second c l a s s of c h a r a c t e r i s t i c s are t r a j e c t o r i e s of s a t e l l i t e p a r t i c l e s . These are p a r t i c l e s which a l s o have i n s u f f i c i e n t energy to escape, but have s u f f i c i e n t energy and angular momentum to stay above the exobase at a l l t i m e s . A t o p i c of c o n s i d e r a b l e i n t e r e s t has been the c a l c u l a t i o n of the number d e n s i t y and the l i f e t i m e of these s a t e l l i t e p a r t i c l e s [R73]. Escaping c h a r a c t e r i s t i c s are the c l a s s of curves d e s c r i b e d by p a r t i c l e s with speeds i n excess of the escape speed and c r o s s the exobase moving upward along h y p e r b o l i c t r a j e c t o r i e s . The corresponding downward moving p a r t i c l e s f o l l o w curves c a l l e d capture c h a r a c t e r i s t i c s . The f i n a l c l a s s of c h a r a c t e r i s t i c s are the t r a j e c t o r i e s of p a r t i c l e s which possess energy i n excess of escape energy but never c r o s s the exobase and are c a l l e d f l y - b y c h a r a c t e r i s t i c s . It should be noted each c h a r a c t e r i s t i c curve may be s p e c i f i e d by E and J . 1 38 FIGURE 5.2 1 39 A t r u n c a t e d Maxwel1-Boltzmann d i s t r i b u t i o n f u n c t i o n at the exobase i s not a s u f f i c i e n t boundary c o n d i t i o n to solve Eg.(5.7) for a l l a l t i t u d e s and v e l o c i t i e s , s i n c e i t only a p p l i e s to b a l l i s t i c , escaping and capture p a r t i c l e s . Boundary c o n d i t i o n s f o r s a t e l l i t e and f l y - b y p a r t i c l e s may be s p e c i f i e d by s e t t i n g the v e l o c i t y d i s t r i b u t i o n f u n c t i o n equal to zero along the s a t e l l i t e and f l y - b y c h a r a c t e r i s t i c s . A s o l u t i o n of Eg.(5.7) was c o n s t r u c t e d in t h i s way by Aamodt and Case [R108]. Using t h i s s o l u t i o n , they showed that the number d e n s i t y must d e v i a t e from the barometric law and go to zero as r — H o w e v e r , i t should be noted that t h i s s o l u t i o n f o r which the s a t e l l i t e p o p u l a t i o n i s empty does not agree with the experimental Lyman-a measurements [R109]. The f l u x of escaping p a r t i c l e s i s e a s i l y found with the boundary c o n d i t i o n s d e f i n e d i n the l a s t paragraph. The f l u x times r2 and the p a r t i c l e e n e r g i e s must be a constant independent of a l t i t u d e . Hence, i f the f l u x i s known at one a l t i t u d e , i t i s known at a l l a l t i t u d e s . The Jeans f l u x , FT, i s c a l c u l a t e d d i r e c t l y from the t r u n c a t e d Maxwellian d i s t r i b u t i o n , that i s , 27T7r°° F7 = nv/2kT /Trm; / J f (y)y3dy cos0 d(cos0)d0 (5.11) J 00 yc = nv/kT/27rm[ 1 + y*3 e x p(-y2), where y=/m/2kTv, n i s the number d e n s i t y of escaping 1 40 p a r t i c l e s with mass m, and yc=VGMm/kTr0 i s the escape speed d i v i d e d by i/2kT/m. The Jeans f l u x overestimates the true p a r t i c l e f l u x s i n c e the l o s s of high energy p a r t i c l e s at the exobase should reduce the number of upward moving p a r t i c l e s with speeds g r e a t e r than the escape speed from that given by a Maxwell-Boltzmann d i s t r i b u t i o n f u n c t i o n . Hence, the r a t i o RT=F/FT, r e f e r r e d to as the the Jeans r a t i o , w i l l depart from u n i t y . In order to estimate how great i s t h i s d e v i a t i o n from the Jeans f l u x , i t i s necessary to c a l c u l a t e the d i s t r i b u t i o n f u n c t i o n i n the t r a n s i t i o n r e gion j u s t below the exobase. The c o l l i s i o n l e s s model of the exosphere i s d e f i c i e n t in s e v e r a l important ways. Since a f u l l Maxwell-Boltzmann p o p u l a t i o n of escaping p a r t i c l e s i s i n c l u d e d , the Jeans r a t i o i s u n i t y . A l s o , the a c t u a l s a t e l l i t e H atom p o p u l a t i o n i s d e p l e t e d owing to charge exchange c o l l i s i o n s with e n e r g e t i c p r o t o n s , p h o t o i o n i z a t i o n , and s o l a r r a d i a t i o n pressure [R19]. A c o l l i s i o n l e s s model cannot take these processes i n t o account. In a d d i t i o n , nonthermal escape processes which i n v o l v e p r o d u c t i o n of e n e r g e t i c s p e c i e s as a r e s u l t of some i n e l a s t i c or r e a c t i v e c o l l i s i o n cannot be c o n s i d e r e d with a model that assumes a c o l l i s i o n l e s s ' exosphere. Such nonthermal escape processes are important l o s s mechanisms on e a r t h , Venus and other p l a n e t a r y bodies [R19,110]. Due to these d e f i c i e n c i e s i n the c o l l i s i o n l e s s model of the exosphere, many re s e a r c h e r s [R74-76,110-121] have 141 attempted to take i n t o account c o l l i s i o n s between atoms in the t r a n s i t i o n r e gion below the exobase. These e f f o r t s have had v a r y i n g s u c c e s s . A d e t a i l e d review and c r i t i q u e of these p r e v i o u s e f f o r t s has been given by Fahr and S h i z g a l [R19]. In t h i s r e g i o n , the c o l l i s i o n s between atoms can no longer be ignored and the c o l l i s i o n l e s s Boltzmann e q u a t i o n , Eq.(5.7), i s not v a l i d . I n s t e a d , the d i s t r i b u t i o n f u n c t i o n must be found as a s o l u t i o n of the c o l l i s i o n a l Boltzmann e q u a t i o n , given by {|t + - Y r + = ^ l f - 3f, (5.12) where J i s the c o l l i s i o n o p e r a t o r . As with Eq.(5.7), Eq.(5.12) i s to be solved s u b j e c t to a p p r o p r i a t e boundary c o n d i t i o n s . T h i s problem was f i r s t c o n s i d e r e d by Hays and L i u [R18]. In t h i s paper the c o l l i s i o n o p e r a t o r , J , was approximated as a s i m p l i f i e d p r o d u c t i o n term minus a l o s s term p r o p o r t i o n a l to the c o l l i s i o n frequency. Hays and L i u f u r t h e r assumed that the system c o u l d be t r e a t e d with a Lorentz model, i . e. the r a t i o between the background gas mass and the escaping p a r t i c l e gas mass i s assumed to be i n f i n i t e . Using t h i s model, they were ab l e to estimate the Jeans r a t i o . Many other r e s e a r c h e r s have attempted to c a l c u l a t e the Jeans r a t i o [R74-76,116,120,121]. I w i l l b r i e f l y d i s c u s s a few of these here and in more d e t a i l in the S e c t i o n 5.6. S h i z g a l and L i n d e n f e l d [R76,116] introduced a model f o r the thermal p r o d u c t i o n of atoms with v e l o c i t i e s g r e a t e r than 1 42 escape v e l o c i t y . Using t h i s model, they were able to solve a l o c a l Boltzmann equation and estimate Rj. T h i s l o c a l model has been c r i t i c i z e d by Fahr and S h i z g a l [R19] s i n c e i t n e g l e c t s the e f f e c t s of d i f f u s i o n and only c o n s i d e r s the i s o t r o p i c p o r t i o n of the d i s t r i b u t i o n f u n c t i o n . Fahr e t . a l . [R120,122] con s i d e r e d the problem from an energy c o n t i n u i t y p o i n t of view. They argued that the r a t e of conduction of heat by the escaping p a r t i c l e s must be equal to the rate that energy i s gained v i a c o l l i s i o n s with the background gas. The heat f l u x i s given by F o u r i e r ' s law, q = (5.13) which depends on the temperature gradient of the escaping gas. The temperature of the escaping gas i s lower than the temperature of the background gas and a t t a i n s a l i m i t i n g v a l u e . The escape f l u x i s given by the Jeans f l u x at the reduced temperature. T h i s work has a l s o been c r i t i c i z e d [R19] since hydrodynamic equations are used where hydrodynamic laws may not be v a l i d . In a d d i t i o n to these s e m i - a n a l y t i c c a l c u l a t i o n s of the Jeans r a t i o , s e v e r a l r e s e a r c h e r s have employed Monte-Carlo type c a l c u l a t i o n s [R72,74,75,121]. In these c a l c u l a t i o n s , a s l a b of atmosphere a few mean fr e e paths t h i c k i s c o n s i d e r e d , with a source of p a r t i c l e s below the bottom boundary. The b a s i c idea here i s to f o l l o w the t r a j e c t o r i e s of a l a r g e number of atoms and count the number which escape 1 43 and the number that don't escape. In t h i s way an estimate of Rj i s o b t a i n e d . It i s important to note, i n connection with the present work d e s c r i b e d below, that the Monte-Carlo c a l c u l a t i o n s ignore the e f f e c t of the g r a v i t a t i o n a l f i e l d i n s o f a r as the p a r t i c l e t r a j e c t o r i e s are assumed to be l i n e a r between c o l l i s i o n s . I w i l l c o n s ider a c o l l i s i o n a l model of the exosphere based on a s o l u t i o n of the Boltzmann equation f o r a geometry s i m i l a r to that chosen i n the Monte-Carlo c a l c u l a t i o n s . The two major approximations that I w i l l use are a plane p a r a l l e l geometry of the atmosphere, and the neglect of the g r a v i t a t i o n a l f o r c e i n the Boltzmann e q u a t i o n . These assumptions w i l l be j u s t i f i e d i n the next s e c t i o n . T h i s plane p a r a l l e l model of the t r a n s i t i o n r e gion w i l l be used to c a l c u l a t e the v e l o c i t y d i s t r i b u t i o n f u n c t i o n i n t h i s r e g i o n . Furthermore, temperature, and number d e n s i t y p r o f i l e s are c a l c u l a t e d as moments of the d i s t r i b u t i o n f u n c t i o n . A d i s c u s s i o n of the v a r i o u s other approaches to t h i s problem w i l l be given i n S e c t i o n 5.6. It i s important to note that t h i s model i s s i m i l a r t o that used i n a number of Monte-Carlo s t u d i e s of the t r a n s i t i o n r e gion [R72,74,75,121]. The s i m i l a r i t i e s and d i f f e r e n c e s i n the two approaches w i l l be d i s c u s s e d l a t e r and comparisons g i v e n . The plane p a r a l l e l model w i l l be c o n s i d e r e d in d e t a i l i n the next s e c t i o n , however i t i s u s e f u l to f i r s t d i s c u s s some of the p h y s i c a l s i m i l a r i t i e s with other problems co n s i d e r e d i n t h i s t h e s i s . 144 There are a number of p h y s i c a l s i m i l a r i t i e s between t h i s atmospheric escape problem and a chemical r e a c t i o n . In each case i t i s necessary th a t an atom or molecule a t t a i n s u f f i c i e n t energy to overcome some b a r r i e r . In the case of a chemical r e a c t i o n , t h i s b a r r i e r i s j u s t the p o t e n t i a l energy b a r r i e r along the r e a c t i o n c o o r d i n a t e between the r e a c t a n t s and p r o d u c t s , while f o r the atmospheric escape problem i t i s a g r a v i t a t i o n a l p o t e n t i a l b a r r i e r . I t i s a l s o i n t e r e s t i n g to note that the Jeans f l u x has an e x p o n e n t i a l dependence on the escape energy, that i s F ~ exp(-mv2 /2kT), (5.14) esc which i s s i m i l a r to the dependence of the rate constant 1/T, T -1 ~ exp(-AE/kT), (5.15) on the a c t i v a t i o n energy AE. Both of these equations are based on s i m i l a r e q u i l i b r i u m assumptions. In order to c a l c u l a t e e i t h e r the rate constant or the escape f l u x , t a k i n g i n t o account the concomitant departure from e q u i l i b r i u m , i t i s necessary to use an a p p r o p r i a t e k i n e t i c equation f o r the d i s t r i b u t i o n f u n c t i o n of r e a c t i n g and/or escaping p a r t i c l e s . For the n-butane i s o m e r i z a t i o n , the k i n e t i c equation used was the Kramers e q u a t i o n . In t h i s c h a p t e r , I w i l l use the Boltzmann equation for the d i s t r i b u t i o n f u n c t i o n of escaping p a r t i c l e s and c a l c u l a t i o n of the escape f l u x . These two problems have not only p h y s i c a l s i m i l a r i t i e s , but a l s o mathematical s i m i l a r i t i e s . The Boltzmann equation 145 i s f o r m a l l y very s i m i l a r to Kramers e q u a t i o n . Each of them d e s c r i b e s the time e v o l u t i o n of a one p a r t i c l e d i s t r i b u t i o n f u n c t i o n and the rate of change of the d i s t r i b u t i o n f u n c t i o n i s given by d r i f t and c o l l i s i o n operators o p e r a t i n g on the d i s t r i b u t i o n f u n c t i o n . The d r i f t p o r t i o n of the equation g i v e s the change of the v e l o c i t y d i s t r i b u t i o n f u n c t i o n per u n i t time as a r e s u l t of a p a r t i c l e moving under an e x t e r n a l f o r c e , while the c o l l i s i o n p o r t i o n gives the r a t e of change of the v e l o c i t y d i s t r i b u t i o n f u n c t i o n as a r e s u l t of random c o l l i s i o n s . In the case of Kramers e q u a t i o n , t h i s l a s t operator r e s u l t s from the modeling of these c o l l i s i o n s as random noise in the system while f o r the Boltzmann t h i s term r e s u l t s from the c o n s i d e r a t i o n of the dynamics of two p a r t i c l e c o l l i s i o n s . I t i s i n t e r e s t i n g to note that the l i n e a r i z e d hard sphere Boltzmann c o l l i s i o n o p e r a t o r , in the l i m i t of l a r g e mass r a t i o between the t e s t p a r t i c l e and heat bath p a r t i c l e , i s i d e n t i c a l to the Kramers c o l l i s i o n o p e r a t o r . 5.2 PLANE PARALLEL MODEL OF THE EXOSPHERE The model to be presented assumes that a s l a b of the atmosphere may be taken to represent the t r a n s i t i o n r e g i o n . T h i s s l a b c o n t a i n s two gases, a background or heat bath gas which i s assumed to be i n thermal e q u i l i b r i u m , and a t e s t p a r t i c l e gas which has a much lower number d e n s i t y than the heat bath gas and i s perturbed from a Maxwellian d i s t r i b u t i o n . On e a r t h , the heat bath gas i s oxygen, while 1 46 hydrogen i s the t e s t p a r t i c l e gas. The o b j e c t i v e i s to f i n d the p a r t i c l e d i s t r i b u t i o n f u n c t i o n f o r the t e s t p a r t i c l e gas. As p r e v i o u s l y mentioned, the a p p r o p r i a t e equation to solve i s the Boltzmann e q u a t i o n , Eq.(5.12). It should be noted that Eq.(5.12) i s not completely adequate s i n c e i t only i n c l u d e s changes in the v e l o c i t y d i s t r i b u t i o n f u n c t i o n caused by d r i f t and thermal c o l l i s i o n s . There are other p h y s i c a l mechanisms for p e r t u r b i n g the d i s t r i b u t i o n f u n c t i o n [Rl12,123—125], such as charge exchange r e a c t i o n s i n the e a r t h ' s upper atmosphere [R123]. Here a high energy proton may c o l l i d e with a hydrogen atom y i e l d i n g a high energy hydrogen atom and a p r o t o n . T h i s w i l l cause a net a d d i t i o n of p a r t i c l e s i n the high energy t a i l of the d i s t r i b u t i o n f u n c t i o n . T h e o r e t i c a l l y , Eq.(5.12) should have other terms to account for such p r o c e s s e s . Although I w i l l not c o n s i d e r such p r o c e s s e s , i t should not be d i f f i c u l t to i n c l u d e them. Since i t i s assumed that the t r a n s i t i o n r e gion i s small compared with the d i s t a n c e from the center of the e a r t h to the bottom of the s l a b of atmosphere, the g r a v i t a t i o n a l f o r c e term in the s l a b i s expected to be s m a l l . In other words, i t i s expected that between c o l l i s i o n s the t e s t p a r t i c l e s w i l l move in almost s t r a i g h t l i n e s . Hence, a u s e f u l approximation to Eq.(5.12) would be to ignore the f o r c e , F i n the d r i f t term. The e f f e c t s of the f o r c e w i l l be modeled by the boundary c o n d i t i o n as d i s c u s s e d l a t e r . With t h i s a pproximation, and assuming that the d i s t r i b u t i o n 1 47 has reached a steady s t a t e , the Boltzmann equation i s s i m p l i f i e d to (5.16) where J i s the c o l l i s i o n operator and M i s the c o s i n e of the angle between the s l a b normal and the v e l o c i t y v. The c o l l i s i o n operator w i l l be d i s c u s s e d i n d e t a i l i n the next s e c t i o n . T h i s model i s s c h e m a t i c a l l y represented i n F i g u r e 5.3 i n terms of reduced v a r i a b l e s d e f i n e d l a t e r . The boundary c o n d i t i o n s are chosen as f o l l o w s . At the bottom of the s l a b , the form of the v e l o c i t y d i s t r i b u t i o n f u n c t i o n f o r hydrogen i s given by where the f u n c t i o n U(v) i s to be determined. T h i s form of the d i s t r i b u t i o n f u n c t i o n was chosen so that i t would be almost Maxwellian yet s t i l l y i e l d a non-zero p a r t i c l e f l u x . At the top of the s l a b , the value of d i s t r i b u t i o n f u n c t i o n r e p r e s e n t i n g incoming p a r t i c l e s with v e l o c i t i e s l e s s than the escape v e l o c i t y i s set equal to the value of the d i s t r i b u t i o n f u n c t i o n r e p r e s e n t i n g outgoing p a r t i c l e s with the same speed. The d i s t r i b u t i o n f u n c t i o n evaluated at v g r e a t e r than the escape v e l o c i t y , ve s c, i s set equal to zero i f ii i s n e g a t i v e , that i s , the p o r t i o n of the d i s t r i b u t i o n f u n c t i o n r e p r e s e n t i n g incoming p a r t i c l e s i s set equal to' z e r o . The top boundary c o n d i t i o n i s c o n v e n i e n t l y given by the formula f ( v , y , r bot ) = fm( v ) + MU(v) (5.17) 148 FIGURE 5.3 0=COS"V • U P P E R BOUNDARY C R I T I C A L L E V E L x = - l L O W E R BOUNDARY Plane P a r a l l e l model of the t r a n s i t i o n r e g i o n . 149 f(v,-n) = 0(v esc - v ) f ( v ,M) M>0, (5.18) where 0(x) i s 1 for x > 0 and zero otherwise. T h i s top boundary c o n d i t i o n models the e f f e c t of the g r a v i t a t i o n a l f o r c e s i n c e upward moving p a r t i c l e s with speeds l e s s than the escape speed have p a r a b o l i c t r a j e c t o r i e s ( i n a plane p a r a l l e l geometry) and must l a t e r c r o s s the top boundary with the same speed, but i n a downward d i r e c t i o n . Furthermore, i n both the s p h e r i c a l and plane p a r a l l e l geometries, the value of M on c r o s s i n g the top boundary in the downward d i r e c t i o n i s equal to minus i t s value on c r o s s i n g i n the upward d i r e c t i o n . I t i s u s e f u l to put Eq.(5.16) i n t o dimensionless form. T h i s i s done e a s i l y by d e f i n i n g dimensionless a l t i t u d e and speed v a r i a b l e s . A convenient v a r i a b l e to use i n place of the a l t i t u d e v a r i a b l e r i s a v a r i a b l e analogous to o p t i c a l depth. T h i s v a r i a b l e i s d e f i n e d by where d /4 i s the d i f f e r e n t i a l hard sphere c r o s s s e c t i o n f o r e l a s t i c s c a t t e r i n g between the t e s t and heat bath p a r t i c l e s and n, i s the d e n s i t y of the background gas. The other change of v a r i a b l e i n v o l v e s the replacement of the speed v with the dimensionless speed d e f i n e d by With these d e f i n i t i o n s , Eq.(5.l6) may be r e w r i t t e n as 7rd2n , ( r ' )dr ' , (5.19) y = »/m/2kT v. (5.20) 150 y u | | = J f , (5.21) where J = {v/m/2kT/7rd2n1 (r) }J. (5.22) The next s e c t i o n s w i l l be d e d i c a t e d to s o l v i n g Eq.(5.21) subject to the a p p r o p r i a t e boundary c o n d i t i o n s , Eqs.(5.17) and(5.l8) i n the atmospheric s l a b . There are a number of approaches to s o l v i n g the Boltzmann equation [R5,126-129 ] . I w i l l seek a s o l u t i o n with the DO method [R77]. P r i o r to s o l v i n g t h i s equation i t i s necessary to c o n s i d e r d e t a i l s of the c o l l i s i o n o p e r a t o r . 5.3 COLLISION OPERATOR The g e n e r a l form of the c o l l i s i o n operator f o r the Boltzmann equation i s given by Jf = ; / [ f ' f T ' - f f ? ] a ( x ) | v - v , | d2x d3v , , (5.23) where v and v, are the v e l o c i t i e s of the t e s t and heat bath p a r t i c l e s r e s p e c t i v e l y , f1? i s the Maxwell-Boltzmann d i s t r i b u t i o n f o r the heat bath p a r t i c l e and a(x) i s the d i f f e r e n t i a l c r o s s s e c t i o n for t e s t p a r t i c l e - h e a t bath p a r t i c l e c o l l i s i o n s , and x i s the s o l i d s c a t t e r i n g a n g l e . T h i s i s one form of the l i n e a r Boltzmann c o l l i s i o n o p e r a t o r . Another u s e f u l form, in terms of a s c a t t e r i n g k e r n e l , i s given by' [ R70 ] Jf(y_) = JK(y_,y/ )f (y_' )d3y_ - »>(y_)f(y_), (5.24) where K i s the s c a t t e r i n g kernel and v i s the c o l l i s i o n 151 frequency. The f i r s t term on the r i g h t hand s i d e of Eq.(5.24) d e s c r i b e s the gain of p a r t i c l e s i n a p a r t i c u l a r region of v e l o c i t y space while the second term d e s c r i b e s the l o s s of p a r t i c l e s . Together they give the t o t a l r a t e of change of the v e l o c i t y d i s t r i b u t i o n f u n c t i o n as a r e s u l t of c o l l i s i o n s . The s c a t t e r i n g k e r n e l i n Eq.(5.24) has been determined by s e v e r a l r e s e a r c h e r s [R14,70] and i s given by 2 K(y'y''*> = ,3/2, 2 V — 7 ^ 7 2 ( 5 ' 2 5 ) 2(777) Ly + y ~ 2yy MJ exp 22 t2\ -v + ^y y ( i - M ) * 9 9 y + y - 2yy M { ( 7 - D2[ y2 + y'2] " 2 ( 7 2 - 1 )yy'U}/4y where 7 = m,/m, m and m, are the masses of the t e s t p a r t i c l e mass and the background p a r t i c l e mass, r e s p e c t i v e l y , and ju i s the c o s i n e of the angle between y and y'. The c o l l i s i o n frequency i n Eq.(5.24) i s given by °°1 r/J" 0-1 P r i o r to proceeding i t i s u s e f u l to w r i t e the c o l l i s i o n o p e r a t o r , Eq.(5.25), in a d i f f e r e n t form. In a b a s i s of s p h e r i c a l harmonic f u n c t i o n s , Y ^ m ( f i ) , the c o l l i s i o n operator i s d i a g o n a l in 1 and independent of the m index [R130]. T h i s i s a consequence of the f a c t that the k e r n e l , K ( y , y ' , £ ) i s only a f u n c t i o n of the angle between y_ and y_' and hence i s i n v a r i a n t with respect to r o t a t i o n s [R130]. The d i a g o n a l v(y) = 2TT;/ K(y',y,M)y'2dMdy'. (5.26) 1 52 matrix elements are given by </m|j|/m> = J7(K(y,y ' ,M ) - i; (y) 8 ( f i-fi') } (5.27) Y* (n)Y.(n')d2nd2n', / m / m where fi and fi' are the o r i e n t a t i o n v e c t o r s for y_ and y_', r e s p e c t i v e l y . Since the values of the matrix elements of the c o l l i s i o n operator are independent of the m index, the value of the matrix element for any p a r t i c u l a r value of m, for f i x e d / , i s equal to the average value over a l l of the values of m for a f i x e d / , thus / </m|j|/m> = J J L ( K ( y , y ' , M ) - v (y) 5 ( f i - f i ' ) } (5.28) m'=-/ Y *M, ( f l ) YL M, (0')d2J2d2fl'/(2/+1 ) . With the use of the angle a d d i t i o n theorem for s p h e r i c a l harmon i c s , I Y *m( S 2 ) Y/ m( J T ) = [(27 + 1 )/4TT]P/ ( f l . f i ' ) , (5.29) m=-/ where P^(x) i s the Legendre polynomial of order / , Eq.(5.28) may be r e w r i t t e n as </m|J|/m> = / / K ( y , y ' , f i - f i ' ) PZ (fi-fi')d2fid2fi'/47r (5.30) - u(y). The i n t e g r a l s i n Eq.(5.30) may be evaluated e a s i l y y i e l d i n g </m|j|/m> = kz(y,y') - *»(y), (5.31) where 1 53 1 k/(y,y') = 2 7 rJ* K(y,y' ,M)P, ( M)C3M (5.32) -1 The e i g e n f u n c t i o n s of the c o l l i s i o n operator may be determined by d i a g o n a l i z i n g the dia g o n a l b l o c k s </m|J|/m> i n the speed v a r i a b l e [R131,132], The k e r n e l s k^(y,y') times (21 + 1)/4TT are the expansion c o e f f i c i e n t s of K(y,y',/n) i n P{(u). For /=0, Ko ( y f Y ' ) y / y ' i s the Wigner-Wilkens kernel [R20] which p l a y s an important r o l e i n energy r e l a x a t i o n problems and was f i r s t i n t r o d u c e d in the d e s c r i p t i o n of neutron t r a n s p o r t . With the r e s u l t Eq.(5.31), the c o l l i s i o n o p e r a t o r , may be w r i t t e n as / co Jf(y,M,x) = J J L I{k.(y,y') - v(y)6(y-y')} (5.33) m=-/ 1=0 1 Y/m( n ) Y*m( n')y'2 f (y',M',x)dy'd2ST , where u i s the angle between y^ ar>d the z a x i s . T h i s i s the form of the c o l l i s i o n operator employed i n the next s e c t i o n so as to express the Boltzmann e q u a t i o n , Eq.(5.21), as a set of coupled i n t e g r o d i f f e r e n t i a l e q u a t i o n s . 5.4 COUPLED SETS OF EQUATIONS In t h i s s e c t i o n , I wish to re w r i t e Eq.(5.2l) i n a form which takes advantage of diag o n a l nature of the c o l l i s i o n operator i n a s p h e r i c a l harmonic b a s i s . To do t h i s , I w i l l seek a s o l u t i o n of Eq.(5.21), f ( y , M , x ) , i n the form of CO f(y,M,x) = L f . , (y,x)P, (5.34) l'=0 1 1 1 54 where P ^ ( M ) i s the / th Legendre polynomial normalized to u n i t y . With the s u b s t i t u t i o n of Eq.(5.34) i n t o E q . ( 5 . 2 l ) , m u l t i p l i c a t i o n by P ^ ( M ) , i n t e g r a t i o n over 0 and fi' I o b t a i n dt[ > (y ,x)1 CO 27ry Z I '=0 (5.35) / CO CO Z Z Z m= - / " / » = o / ' = 0 A , ( y , y ' ) y ' 2 f ( y ' , x ) d y ' -v ( y ) f ( y , x ) 1 2TT ; ; p . ( M ) Y , ( M , 0 ) d M d 0 j-io 1 ' m 1 2TT J / P, » U ' ) Y . ) d M ' d 0 ' -10 The i n t e g r a t i o n s in Eq.(5.35) may be done e a s i l y using the o r t h o n o r m a l i t y of the s p h e r i c a l harmonics and the f a c t that Y^ Q ( u, 0 ) =P j ( M) /V/2TT . The f i r s t i n t e g r a l on the r i g h t hand side of Eq.(5.35) y i e l d s 6, ,„5 n, while the second J / , / m, 0 i n t e g r a l y i e l d s 6^ , „ ^ , 5m Q. With the use of the Kronecker 8 f u n c t i o n s , the three summations are evaluated e a s i l y . S i m i l a r l y , the i n t e g r a l s on the l e f t hand s i d e of Eq.(5.29) may be evalu a t e d with the use of the three term recurrence r e l a t i o n f o r P ^ ( M ) , given by MP ; U ) a / + 1P/ + 1 U > a / P / _ ^ M ) , (5.36) where (21 - 1)(2/ + 1 ) 1/2 (5.37) 1 55 I t h e r e f o r e have the f o l l o w i n g set of coupled i n t e g r o d i f f e r e n t i a l equations to solve y9 x ^a/f/ - 1( Y , X ) + A/ + 1f/ + 1( y , x )^ = J/f/(y 'x )' (5.38) where 2 J . f , ( y , x ) = J k , ( y , y ' ) f . ( y ' , x ) y ' ^ d y ' - (5.39) 0 ^ ( y )f/ ( y r x ) T h i s y i e l d s an i n f i n i t e set of coupled equations f o r the f j ( y , x ) which i s t r u n c a t e d by s e t t i n g f^(y,x)=0 for a l l />L. The method f o r s o l v i n g these s e t s of coupled equations w i l l be introduced i n the next s e c t i o n . 5.5 ITERATIVE SOLUTION TO BOLTZMANN EQUATION The h i e r a r c h y of equations i s solved by an i t e r a t i v e technique in t r o d u c e d by McMahon [R128]. The b a s i c idea i s to c r e a t e a time d e r i v a t i v e , choose an i n i t i a l v e l o c i t y d i s t r i b u t i o n f u n c t i o n and i t e r a t e u n t i l the time d e r i v a t i v e v a n i s h e s . In a d d i t i o n , i t i s necessary to s l i g h t l y modify the Boltzmann equation to ensure convergence of the i t e r a t i o n . The m o d i f i e d Boltzmann equation (MBE) i s given by dt, a H t " = V/ " y! l [ a / f / - l + a / + lf/+1] " s ( f/ " ^/) (5-40) where the convergence f a c t o r and f u n c t i o n , s and g^ r e s p e c t i v e l y , have been i n t r o d u c e d . With the use of a small f i n i t e time i n t e r v a l , Eq.(5.40) may be i n t e g r a t e d as 1 56 + a f 1 /+1 /+1 ]}At - (5.41) (fJ - ql)sAt. S e t t i n g (5.42) d e f i n e s the i t e r a t i o n . The l a s t term on the r i g h t hand side of Eq.(5.41) i s added to prevent the i t e r a t i o n from d i v e r g i n g . To ensure that the s o l u t i o n converges to a s o l u t i o n of E q . ( 5 . 2 l ) , g^ must be p e r i o d i c a l l y set equal to fj , u s u a l l y a f t e r between 10 and 50 i t e r a t i o n s . A l s o f o r a reasonable r a t e of convergence the product sAt must be on the order of 0.05 - 0.1. The space dependence of the d i s t r i b u t i o n f u n c t i o n w i l l be approximated with the d i s c r e t e o r d i n a t e method for which an Mth order Gauss-Legendre quadrature procedure i s employed. The s l a b of atmosphere has a t h i c k n e s s of X mean f r e e paths with f^(y,-X) d e f i n i n g the bottom s u r f a c e and f^(y,0) d e f i n i n g the top surface of the s l a b . Consequently, i t i s convenient to d e f i n e a set of M quadrature p o i n t s which f a l l on the i n t e r v a l [-X,0] with XQ = -X and xM_1 = 0. The usual Gauss-Legendre p o i n t s are d e f i n e d on the i n t e r v a l [-1,1] and hence i t i s convenient to parameterize the x v a r i a b l e i n terms of another v a r i a b l e x. Let {x\,i = 0,M-l} and {w.,i=0,M-l} be a set of M Gauss-Legendre quadrature p o i n t s and weights. Consider the v a r i a b l e change x(x) = - (x - x0)X/(x. M-1 (5.43) 1 57 where X Q and xM_1 are the f i r s t and l a s t p o i n t s of an M p o i n t Gauss-Legendre quadrature r u l e , r e s p e c t i v e l y . By e v a l u a t i n g x(x) at the set of p o i n t s {x^}, I o b t a i n a set of quadrature p o i n t s f a l l i n g over the range of i n t e r e s t and with the proper e n d p o i n t s . Thus, any f u n c t i o n of x may have i t s d e r i v a t i v e at the set of p o i n t s (x(x\)} evaluated by rdxi rM~1 - 1 ,/wif/'(y,x(xi)) = Ijgj L Di jf/( y , x ( xj) ) / w j J , (5.44) where D i s the DO d e r i v a t i v e operator based on Gauss-Legendre quadrature p o i n t s and weights, d e r i v e d i n Appendix A, and the f i r s t term in the brackets r e s u l t s from the use of the chain r u l e . For convenience, I w i l l use the n o t a t i o n x^ f o r x ( x ^ ) . The c o l l i s i o n operator was s i m i l a r l y approximated by a f i n i t e m a t r i x . The i n t e g r a l i n Eq.(5.32) i s evaluated by a f i f t y p o i n t Gauss-Legendre quadrature r u l e , thus 49 k; <Y,y' ) * 2TT Z K(y,y' )u.P, '(». ) , (5.45) ' i=0 i i * i where (M^I and {u^} are a set of Gauss-Legendre p o i n t s and weights r e s p e c t i v e l y . The y' i n t e g r a t i o n of Eq.(5.26) and Eq.(5.38) may a l s o be e v a l u a t e d with a Gaussian quadrature r u l e . Two s e t s of quadrature p o i n t s and weights were generated. The f i r s t i s used to e v a l u a t e i n t e g r a l s of the form y _ 2 N'-1 ;cy2e y h(y)dy = Z v : h (y: ) (5.46) 0 i=0 1 1 p r o v i d e d that h(y) i s a polynomial of degree l e s s than 2N', 1 58 while the second i s used with i n t e g r a l s of the form « 2 N"-1 / y2e y h(y)dy = Z VTMy?) (5.47) yc i=o The d e t a i l s of the c a l c u l a t i o n of these quadrature r u l e s are given i n Appendix C. These quadrature r u l e s were chosen to f a c i l i t a t e the i m p o s i t i o n of the boundary c o n d i t i o n s , as w i l l be d i s c u s s e d l a t e r . A set of N p o i n t s and weights, denoted by {y^} and {V\| r e s p e c t i v e l y , with N=N' + N" are generated from the two broken range quadrature r u l e s , thus i<N' (5.48) i>N'. A s i m i l a r d e f i n i t i o n holds f o r the weights. The i n t e g r a l p o r t i o n of the c o l l i s i o n operator may be approximated as a matrix operator based on t h i s quadrature r u l e , thus oo N-1 A , ( y , y ' ) y ' 2 f / d y - Z V.k, (y,y . )exp(y 2.-y 2)f ( y . , x ) . (5.49) 0 j=0 J 3 ] ] Thus the / t h component of the c o l l i s i o n o p e r a t o r , d e f i n e d by Eq.(5.39), may be approximated by — N _ 1 / — v/V.J f - Z B . . f . (y • , x ) |/V . , (5.50) j_Q •*•_)' J J where B[. = / v T k / ( y i , y j ) e x p ( y ^ - y ? ) - viy^S^, (5.51) where the Kronecker 5 f u n c t i o n has been int r o d u c e d so that the c o l l i s i o n frequency i s s u b t r a c t e d out p r o p e r l y . 159 The i t e r a t i o n Eq.(5.4l) and (5.42), may be approximated by rN-1 i/w, v A f ^ ( y ,x, ) = Z /v.w.B . f j ( y . , x . ) -k n / J n' k _. n ] k n] / Jj ' k j = 0 (5.52) M-1 y Z /w.v D, .[a , f ^ , (y ,x.) + nj _ 0 j n k] / /-1  Jn' ] a/+if/ + i(yn'xj) ] " s ( f/ " g/) At where Eqs.(5.44) and (5.50) have been used. The i t e r a t i o n was begun with the i n i t i a l d i s t r i b u t i o n f u n c t i o n d e f i n e d by f°(y,M,x) = fm( y ) { - [ ! + 3HM/2]x/X + (5.53) [1 + x / X ] [ 0 ( yc - y)d(u) + e(-n)]} H=F T/ ( ni/2kT/m) i s the dimensionless Jeans f l u x and fm( y ) = 7r 3 / / 2e y i s the Maxwellian d i s t r i b u t i o n f u n c t i o n normalized to u n i t y over d3y . This form of the i n i t i a l d i s t r i b u t i o n f u n c t i o n was chosen so that i t would g r a d u a l l y change from a Maxwellian with c o r r e c t i o n term to give a Jeans' p a r t i c l e f l u x at the bottom s u r f a c e of the s l a b , to a t r u n c a t e d Maxwellian d i s t r i b u t i o n at the top s u r f a c e . The i t e r a t i o n , d e f i n e d by Eqs.(5.42) and (5.52), was s t a r t e d by r e p r e s e n t i n g the i n i t i a l d i s t r i b u t i o n f u n c t i o n , Eq.(5.53), i n the normalized Legendre polynomial b a s i s . The f u n c t i o n s g^ are i n i t i a l l y set equal to the r e s u l t i n g f ° . 160 A f t e r each i t e r a t i o n , the boundary c o n d i t i o n s were imposed in the f o l l o w i n g manner. At the bottom s u r f a c e , the d i s t r i b u t i o n f u n c t i o n i s set equal to f o ( yi f- X ) = v / 2 fm(Y i ) (5.54) and f j ( yi f- X ) = 0 1>1, (5.55) The top boundary c o n d i t i o n i s imposed with the convenient t r a n s f o r m a t i o n from the polynomial b a s i s to the DO b a s i s in a n g l e . Thus _ L • ukf (yi,Mk,0) = L T j J/f/ (y i, 0 ) , (5.56) where the c a l c u l a t i o n of TT i s given i n Appendix A. With the f u n c t i o n i n t h i s form the top boundary c o n d i t i o n i s imposed by s e t t i n g • f ( yi 'ML - k - 1 '0 ) = f ( yi ^k'O ) 0 ( yc "yi) ( 5*5 7 ) k=0,1,2...(L-1)/2 where L i s chosen to be odd so that the come i n p o s i t i v e and negative p a i r s . T h i s f o r c e s the d i s t r i b u t i o n f u n c t i o n for values of n r e p r e s e n t i n g incoming p a r t i c l e s to be equal to the d i s t r i b u t i o n f u n c t i o n f o r outgoing p a r t i c l e s , except when the value of the p a r t i c l e speed, y, i s g r e a t e r than the escape speed y . In t h i s l a t t e r c a s e , the p o r t i o n of the d i s t r i b u t i o n f u n c t i o n r e p r e s e n t i n g incoming p a r t i c l e s has been set equal to z e r o . I t should be noted that the d i s t r i b u t i o n f u n c t i o n at the top boundary has the form of a 161 step f u n c t i o n as a f u n c t i o n of y for values of M<0. The quadrature r u l e d e f i n e d by the p o i n t s and weights {y^} and {V\} i s i d e a l f o r the i n t e g r a t i o n of t h i s d i s t r i b u t i o n f u n c t i o n s i n c e the quadrature r u l e i s broken i n t o two ranges at the p o i n t , y , where any d i s c o n t i n u i t y may o c c u r . A f t e r the top boundary c o n d i t i o n s have been imposed i n t h i s way, the d i s t r i b u t i o n f u n c t i o n i s transformed back to the polynomial b a s i s i n angle and the next i t e r a t i o n , Eq.(5.42), i s performed. It was found that the i t e r a t i o n proceeded more r a p i d l y i f the At i s made a f u n c t i o n of speed. T h i s only changes the rate of convergence s i n c e the time d e r i v a t i v e must van i s h at the end of the i t e r a t i o n . Normally a converged d i s t r i b u t i o n f u n c t i o n may be obtained i n 500-1000 i t e r a t i o n s , s t a r t i n g with the d i s t r i b u t i o n f u n c t i o n given i n Eq.(5.53), by s e t t i n g sAt=0.05 and 9^=^ a f t e r every 50 i t e r a t i o n s . The number of i t e r a t i o n s r e q u i r e d f o r a converged s o l u t i o n may be s u b s t a n t i a l l y reduced by s t a r t i n g with a p r e v i o u s l y converged s o l u t i o n with a d i f f e r e n t number of quadrature p o i n t s and/or normalized Legendre p o l y n o m i a l s . Once a converged d i s t r i b u t i o n has been obtained the f l u x , number d e n s i t y and temperature p r o f i l e s are obtained from the f o l l o w i n g i n t e g r a l s CO N(x) = /87rN/f0(y,x)y2dy (5.58) 0 1 62 CO F(x) = 47r/kT/3mN/f , (y rx ) y3d y 0 (5.59) oo 00 T(x) = i Tb/ f0( y , x ) y " d y / ; f0y2d y . D0 0 (5.60) 5.6 DISCUSSION OF NUMERICAL RESULTS T h i s model was a p p l i e d to the d e s c r i p t i o n of the l o s s of hydrogen from e a r t h and Mars as we l l as the escape of helium from e a r t h . In each of these c a s e s , there i s a l i g h t minor atom escaping from a heavy moderator gas. On Mars, the moderator gas i s C02, while on e a r t h i t i s 0. It i s p o s s i b l e to c o n s i d e r the escape of atoms from other atmospheres as w e l l , although t h i s wasn't done. Table 5.1 gives a summary of p o s s i b l e escaping gas moderator gas combinations i n the s o l a r system. It should be noted that in some c a s e s , such as escape of hydrogen from Venus, non-thermal escape mechanisms play an important r o l e . The three c a s e s , escape of H and He from earth and H from Mars, were chosen because these three cases have been w e l l s t u d i e d and a l a r g e p o r t i o n of the escaping gas on these p l a n e t s may be e x p l a i n e d by a thermal escape model. P r i o r to d i s c u s s i n g the r e s u l t s , i t i s necessary to show that converged s o l u t i o n s to the Boltzmann equation have been found. The convergence of the p a r t i c l e f l u x i s employed as a c r i t e r i a f o r the convergence of the d i s t r i b u t i o n f u n c t i o n . The p a r t i c l e f l u x converges more slowly than e i t h e r the number d e n s i t y or the temperature. 1 63 TABLE 5.1 Planet or S a t e l l i t e Major Gas (2) Minor Gas (1 ) m2/mH v ,Km/s e s c ' / E a r t h 0 H, He 1 6 11.2 Venus co2 H, He 1 6 10.3 Mars CO 2 H,He,N,0 16, 5. 10 J u p i t e r H2 ,He H 2 60.0 Io so2 H,Na,K 64 2.60 Saturn H2 ,He H 2 35.0 T i t a n N2 H 28 4.50 T h i s i s p r i m a r i l y because the major c o n t r i b u t i o n to the p a r t i c l e f l u x i s from that p o r t i o n of the d i s t r i b u t i o n f u n c t i o n which i s most perturbed from a normal Maxwell-Boltzmann d i s t r i b u t i o n , that i s the high energy t a i l . In a d d i t i o n , N(x) and T(x) r e q u i r e only f0(y_) f o r t h e i r d e t e r m i n a t i o n , while F(x) r e q u i r e s f 1 (y^) which i s probably l e s s a c c u r a t e l y o b t a i n e d . I t w i l l be r e c a l l e d that the d i s t r i b u t i o n f u n c t i o n at the su r f a c e i s a step f u n c t i o n i n ix i f y>Yc • Consequently, the d i s t r i b u t i o n f u n c t i o n cannot be expected to converge e x a c t l y s i n c e the top boundary c o n d i t i o n cannot be w e l l represented with a small number of Legendre p o l y n o m i a l s . There are two d i s t i n c t s e t s of approximations used to c a l c u l a t e the d i s t r i b u t i o n f u n c t i o n . The f i r s t i s the DO approximation to the d r i f t and c o l l i s i o n p o r t i o n s of the Boltzmann o p e r a t o r . T h i s f i r s t approximation allows us to 1 64 w r i t e the Boltzmann equation as a set of l i n e a r a l g e b r a i c e q u a t i o n s . The second approximation was an i t e r a t i v e procedure f o r s o l v i n g t h i s set of e q u a t i o n s . I w i l l c o n s i d e r f i r s t the second approximation and l a t e r the f i r s t a p p r oximation. Table 5.2 shows a t y p i c a l sequence of the convergence f o r the f l u x as a f u n c t i o n of number of i t e r a t i o n s . Both sequences are for helium escaping from the e a r t h at two d i f f e r e n t temperatures, 4000K and 4400K, r e s p e c t i v e l y . In each case the f l u x e s show a steady d e c l i n e to some f i x e d v a l u e . Table 5.3 shows the convergence of the f l u x as a f u n c t i o n of the number of Legendre polynomials L, and as a f u n c t i o n of the number of speed quadrature p o i n t s , N, used. The f i r s t t a b l e s t u d i e s the convergence of the Jeans r a t i o , Rj = 7/Fj, f o r H escaping from e a r t h with an exospheric temperature of 1500K, while Table 5.3b shows the convergence of R, f o r helium escaping from e a r t h at T=2000K. These r e s u l t s show remarkable convergence as a f u n c t i o n of number of speed quadrature p o i n t s . T h i s i s p r i m a r i l y due to the f a c t that the speed range was broken, a l l o w i n g one set of quadrature p o i n t s to f a l l p r e c i s e l y where the d i s t r i b u t i o n f u n c t i o n i s most p e r t u r b e d . The convergence as a f u n c t i o n of number of Legendre polynomials i s a l s o very good, e s p e c i a l l y when c o n s i d e r i n g the step f u n c t i o n nature of the d i s t r i b u t i o n f u n c t i o n at the upper boundary. 165 TABLE 5.2: R.. as Function of Iterations Iterat ions *J R3 20 1.0100 1.0099 40 0.9867 0.9944 60 0.9581 0.9715 80 0.9412 0.9494 1 00 0.9368 0.9407 120 0.9363 0.9381 1 40 0.9364 0.9375 1 60 0.9367 0.9375 180 0.9369 0.9375 200 0.9370 0.9376 220 0.9371 240 0.9372 260 0.9373 280 0.9374 300 0.9374 320 0.9375 340 0.9375 360 0.9376 380 0.9376 1. Helium on Earth; T=4000K, L=9, M=10, N=28 2. Helium on Earth; T=4400K, L=9, M=10, N=28 It i s interesting to study the exact nature of the di s t r i b u t i o n function as a function of depth in the atmosphere. This gives an indication of how the surface discontinuity smooths i t s e l f out. The d i s t r i b u t i o n function is found'by summing up the L+1 terms in the polynomial 166 TABLE 5.3: Convergence of R Hydrogen escape from e a r t h ; T = 1 5 0 0 , X= 5 . , M= 10 N \ L 3 5 7 9 1 1 1 3 16 0 . 7 4 2 2 0 . 7 1 3 4 0 . 7 0 8 9 0 . 7 0 1 6 0 . 7 0 0 1 0 . 6 9 9 2 20 0 . 7 1 9 3 0 . 71 00 0 . 7 0 3 4 0 . 7 0 0 2 0 . 6 9 9 4 0 . 6 9 7 6 24 0 . 7 1 5 5 0 . 7 0 4 7 0 . 6 9 9 3 0 . 6 9 6 7 0 . 6 9 5 2 0 . 6 9 4 4 28 0 . 7 2 0 7 0 . 7 0 7 2 0 . 7 0 2 0 0 . 6 9 2 0 0 . 6 9 0 8 0 . 6 9 0 1 Helium escape from e a r t h ; T = 2 0 0 0 , X = 5 . , M=10 N \ L 3 5 7 9 1 6 0 . 8 7 6 6 0 . 8 9 2 5 0 . 8 9 1 2 0 . 8 9 3 3 20 0 . 8 9 0 2 0 . 8 5 4 1 0 . 8 8 4 9 0 . 8 8 7 9 24 0 . 8 9 8 8 0 . 8 5 4 4 0 . 8 8 4 9 0 . 8 8 7 9 28 0 . 8 7 6 7 0 . 8 5 4 4 0 . 8 8 4 9 0 . 8 8 7 9 expansion of f(y, y , x ) f o r p a r t i c u l a r values of x and y. F i g u r e s 5.4 - 5.6 shows the d i s t r i b u t i o n f u n c t i o n versus angle at y=2.94 f o r three d i f f e r e n t depths of the atmosphere. In each case the temperature of the escaping gas was chosen so that the escape speed i s approximately 2. 1 7v/2kT/m. These temperatures correspond to 1500. K i n the case of hydrogen escaping from e a r t h , 6000.K for helium escaping from e a r t h , and 322.K f o r hydrogen escaping from Mars. In a l l three c a s e s , f o r speeds l e s s than the escape speed, the d i s t r i b u t i o n f u n c t i o n i s almost i s o t r o p i c . That i s , there appears to be only a small amount of c o u p l i n g of 167 FIGURE 5.4 V e l o c i t y d i s t r i b u t i o n f u n c t i o n s f o r escape of hydrogen from e a r t h , y=2.94; (a) X=-1/2; (b) X=-1/4; (c) x=0. 1 68 FIGURE 5.5 V e l o c i t y d i s t r i b u t i o n f u n c t i o n s f o r escape of helium from e a r t h , y=2.94; (a) X=-1/2; (b) x=-1/4; (c) x=0. FIGURE 5.6 169 -0 02' 1 1 1 1 1 1 1 1 -1 -0.5 0 0.5 1 V e l o c i t y d i s t r i b u t i o n f u n c t i o n s f o r escape of hydrogen from Mars, y=2.94; (a) x=-1/2; (b) x=-1/4; (c) x=0. 170 the d i s t r i b u t i o n f u n c t i o n f o r the speeds l e s s than the c r i t i c a l speed and speeds g r e a t e r than the c r i t i c a l speed. The v e l o c i t y a n i s o t r o p y , f o r speeds g r e a t e r than the escape speed has a l s o been c o n s i d e r a b l y reduced. T h i s i n d i c a t e s that o r i e n t a t i o n of the p a r t i c l e v e l o c i t y i s e f f i c i e n t l y randomized. There i s , however, a s i g n i f i c a n t d i f f e r e n c e i n the magnitude of the d i s t r i b u t i o n f u n c t i o n f o r the three d i f f e r e n t c a s e s . The d i s t r i b u t i o n f u n c t i o n i s gr e a t e s t i n the case of helium escaping from e a r t h . The mass r a t i o in t h i s case i s 4, t h i s i s a small enough value that the energy exchange i s e f f i c i e n t . Consequently, the high enery t a i l of the d i s t r i b u t i o n f u n c t i o n i s r e p l e n i s h e d e a s i l y . As the mass r a t i o i n c r e a s e s , f i r s t t o 16 (escape of hydrogen from earth) and then to 44 (escape of hydrogen from Mars), the energy exchange between the escaping p a r t i c l e s and background p a r t i c l e s becomes i n c r e a s i n g l y i n e f f i c i e n t and hence the hig h enery t a i l of the d i s t r i b u t i o n f u n c t i o n i s not f i l l e d . T h i s e f f e c t i s e s p e c i a l l y n o t i c e a b l e f o r H escaping from Mars. It i s a l s o i n t e r e s t i n g to point out how r a p i d l y the o r i e n t a t i o n of the d i s t r i b u t i o n f u n c t i o n i s randomized as a f u n c t i o n of depth. A f t e r only a h a l f of a mean free path the d i s t r i b u t i o n f u n c t i o n i s almost l i n e a r f u n c t i o n of UL. T h i s suggests that the d i s t r i b u t i o n f u n c t i o n should approach the form g i v e n by Eq.(5.l7) i n a few mean fr e e paths. The present c a l c u l a t i o n s used a s l a b t h i c k n e s s of 5 mean f r e e p a t h s . T h i s i s s i m i l a r to the values used by Brinkmann [R74,75] and around the value of 4 use by 171 Chamberlain and Smith [R121]. V a r i o u s models of the upper atmosphere y i e l d d i f f e r e n t v a l u e s f o r the Jeans' r a t i o . Table 5.4 shows the r e s u l t s of the present c a l c u l a t i o n s along with some of the Monte-Carlo c a l c u l a t i o n s . The agreement between the present r e s u l t s and those of Chamberlain and Smith [R121] i s to w i t h i n a few p e r c e n t . Where there are d i s c r e p a n c i e s , these are probably due the d i f f e r e n c e i n the s l a b width used i n the two c a l c u l a t i o n s . Chamberlain used a t h i c k n e s s of 4 mean fr e e p a ths, while 5 mean fr e e paths were used in the present study. The agreement i s poorest f o r the Hydrogen escape from the Martian atmosphere. These r e s u l t s are a l s o presented g r a p h i c a l l y along with the Jeans r a t i o c a l c u l a t e d by other r e s e a r c h e r s i n F i g u r e s 5.7 - 5.9 The present r e s u l t s c o n f i r m the Monte-Carlo work. It should be noted that there has only r e c e n t l y been agreement between v a r i o u s Monte-Carlo c a l c u l a t i o n s . As expected the r e s u l t s obtained by S h i z g a l and L i n d e n f e l d are lower than the present Jeans r a t i o s . It was p r e v i o u s l y noted that t h e i r p r o duction model for escaping atoms ignored the p r o d u c t i o n of hot atoms by d i f f u s i o n from lower l e v e l s of the atmosphere. T h e i r v a l u e s agree best with those of helium escaping from e a r t h . In t h i s case the r e l a t i v e l y small mass r a t i o , and hence e f f i c i e n t energy exchange a f t e r c o l l i s i o n s , ensures that the p r o d u c t i o n of hot atoms w i l l be much l a r g e r than the d i f f u s i o n from lower l e v e l s . On the other hand, t h e i r r e s u l t s d i f f e r g r e a t l y from the present 1 72 Table 5.4: Comparisons of Present Results with Monte-Carlo Calculations : Jeans' Ratio, RJ Temperature Monte-Carlo Present1 Present2 Present3 Hydrogen Escape from Earth 1000. 0.758 0.754 0.755 0.762 1 500. 0.680 0.693 0.694 0.696 2700. 0.670 0.661 0.644 0.655 3570. 0.669 0.677 0.648 0.652 Helium escape from Earth 4400 . 0.928 0.930 0.936 0.939 6000. 0.913 0.894 0.906 0.912 10800. 0.916 0.855 0.872 0.871 14280. 0.939 0.849 0.881 0.869 Hydrogen escape from Mars 230. 0.549 0.582 0.557 0.567 310. 0.5.14 0.534 0.532 0.526 560. 0.472 0.519 0.471 0.466 730. 0.506 0.541 0.472 0.459 "1. Slab thickness i s 2.5 mean free paths 2. Slab thickness i s 5.0 mean free paths 3. Slab thickness i s 7.5 mean free paths cal c u l a t i o n for hydrogen escape from Mars. Here, energy exchange is i n e f f i c i e n t and consequently hot atoms d i f f u s i n g from lower altit u d e s provide an important contribution to FIGURE 5.7 173 i • • • J _ 1 1 1 1 1 L 0.41 , n 0.8 1 1.2 1.4 1.6 1.8 Tr(l03K) Jeans r a t i o R, f o r Hydrogen escape from e a r t h . S o l i d l i n e i s present c a l c u l a t i o n , d otted l i n e i s that of S h i z g a l and L i n d e n f e l d [R76]. The open c i r c l e s , c l o s e d c i r c l e s , squares and t r i a n g l e s are given by Brinkmann [R75], Chamberlain and Smith [R121], Chamberlain and Campbell [R72] and Lew [R133], respect i v e l y . 1 FIGURE 5.8 Jeans r a t i o Rj f o r Helium escape from e a r t h . S o l i d l i n e i present c a l c u l a t i o n , d o t t e d l i n e i s that of S h i z g a l and L i n d e n f e l d [R76]. The open c i r c l e s and c l o s e d c i r c l e s are given by, r e s p e c t i v e l y , Brinkmann [R75], Chamberlain and Smith [R121 ] . FIGURE 5.9 0.6 8 TC(I02K) Jeans r a t i o R, f o r Hydrogen escape from Mars. S o l i d l i n e present c a l c u l a t i o n , dotted l i n e i s that of S h i z g a l and L i n d e n f e l d [R76], the c l o s e d c i r c l e s are given by Chamberlain and Smith [R121]. 176 the number of escaping atoms. The other model which was presented in the i n t r o d u c t i o n was that developed by Fahr [R120]. T h e i r escape f l u x i s the Jeans f l u x but c a l c u l a t e d with the temperature of the escaping gas. The present r e s u l t s suggest that he underestimates the true escape f l u x , e s p e c i a l l y i f the temperature drop of the escaping atom i s s u f f i c i e n t l y l a r g e . If the escape speed i s s u f f i c i e n t l y low i n comparison to the average thermal energy then there w i l l be a l a r g e temperature drop, due mainly to the absence of incoming p a r t i c l e s i n the high energy p o r t i o n of the d i s t r i b u t i o n f u n c t i o n . There i s no c l e a r reason that t h i s temperature drop should be simply r e l a t e d to the f l u x . Fahr [R120] a l s o suggested that the temperature p r o f i l e c o u l d be r e l a t e d to the depth i n mean f r e e paths by the r e l a t ion T(x) = Tb - AT^expC-Ax), (5.61) where A i s some c o n s t a n t . F i g u r e s 5.10 -5.12 show the temperature p r o f i l e s f o r the three cases c o n s i d e r e d . Although Eq.(5.6l) was d e r i v e d with F o u r i e r ' s law, which i s not n e c e s s a r i l y v a l i d i n the present c a s e , i t i s i n t e r e s t i n g t o note that the temperature p r o f i l e appears to have approximately t h i s form. The present numerical r e s u l t s do not give any meaningful estimate of the heat c o n d u c t i v i t y c o e f f i c i e n t K, d e f i n e d i n Eq.(5.13). Whether t h i s i s a consequence of numerical problems or of there not being a d e f i n i t e heat c o n d u c t i v i t y c o e f f i c i e n t i s d i f f i c u l t to 1 77 FIGURE 5.10 Temperature p r o f i l e s : Hydrogen (b) T=1500; (c) T=1900. escape from e a r t h ; (a) T=1100 1 78 FIGURE 5.11 X Temperature p r o f i l e s : Helium escape from e a r t h ; (a) T= 3 0 0 0 (b) T = 4 0 0 0 ; (c) T = 6 0 0 0 . 1 79 FIGURE 5.12 Temperature p r o f i l e s : (b) T=500; (c) T=650. Hydrogen escape from Mars; (a) T=310 FIGURE 5.13 3 I 1 1 1 1 r Q 51 1 1 1 1 1 1 0.8 1.2 1.6 2 TC(I03K) Temperature drop at x=0 as a f u n c t i o n of heat bath gas temperature. S o l i d l i n e i s present r e s u l t , d o t t e d l i n e o btained by L i n d e n f e l d and S h i z g a l [R117] and the c l o s e d c i r c l e s are obtained by Atreya e t . a l . [R134] 181 FIGURE 5.14 De n s i t y p r o f i l e s , N(x)/N(-X) f o r Hydrogen escape from e a r t h ; (a) T=1100; (b) T=1500; (c) T=1900. 182 FIGURE 5.15 De n s i t y p r o f i l e s , N(x)/N(-X) f o r Helium escape from e a r t h ; (a) T = 3 0 0 0 ; (b) T = 4 0 0 0 ; (c) T = 6 0 0 0 . 183 FIGURE 5.16 Density p r o f i l e s , N(x)/N(-X) f o r Hydrogen escape from Mars; (a) T=310; (b) T=500; (c) T=650. 184 a s c e r t a i n . It was a l s o mentioned in the i n t r o d u c t i o n that one of the few o b s e r v a t i o n s which may be r e l a t e d to the d i s t r i b u t i o n f u n c t i o n was the temperature of the hydrogen i n f e r r e d from Balmer-a l i n e width measurements [ R 1 3 4 ] , The previous models d i s c u s s e d a l s o provide e s t i m a t e s of t h i s temperature drop. These r e s u l t s and the present c a l c u l a t i o n s are given i n F i g u r e 5 . 1 3 . These r e s u l t s are s i m i l a r to those suggested by other r e s e a r c h e r s [ R 1 1 3 , 1 1 5 , 1 1 7 ] Density p r o f i l e s are given i n F i g u r e s 5 . 1 4 - 5 . 1 6 . The f l u x i s r e l a t e d to the d e n s i t y g r a d i e n t by a d i f f u s i o n constant i n the region where the d i s t r i b u t i o n f u n c t i o n i s almost Maxwell-Boltzmann. T h i s r e l a t i o n s h i p i s given by, D = awd? ( 5-6 2 ) These d i f f u s i o n constants for hard sphere c r o s s s e c t i o n s have been estimated [ R 1 3 5 ] to be ^VT/m = ^ T E I v0l0, 2\/yn n ^ d ^ 2\/yn ° u where v0 i s the most probable v e l o c i t y and 10 i s the mean fre e p a t h . In these u n i t s the d i f f u s i o n c o n s t a n t s come out to be 0 . 4 2 0 , 0 . 3 8 8 and 0 . 3 8 0 f o r the cases of helium escaping from e a r t h , hydrogen escaping from e a r t h and hydrogen from Mars. Estimates of the d i f f u s i o n constant taken from the slopes of the s t r a i g h t p o r t i o n of the d e n s i t y p r o f i l e s range from about 0.32 to 0.49 depending on the temperature and escaping gas. Such comparisons are important to show that the s o l u t i o n obtained i s p h y s i c a l l y 185 r e a s o n a b l e . The o b j e c t i v e of t h i s chapter was to c o n s t r u c t a model of the t r a n s i t i o n r e gion of upper atmospheres based on the Boltzmann e q u a t i o n . T h i s model was then used to c a l c u l a t e d escape f l u x e s of l i g h t c o n s t i t u e n t s of these atmospheres as w e l l as temperature and d e n s i t y p r o f i l e s . There i s , however, c o n s i d e r a b l e work to be done. I have only c o n s i d e r e d thermal escape y e t , as was p r e v i o u s l y pointed out, there are non-thermal escape mechanisms. These mechanisms may be i n c l u d e d by r e d e f i n i n g the c o l l i s i o n operator to i n c l u d e other processes such as charge exchange react i o n s . 6. SUMMARY The major t o p i c of t h i s t h e s i s was the development and a p p l i c a t i o n of numerical methods to s t o c h a s t i c problems. These a p p l i c a t i o n s i n c l u d e d the d e s c r i p t i o n of d i f f u s i o n in a double w e l l p o t e n t i a l , i s o m e r i z a t i o n of n-butane and the escape of gases from a p l a n e t a r y atmosphere. As was p o i n t e d out i n the i n t r o d u c t i o n , there i s a p l e t h o r a of p h y s i c a l systems to which s t o c h a s t i c equations may be a p p l i e d p r o v i d e d there i s an e f f i c i e n t method of s o l v i n g them. The d i s c r e t e o r d i n a t e method provides an e f f i c i e n t method and may, with f u r t h e r development, be even more f l e x i b l e . The DO method may be extended in a number of ways. This method i s p r i m a r i l y e f f i c i e n t s i n c e i t i s p o s s i b l e to choose a b a s i s s e t , i . e . a s p e c i f i c set of quadrature p o i n t s and weights, a p p r o p r i a t e for each problem. Work i s p r e s e n t l y i n progress to develop an e f f i c i e n t means of g e n e r a t i n g sets of p o l y n o m i a l s , and hence quadrature p o i n t s and weights, on a r b i t r a r y i n t e r v a l s and weight f u n c t i o n s . The method has been extended to sums of Gaussian weights [ R 3 9 ] . Although there are techniques for g e n e r a t i n g an a r b i t r a r y quadrature r u l e [ R 8 0 ] , they have l i m i t e d a p p l i c a b i l i t y because of round o f f e r r o r i n the computation of recurrence c o e f f i c i e n t s for orthonormal p o l y n o m i a l s . T h i s method would a l s o be more f l e x i b l e i f i t were p o s s i b l e to impose boundary c o n d i t i o n s on matrix r e p r e s e n t a t i o n s of d i f f e r e n t i a l o p e r a t o r s e a s i l y . For example, the prbper way to t r e a t the chemical i s o m e r i z a t i o n 186 187 problem i s with a Kramers or Smoluchowski equation whose independent v a r i a b l e s are the angle of r o t a t i o n about the c e n t r a l bond and the angular momentum of the methyl group. These equations would then be subject to p e r i o d i c boundary c o n d i t i o n s . While there has been some progress i n d e v i s i n g such a r e p r e s e n t a t i o n , i t i s d i f f i c u l t to generate a symmetric r e p r e s e n t a t i o n of a s e l f - a d j o i n t operator i n a DO b a s i s with such boundary c o n d i t i o n s . There i s a l s o some d i f f i c u l t y in using t h i s method when s o l u t i o n s to equations have d i s c o n t i n u i t i e s . T h i s i s true when one t r i e s to sol v e the Boltzmann equation with a s p h e r i c a l l y symmetric geometry. T h i s equation a r i s e s i n connection with a model of p l a n e t a r y gas escape, i n c l u d i n g a g r a v i t a t i o n a l f i e l d . In t h i s l a t t e r case, the d i s c o n t i n u i t i e s a r i s e from the e x c l u s i o n of s a t e l l i t e and f l y b y p a r t i c l e s at the upper boundary. In t h i s c a s e , the problem cannot be avoided as i n the plane p a r a l l e l geometry by d e f i n i n g two se t s of quadrature p o i n t s s i n c e the l o c a t i o n of the d i s c o n t i n u i t y i n v e l o c i t y space i s a f u n c t i o n of a l t i t u d e . The plane p a r a l l e l model of the atmosphere may be developed f u r t h e r with the i n c l u s i o n of sources and s i n k s of hot atoms. 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The DO approximation of a f u n c t i o n i s an approximate r e p r e s e n t a t i o n which i s e s s e n t i a l l y e q u i v a l e n t to the tr u n c a t e d polynomial r e p r e s e n t a t i o n of the f u n c t i o n . A general method of gen e r a t i n g DO r e p r e s e n t a t i o n s , based on a r b i t r a r y p o l y n o m i a l s , of both f u n c t i o n s and of d i f f e r e n t i a l o p e r a t o r s i s developed i n t h i s Appendix. 7.1 INTRODUCTION It i s w e l l known that i n t e g r a l s may be approximated by a quadrature r u l e of the form [R80,136,137] b N-1 Jf (x)w(x)dx =* I f (x. )w. , (Al ) a i=0 1 1 where {x^} and {w^} are se t s of quadrature p o i n t s and weights r e s p e c t i v e l y . The d i s c r e t e o r d i n a t e method developed by S h i z g a l and Blackmore [R77] i s based on a g e n e r a l i z a t i o n of t h i s quadrature i n t e g r a t i o n to cover numerical d i f f e r e n t i a t i o n . Thus N-1 df (x) * L D. .f(x . ) . (A2) x = xi i=0 1 ] 3 dx A general procedure f o r c a l c u l a t i n g the matrix D w i l l be 196 1 97 developed i n t h i s Appendix. There are two reason that one may wish to develop a quadrature procedure f o r numerical d i f f e r e n t i a t i o n . The f i r s t i s that the a l g o r i t h m should be much more accurate than other numerical procedures such as f i n i t e d i f f e r e n c e method. The other reason i s that i n t e g r a l o p e r a t o r s are a l s o e a s i l y represented by a matrix i n t h i s quadrature r e p r e s e n t a t i o n [R138,139], Hence, i t i s u s e f u l to develop a procedure whereby the i n t e g r a l and d i f f e r e n t i a l p o r t i o n s of such op e r a t o r s may be t r e a t e d by the same approximation to sol v e i n t e g r o d i f f e r e n t i a l e q u a t i o n s . For .example, as has been seen i n the l a s t c h a p t e r , the s p h e r i c a l component of the c o l l i s i o n operator may be represented by [R5] CO J f ( x ) = /K(x,x')f (x')dx' - j>(x)f(x). (A3) 0 The value of g(x)=Jf(x) may be approximated by the sum •N-1 g(x.)i/w. « L K(x . ,x . )/w. w .f (x . )- (A4) 1 1 j = 0 3 3 3 or g(x.)i/w\ u(x•)f(x.)/w., l l l ' N- 1 r L j=o K(x.,x.)i/w.w.-5. • ( x . ) l ] l D i ] l (A5) f U j V w . , where I have chosen to eva l u a t e the c o l l i s i o n operator at the quadrature p o i n t s 198 x^. Equation (A5) i s c o n v e n i e n t l y w r i t t e n as 2 J._f (A6) where £ i s a vector c o n t a i n i n g f evaluated at the quadrature p o i n t s times the square root of the weight, g_ i s s i m i l a r l y d e f i n e d and J d e f i n e d by the ex p r e s s i o n in brackets i n Eq.(A5). I f the i n t e g r a l and d i f f e r e n t i a l p o r t i o n s of an i n t e g r o d i f f e r e n t i a l operator are approximated by matrices anologous to the one given by Eq.(A5), then t h e i r sum may be approximated by the element by element sum of the two m a t r i c e s . The DO r e p r e s e n t a t i o n of d i f f e r e n t i a l o p e r a t o r s i s developed i n the next s e c t i o n . 7.2 REPRESENTATION OF DIFFERENTIAL OPERATORS An a r b i t r a r y l i n e a r d i f f e r e n t i a l operator may be w r i t t e n in the form g(x) = L f ( x ) , (A7) where L = ZH ( x ) 2 _ . (A8) m , m m dx I wish to f i n d a d i s c r e t e o r d i n a t e approximation f o r L, analogous to the r e p r e s e n t a t i o n of J , such that Eq.(A7) may be w r i t t e n in the form a ^ L-f_, (A9) where g_ and t are as d e f i n e d p r e v i o u s l y and L i s the d i s c r e t e o r d i n a t e r e p r e s e n t a t i o n of the d i f f e r e n t i a l 199 o p e r a t o r . Once t h i s approximate r e p r e s e n t a t i o n of the operator L has been found, a d i f f e r e n t i a l equation may be s o l v e d by i n v e r t i n g the matrix L i n Eq.(A8), subject to the a p p r o p r i a t e boundary c o n d i t i o n s . S i m i l a r l y , the eigenvalues of the d i f f e r e n t i a l operator may be approximated by the eigenvalues of the matrix L. The method for f i n d i n g the matrix approximation of L with an a r b i t r a r y set of quadrature p o i n t s and weights i s the subject of t h i s Appendix. The d i s c r e t e o r d i n a t e r e p r e s e n t a t i o n of the d i f f e r e n t i a l operator L, Eq.(A9), i s based on the t r a n s f o r m a t i o n between the r e p r e s e n t a t i o n of a f u n c t i o n i n a polynomial b a s i s set and the corresponding d i s c r e t e o r d i n a t e r e p r e s e n t a t i o n . Thus, the d i s c r e t e o r d i n a t e r e p r e s e n t a t i o n of the d e r i v a t i v e o p e r a t o r , d/dx, i s generated from i t s f i n i t e matrix r e p r e s e n t a t i o n i n some polynomial b a s i s by simple matrix m u l t i p l i c a t i o n . The r e p r e s e n t a t i o n of the d e r i v a t i v e operator L i n Eq.(A8) i s then e a s i l y w r i t t e n . I begin the development with a s e r i e s of d e f i n i t i o n s . 7.3 DEFINITIONS A set of polynomials Rn( x ) , orthonormal with respect to the weight f u n c t i o n w(x) on the i n t e r v a l [ a , b ] , form a complete b a s i s of the L2[ a , b ] H i l b e r t space [R137]. If R i s a polynomial of degree n, then the set i s f u l l y s p e c i f i e d and unique [R137], The f i r s t N of these polynomials form a 200 subspace of the H i l b e r t space which i s isomorphic with the JR E u c l i d e a n space. The elements of the b a s i s v e c t o r s of t h i s polynomial b a s i s , r e f e r r e d to as the e - b a s i s , are d e f i n e d by where e^ i s the i t h element of the nth b a s i s v e c t o r . The e b a s i s w i l l be r e f e r r e d to as the polynomial b a s i s . M Let S be the set of a l l polynomials of degree l e s s than or equal to M. The inner product between two v e c t o r s in t h i s subspace w i l l be denoted by the dot product between the two v e c t o r s . It w i l l be assumed throughout t h i s Appendix tha t a l l v e c t o r s and o p e r a t o r s are i n the N dimensional s p a c e i f tN. Since (Rn(x)} i s a set of orthonormal p o l y n o m i a l s , i t i s p o s s i b l e to f i n d a set of p o i n t s {x^,i=0,1,...N-1} and weights {w^,i=0,1,...N-1} such that N-1 b I g(x.)w. = Jw(x)g(x)dx, (A11) i = 0 a 2 N ~ 1 p rovided that gsS and {x^} are the roots of RN(x) [R80]. With the use of Eq.(A11), i t i s p o s s i b l e to d e f i n e a u n i t a r y t r a n s f o r m a t i o n which w i l l allow a change to the d i s c r e t e o r d i n a t e b a s i s . The matrix T, whose elements are given by Jw(x)Rn(x)Ri(x)dx = 6i (A10) (n) T• • = R. (x . )/w . , I D l 3 3 (A12) i s a u n i t a r y m a t r i x , that i s 201 N-1 (T • TT ) . . = Z R • (x, ) R . ( x, ) w, , (A13a) - - 1 J k=0 3 = 6 i y (M3b) 2 n —* 2 s i n c e R.R.CS and R. i s orthonormal to R-. Hence TT i s a 1 1 i D matrix that d e f i n e s a new b a s i s , r e f e r r e d to as the f b a s i s . The nth v e c t o r , f / n \ of t h i s b a s i s , s a t i s f i e s •£<n>.e<m) = /wTR ( x ) , (A14) — — n m n where the rhs of Eq.(A14) i s (TT) =T ^ nm mn Th i s new b a s i s w i l l be r e f e r r e d to as the d i s c r e t e o r d i n a t e (DO) b a s i s . An a r b i t r a r y f u n c t i o n g SN 1 may be represented e x a c t l y i n t h i s b a s i s by a vector c j ^ ^ with the elements ,n ' N-1 , , g j  = Z T j . g !e' , (A15) where ( ) b g. = Jw(x)R.(x)g(x)dx. (A16) a With Eq.(A11) and the d e f i n i t i o n Eq.(A12), I f i n d that (f) N~1 N-1 _ g;r' = Z Z Tj .T. Vw .g(x . ) , (A17) k i-0 j-0 k l 1 3 ] or gkf ) = , /* kg ( xk)' ( A 1 8 ) where Eq.(A13) has been used to perform the sum over i . T h i s r e s u l t i s the working d e f i n i t i o n of the r e p r e s e n t a t i o n of f u n c t i o n s i n the DO b a s i s . 202 It i s important to note that there i s an i n t e r p o l a t i o n scheme inherent in the DO r e p r e s e n t a t i o n of a f u n c t i o n . If one has the DO r e p r e s e n t a t i o n , then the approximate polynomial r e p r e s e n t a t i o n i s obtained by a p p l y i n g the tra n s f o r m a t i o n matrix to i t . Once the f u n c t i o n i s in the polynomial b a s i s , the value of the f u n c t i o n at p o i n t s other than quadrature p o i n t s are c a l c u l a t e d e a s i l y . Now i t i s only necessary to express a r b i t r a r y d i f f e r e n t i a l o p e r a t o r s i n t h i s b a s i s so that d i f f e r e n t i a l equations of the form of Eq.(A7) can be s o l v e d . To do t h i s , I w i l l f i r s t f i n d the polynomial r e p r e s e n t a t i o n of the d e r i v a t i v e operator and then transform i t to the DO r e p r e s e n t a t i o n . The matrix elements of the d e r i v a t i v e operator i n the polynomial b a s i s are given by l ) b D.. = /w(x)R.(x)R'(x)dx. (A19) 1 a 1 T h i s matrix i s upper t r i a n g u l a r s i n c e f o r i > j , R^(x) i s orthogonal to R\ ( x ) . The r e p r e s e n t a t i o n of t h i s operator i n the DO b a s i s i s given by r j( f) = TT.D( e )-T. (A20) N_1 • • • • If g6S , then the d i f f e r e n t i a t i o n o p e r a t i o n in the d i s c r e t e o r d i n a t e b a s i s i s given simply by, a'( f ) = D( f ).3( f ). (A21) The approximate DO r e p r e s e n t a t i o n of the f u n c t i o n s H(x) in the d i f f e r e n t i a l operator L of Eq.(A8) i s given by the diag o n a l matrix with the elements 203 U^V = H(x. ) 5. . , (A22) 13 i i ] where the m index has been om i t t e d . The d i s c r e t e o r d i n a t e approximation of the d i f f e r e n t i a l operator i n Eq.(A7) i s now w r i t t e n i n the form, L( f ) - E [ H( f )]m- [ D( f )]m, (A23) m where [ p / ^ ] ^ i s the u n i t m a t r i x . T h i s i s not an exact r e p r e s e n t a t i o n of the operator i n the DO b a s i s . However, i f a l l the Hm f u n c t i o n s of Eq.(A8) are p o l y n o m i a l s of degree m+1 or l e s s , then i t may be shown that the DO r e p r e s e n t a t i o n i s e q u i v a l e n t to the polynomial r e p r e s e n t a t i o n of the some or d e r . 7.4 DIFFERENTIAL OPERATORS (e) The b a s i c procedure i n v o l v e s the d e t e r m i n a t i o n of D^j d e f i n e d by Eq.(Al9) and performing the t r a n s f o r m a t i o n given by Eq.(A20). A polynomial b a s i s set must be chosen and t h i s choice depends on the problem to be c o n s i d e r e d . The methods developed here are a p p l i c a b l e to any b a s i s s e t . In p a r t i c u l a r the DO b a s i s based on Legendre p o l y n o m i a l s , P^.(x), orthogonal on the i n t e r v a l [-1,1] with unit weight f u n c t i o n i s c o n s i d e r e d i n t h i s Appendix. In a d d i t i o n , the new speed polynomials Bn(x) [R140], orthonormal on [ 0 , » ] 2 -x 2 with the weight f u n c t i o n w(x)=x e are t r e a t e d . F i n a l l y , a general numerical procedure for f i n d i n g the matrix (e) elements of D i s a l s o c o n s i d e r e d . 204 (e) The matrix elements D^_j in any polynomial b a s i s , may be r e a d i l y evaluated from Eq.(Al9) with an i n t e g r a t i o n by p a r t s , that i s DJ5> - 0, i>j w(x)Ri(x)Rj(x) J - (A24) CI b Jw'(x)R,(x)R.(x)dx, i < j . a 3 In the case of normalized Legendre polynomials P ^ ( x ) , t h i s reduces to = (/(2m+1 ) (2n+1 ) m>n,m+n odd = 0, ot h e r w i s e . (A25) T h i s was transformed i n t o the DO b a s i s using the a p p r o p r i a t e l y d e f i n e d t r a n s f o r m a t i o n o p e r a t o r . For the speed polynomial i t i s convenient to proceed i n an a l t e r n a t e way. Any set of orthogonal p o l y n o m i a l s , Rn, may be generated by the three term recurrence r e l a t i o n ^n+ 1Rn + 1( x ) = (x- "n) Rn( x ) " ^ n V 1 ( x > ' ( A 2 6 ) In p a r t i c u l a r , the r e l a t i o n Eq.(A26) i s v a l i d f o r the polynomials Bn( x ) , orthonormal on [0,°°] with respect to the -x2 weight f u n c t i o n x2e . The c a l c u l a t i o n of the a and 0 i n t h i s p a r t i c u l a r case i s d i s c u s s e d elsewhere [R140]. The matrix r e p r e s e n t a t i o n of the d e r i v a t i v e operator i n the 205 polynomial b a s i s may be found by the use of the c o n f l u e n t form of the C h r i s t o f f e l - D a r b o u x i d e n t i t y [R80], N-1 0 Z ( Bk( x ) r = / ^ [ B ^ ( x ) Bn_1 (x) - B (x)B^_-(x) (A27) k = 0 -x 2 If Eq.(A26) with R(x)=B(x) i s m u l t i p l i e d by x2e and i n t e g r a t e d over [0,°°], I f i n d t h a t , Di - i , „ - <A 2 8> -x2 S i m i l a r l y , the m u l t i p l i c a t i o n of Eq.(A27) by x3e and the use of the recurrence r e l a t i o n Eq.(A26), followed by i n t e g r a t i o n over x, y i e l d s , N-1 , x ' , » Z a, = / f l a .D,:e n + \/BB ,D^ e' (A29) k_Q k n n-1 n~1,n n n-1 n-2,n T h i s equation may be rearranged with the use of Eq.(A28) to give rN- 1 D( e» n-2, n Z a, - na , Lk = 0 k n"1 / ^ n ^ n - l ( A 3 0 ) The remaining matrix elements may be found by m u l t i p l i c a t i o n of the recurrence r e l a t i o n , Eq.(A26), by -x2 x2e B^ followed by i n t e g r a t i o n . T h i s y i e l d s D( e )  uk-1,n ; Bk( x ) x 3 e -x 2B i; ( x ) d x - Pk + 1D^Jfn (A31) n( e ) ' a, D, k k, n The i n t e g r a l i n Eq . ( 3 l ) may be evalua t e d by i n t e g r a t i o n by p a r t s , and r e w r i t t e n as 206 Dk - i , n = [ 2^ k+A + 25k + 2,n " ^ W ^ ! , n " ( A 3 2 ) provided k+2 < n. Thus a l l of the nonzero matrix elements are thus e v a l u a t e d by r e c u r s i o n . The method o u t l i n e d above i s a p p l i c a b l e f o r weight f u n c t i o n s of the form x^e x as used i n r e f e r e n c e [R140]. As before the operator i s transformed to the DO b a s i s . Even i f there i s no simple e x p r e s s i o n f o r the matrix elements of the d e r i v a t i v e operator i n some polynomial b a s i s , i t i s always p o s s i b l e to c o n s t r u c t i t i f the recurrence c o e f f i c i e n t s of the polynomials are known. The general recurrence r e l a t i o n f o r a a set of polynomials i s given by Eq.(A26) If the d e r i v a t i v e i s taken on both s i d e s of Eq.(A26) for an a r b i t r a r y set of p o l y n o m i a l s , RN then I ob t a i n / / W *n + 1(x ) = Rn( x ) + ( x"an) Ri( x ) ( A 3 3 ) - ^ R ^ U ) , which, together with Eq.(A26), p r o v i d e a simple recurrence r e l a t i o n f o r the d e r i v a t i v e s of the p o l y n o m i a l s . The matrix elements of D are given simply by N-1 Dn m = I R ( x . )R' (x, )w, , (A34) nm , n n k m K k k = 0 where the sum i s exact i f the proper set of p o i n t s and weights are used, s i n c e Rn(x)R^x) must be a polynomial of degree l e s s than 2N-1. The proper p o i n t s and weights may 207 always be generated from the recurrence c o e f f i c i e n t s , an and Pn [R80]. As an i l l u s t r a t i o n of the u s e f u l n e s s of t h i s .representation of the d e r i v a t i v e o p e r a t o r , I c o n s i d e r the d i f f e r e n t i a t i o n of the o s c i l l a t o r y f u n c t i o n f ( x ) = s i n [ 3 ( s i n h ( x ) + (1 + x )2) ] , chosen a r b i t r a r i l y , Since I am c o n s i d e r i n g the i n t e r v a l [ 0 , 1 ] , Gauss-Legendre quadrature p o i n t s are employed. The second d e r i v a t i v e of t h i s f u n c t i o n was determined n u m e r i c a l l y by repeated a p p l i c a t i o n of Eq.(A2l) with , c o n s t r u c t e d as d e s c r i b e d above. For comparison, the f o u r t h order f i n i t e d i f f e r e n c e approximation of the second d e r i v a t i v e , that i s , f"(x) = [-f(x-2h) + 16f(x-h) + 30f(x) + (A35) 16f(x+h) - f (x + 2h) ] / l 2 h2 + 0 ( hf l) , was a l s o c a l c u l a t e d . The r e s u l t s of t h i s comparison with N=30 and h=0.000l together with f ( x ) a r e shown in Table A1. To generate a matrix d e r i v a t i v e operator based on f i n i t e d i f f e r e n c e s that even approached the accuracy of the DO o p e r a t o r , i t would be necessary to use a matrix of order 300 times that of the DO m a t r i x . Other simple a p p l i c a t i o n s of the DO method have been given by S h i z g a l and Blackmore [R77]. P r i o r to s o l v i n g d i f f e r e n t i a l equations or eigenvalue problems i t i s necessary to impose the boundary c o n d i t i o n s of the problem. T h i s i s done e a s i l y by t a k i n g the d i f f e r e n t i a l operator L a'nd r e p l a c i n g the a p p r o p r i a t e number TABLE A1: Numerical Derivative Comparslon X f(x) f " (x) E 1 f " ( x ) E' 0 0O155326 0 12726007 -16 28527098 -0 17(-10) -16 28527097 0 80 ( -8) 0 00816594 0 06784673 -11 56632236 -0.58(-9) -11 56632235 0 91 ( -8) 0 01998907 -0 03950136 -2 76935530 0 6K-10) -2 76935530 0 4 1 ( -8) 0 03689998 -0 19339086 10 45676067 -o 15(-10) 10 45G76067 0 73( -8) 0 05871973 -0 38695791 28 18794863 0 52(-11) 28 18794864 0 73( -8) 0 08521712 -0 60315987 49 70003239 -0 42(-11) 49 70003239 0 48( -8) 0 11611128 -0 81028462 72 78496383 0 37(-11) 72 78496383 0 99 ( -9) 0 15107475 -0 96035612 93 10145744 -0 25(-11) 93 10145744 -0 50 ( -8) 0 18973691 -0 99431669 103 99807283 0 22C-11) 103 99807281 -0 1 1 ( -7) 0 23168793 -0 85766993 97 48245360 -0 22(-11) 97 48245359 -0 16( -7) 0 27648312 -0 52685871 66 97435252 0 3K-11) 66 97435250 -0 18( -7) 0 32364764 -0 03906356 1 1 79600346 -0 26(-11) 1 1 79600344 -0 21( -7) 0 37268154 0 49138963 -58 16487063 0 18(-11) -58 16487064 -0 99( -8) 0 42306504 0 89100469 -120 97517431 -0 72(-12) -120 97517430 0 1 K -7) 0 47426408 0 98977657 -148 12752992 -0 72(-12) -148 12752989 0 35 ( -7) 0 52573592 0 71055940 -117 80989495 0 57(-13) -117 80989490 0 50 ( -7) 0 57693496 0 13575281 -30 60966056 0 10(-12) -30 60966052 0 44( -7) 0 62731846 -0 50342574 83 12313611 -0 37(-12) 83 12313612 0 74( -8) 0 67635236 -0 92948994 173 24524007 0 15(-11) 173 24524003 -0 45( -7) 0 72351688 -0 96080670 194 99796879 -0 10(-11) 194 99796870 -0 94( -7) 0 76831207 -o 60287125 134 26194964 0 19(-11) 134 26194953 -0 1 1 ( -6) 0. .81026309 -0.02957128 15, .26821907 -0. 2K-11) 15, .26821899 -0. 77(-7) 0. 84892525 0. .52532054 -114. ,48439921 0. 10(-11) -114, .48439922 -0. 23(-8) 0. .88388872 0, .88728264 -210. 48866829 0. 98(-12) -210 .48866820 0. 88(-7) 0. 91478288 0 .99998511 -250. 76022160 -0. 60(-11) -250. .76022144 0. 16(-6) 0. 94128027 0 90969816 -238 . 99727696 0. 13(-10) -238. ,99727676 0. 20(-6) 0. 96310002 0. 71002361 -194. 79272701 -0. 87(-11) -194. , 79272680 0. 21 (-6) 0. 98001093 0. 49067637 -140. 83294025 -0. 41(-10) -140. 83294005 0. 20(-6) 0. 99183406 0. 31251156 -94 . 62706369 -0.27(-9) -94 . 62706351 O. 18(-6) 0. 99844674 0. .20667729 -66 . 36410768 -c ). 22( -8) -66. 36410752 0. 16(-6) 1 . DO result 2. Finite difference result 210 of rows with equations for the boundary c o n d i t i o n s . The a p p r o p r i a t e number depends on the order of equation c o n s i d e r e d . 7.5 SUMMARY The d i s c r e t e o r d i n a t e method developed here g i v e s a high order a l g o r i t h m f o r the numerical d e t e r m i n a t i o n of d e r i v a t i v e s . Once the d e r i v a t i v e operator has been generated, i t may be used to approximate d i f f e r e n t i a l o p e r a t o r s . This approximate matrix r e p r e s e n t a t i o n of the operator may be used to solve Eq.(A7) by a simple matrix i n v e r s i o n . Furthermore, once the s o l u t i o n v e c t o r has been found, the s o l u t i o n i s given at a l l p o i n t s s i n c e , using the t r a n s f o r m a t i o n o p e r a t o r , the polynomial r e p r e s e n t a t i o n of the s o l u t i o n vector may be c a l c u l a t e d . I t should be noted that t h i s proceduce i s very s i m i l a r to the c o l l o c a t i o n method [R141] in that the s o l u t i o n i s found at a set of quadrature p o i n t s . Other methods include f i n i t e d i f f e r e n c e s [R142-144], and i n v e r s i o n s of Green's f u n c t i o n s [R145], Each of these methods has t h e i r own p i t f a l l s . F i n i t e d i f f e r e n c e c a l c u l a t i o n s u s u a l l y r e q u i r e f i n e mesh of p o i n t s i f accuracy i s r e q u i r e d and Green's f u n c t i o n s are o f t e n d i f f i c u l t to c a l c u l a t e The numerical a p p l i c a t i o n s of t h i s operator haven't been presented in t h i s s e c t i o n s i n c e t h i s method has been e x t e n s i v e l y used throughout Chapters 1 through 5. The DO method f o r s o l u t i o n of d i f f e r e n t i a l equations has v a r i o u s advantages i n c l u d i n g i t s f l e x i b i l i t y with the use of a r b i t r a r y b a s i s s e t s . In a d d i t i o n , once the d e r i v a t i v e operator has been generated, a r b i t r a r y d i f f e r e n t i a l o p e r a t o r s are approximated e a s i l y . F i n a l l y , s o l u t i o n v e c t o r s are obtained at the quadrature p o i n t s of an i n t e g r a t i o n r u l e , and hence, numerical i n t e g r a l s over the s o l u t i o n are performed e a s i l y . 8. APPENDIX B In t h i s Appendix, I develop a recurrence r e l a t i o n f o r the recurrence c o e f f i c i e n t s f o r a set of polynomials which are orthonormal over the bimodal weight f u n c t i o n given by Eq.(3.2). The corresponding quadrature weights and p o i n t s , and the d e r i v a t i v e operator i n the DO b a s i s are a l s o c a l c u l a t e d . The set of polynomials are d e f i n e d such that Rn(x) i s a polymomial of degree n. These polynomials may be generated from a three term recurrence r e l a t i o n of the form ^ W V l ( x ) = ( x-an) Rn( x )- ^ V l( x )- ( B 1 ) The an are r e l a t e d to odd moments of w(x) by CO an = J w ( a , 7 ; x ) x [ Rn( x ) ]2d x . (B2) — CO The r e l a t i o n Eq.(B2) was d e r i v e d by m u l t i p l y i n g Eq.(B1) by w(x)Rn(x) and i n t e g r a t i n g . The an must a l l vanish s i n c e the weight f u n c t i o n i s an even f u n c t i o n and hence I am l e f t with the simpler recurrence r e l a t i o n , /^77Rn+1(x) = *V*>-/*^7Vi( x )- ( B 3 ) The B were generated by a method s i m i l a r to one p r e v i o u s l y employed [R140], based on the C r i s t o f f e l - D a r b o u x i d e n t i t y , j 0 Rk( x ) = ^ f Vx ) Rn+1( x )- V l( x )Vx ) ]- ( B 4 ) When Eq.(B4) i s m u l t i p l i e d by w(a,y;x) and i n t e g r a t e d I o b t a i n 212 213 n+1 = v//3n+i; w(a,7;x)Rn(x)R^+ 1 (x)dx, (B5) — 00 where the second i n t e g r a l vanishes s i n c e R^ i s orthogonal to Rn. The r i g h t hand side of Eq.(B5) may be i n t e g r a t e d by par t s y i e l d i n g , n+ 1 = / 0 n + 1 J ( 7 X3- a x ) w ( a , 7 ; x ) Rn( x ) Rn + 1( x ) d x . (B6) — 00 F i n a l l y , with the repeated a p p l i c a t i o n of Eq.(B3), the i n t e g r a l i n Eq.(B6) may be evalua t e d g i v i n g with some rearrangment "n+2 " f " " » „ • <B 7» n+1 Eq.(B7) i s the d e s i r e d r e s u l t . A l l the 0 may be found by recurrence i f j3, i s known s i n c e /30 i s equal to 0. It may be determined by s e t t i n g n=0 i n Eq.(B3), squaring and i n t e g r a t i n g . T h i s g i v e s , with some rearrangement 00 j3! = J* w(a,7;x)x2dx, = / e("7 x V 2 + a x 2 )x2d x / / e(-7 x V 2 + a x 2 )d x . (B8) — 00 —00 As i n the p r e v i o u s paper, the recurrence r e l a t i o n , Eq.(B7) for the re c u r r e n c e c o e f f i c i e n t s s u f f e r s from round o f f e r r o r and i t i s t h e r e f o r e necessary to use high p r e c i s i o n when c a l c u l a t i n g the 0 . Thus the two i n t e g r a l s Km) = / xme(^x V 2 + a x 2 )d x , (B9) — 00 where m=0,2 must be evaluated e s s e n t i a l l y e x a c t l y . 214 These i n t e g r a l s were evaluated i n the f o l l o w i n g manner. The exp(ax2) f a c t o r in the integrand was expanded i n a T a y l o r ' s s e r i e s expansion and the r e s u l t i n g i n t e g r a l s were evaluated term by term, y i e l d i n g Km) = 2 -i (m+1 )/4 n = 0 n! "2a2" n/2 r "m+2n+l" 7 4 (B10) With Eq.(BlO) w r i t t e n out term by term and the use of the recurrence r e l a t i o n f o r the gamma f u n c t i o n , I have t h a t , Km) = 2 ~2 (m+1)/4 r "m+ r + a "2" 1/2 r "m+3" 7 4 7 * 4 (B1 1 ) m+1 00 z -I n=1 2n! 2a- n ( k - i + 2 i i ) Lk=i 4 "2" 1/2 r "m+3" .7. 4 n= 1 1 (2n+1)! 2a-7 n r n , n ( k - i+5»±3) Lk=1 • By i n c l u d i n g enough terms i n t h i s expansion, the f i r s t two moments c o u l d be e v a l u a t e d to any degree of acc u r a c y . The gamma f u n c t i o n was e v a l u a t e d by breaking i t up i n t o two p i e c e s , x r ( a ) = J y( a 1 )e yd y +7( a , x ) . 0 (B1 2) The f i r s t i n t e g r a l of Eq.(Bl2) was evaluat e d by expanding the i n tegrand and i n t e g r a t i n g term by term. The incomplete gamma f u n c t i o n has a con t i n u e d f r a c t i o n expansion[R146], / \ -x a y(a,x) = e x 1 1 -a 1 x+ 1+ x+ 1+ 2-a 2_ x + (B13) With the use of Eqs.(B11-13) the i n t e g r a l s i n Eq.(B9) were 215 e v a l u a t e d to 115 decimal p l a c e s and the B were the c n evualated with the use of Eq.(B7). I now wish to develop a d i s c r e t e o r d i n a t e d e r i v a t i v e operator as d e s c r i b e d p r e v i o u s l y . To do t h i s I f i r s t f i n d the matrix r e p r e s e n t a t i o n of the d e r i v a t i v e operator i n the polynomial b a s i s , that i s , CO D L , = / w(a,7;x)R (x)R'(x)dx. (B14) nm J n m — Co From Eq.(B5) i t f o l l o w s d i r e c t l y that - 7 ^ 7 , ' m = n + ' - ( B , 5 > The other matrix elements may be evaluated by i n t e g r a t i n g Eq.(B14) by p a r t s . With repeated use of the recurrence r e l a t i o n , Eq.(B3), i t may be shown that the only other non-zero matrix elements are those for which m=n+3. Thus, I have that Dnm = 2^n + 1/ 3n + 2A + 3, m=n + 3, (B16) and D*J = 0, oth e r w i s e . (B17) Eqs.(B15-B17) d e f i n e the polynomial r e p r e s e n t a t i o n of the d e r i v a t i v e o p e r a t o r . The corresponding DO r e p r e s e n t a t i o n may be found with Eq.(3.22). The quadrature p o i n t s and weights were found by d i a g o n a l i z i n g the symmetric t r i d i a g o n a l J a c o b i matrix d e f i n e d by X = |3 n=m+1 , (Bl8a) nm n 216 X „m = B , m=n+1, (Bl8b) nm m Xn m = 0, otherwise, ( B 1 8 C ) where the quadrature p o i n t s are the eigenvalues of t h i s matrix and the weights are the square of the f i r s t element of the e i g e n v e c t o r s [R80]. 9. APPENDIX C The o b j e c t i v e of t h i s Appendix i s to generate the 0 **• X ^ i n t e g r a t i o n r u l e for the weight f u n c t i o n x^e and the i n t e r v a l s [0,y] and [y,°°]. That i s , I wish to generate two se t s of p o i n t s ({x^} and {x^}) and weights ({V^} and {V^}) such that Y D _ X 2 N-1 f xPe X f ( x ) d x = 2 V^f(x^) (CD 0 j=0 3 3 and oo _ 2 N-1 Jxpe x f(x)dx = Z V".f(x".) (C2) y j=0 3 3 p r o v i d e d f ( x ) i s a polynomial of degree l e s s than 2N-1. As was seen i n Appendix B, t h i s may be done by d i a g o n a l i z i n g the J a c o b i m a t r i x . The elements of t h i s matrix are the recurrence c o e f f i c i e n t s f o r the polynomials orthonormal over o — x2 the weight f u n c t i o n x^e and the two i n t e r v a l s given above. These recurrence c o e f f i c i e n t s w i l l be d e r i v e d i n t h i s Appendix. These c o e f f i c i e n t s should have l a b e l s i n d i c a t i n g which range they r e p r e s e n t , however, f o r the sake of n o t a t i o n a l s i m p l i c i t y , they w i l l remain u n l a b e l e d . The method used to c a l c u l a t e the recurre n c e c o e f f i c i e n t s i s s i m i l a r to the method used i n a previous paper [R140] to f i n d these c o e f f i c i e n t s f o r the set of polynomial orthonormal over the same weight f u n c t i o n but on the i n t e r v a l [0,°°]. The d e r i v a t i o n in t h i s Appendix w i l l c e n t e r on the set of recurrence c o e f f i c i e n t s f o r the polynomials orthonormal on [0,y] with respect to the weight 217 218 D — X 2 x*e . The d e r i v a t i o n of the recurrence r e l a t i o n f o r the other set of polynomials i s s i m i l a r and the f i n a l r e s u l t w i l l be noted for that s e t . Every set of orthonormal polynomials has the recurrence r e l a t i o n given by xBn = ^ n + ] B n + ] + anBn + S ( tNBN_R (C3) T h i s r e l a t i o n , in c o n j u n c t i o n with the C r i s t o f f e l - D a r b o u x , given by * Bk = ^ n+1t Bn+1Bn " Bn+1Bn]' ( C 4 ) k = 0 form the b a s i s for the d e r i v a t i o n of the recurrence c o e f f i c i e n t s . Two equations immediately f o l l o w from the m u l t i p l i c a t i o n of Eq.(C4) by x^e x or xp + 1e X f o l l o w e d by i n t e g r a t i o n , thus ( n + l ) = ^ n+1< Bn + l V ' { C 5 ) and * ak = ^ n+1< x B; + l V <C6) k = 0 where the angular brackets i n d i c a t e s an i n t e g r a l over [0,y] of the bracketed e x p r e s s i o n times the weight f u n c t i o n . I t immediately f o l l o w s form the recurrence r e l a t i o n Eq.(C3) and Eq.(C5) that (n+1) = <xBn + 1B;+ l>. (C7) The i n t e g a l 219 <xBn(x)B^+ 1 (x)> = J xP + 1e x 2B j x ) B ; + 1 (x)dx, (C8) Y /: 0 nv ' n+1 i s e v a l u a t e d by i n t e g r a t i o n by p a r t s to give <xB (x)B' (x)> = yC + n n+1 J n,n+1 (C9) where 2 < x2Bn( x ) Bn + 1( x ) > , 2 C m = yPe y B ( y ) B ( y ) . (C10) n J m and the terms v a n i s h i n g as a r e s u l t of the o r t h o g o n a l i t y of the polynomials have been removed. The i n t e g r a l on the l e f t hand s i d e of Eq.(C9) i s given by Eq.(C5), while the i n t e g r a l on the r i g h t hand side may be e v a l u a t e d by repeated a p p l i c a t i o n of Eq.(C3) i n s i d e the i n t e g r a l to give < x2B „ ( x ) B _ ( x ) > = B (a +a .,) n n+1 n+1 n n+1 (CI 1 Thus, with Eqs.(C5) and (C11), I am able to write Eq.(C9) as 2^n+1( an+ an+1} = 2 a, - //3n + 1yC Lk = 0 n+1J n,n+1 (C12) T h i s i s the f i r s t of two r e l a t i o n s between the a and 8 n n necessary f o r t h e i r c a l c u l a t i o n . To f i n d another r e l a t i o n s h i p , c o n s i d e r the i n t e g r a l <xBn + 1B^+ l>. T h i s may be i n t e g r a t e d by part to give <xBn + 1B;+ 1> = [ y Cn + 1 f P + 1 - (p+1) <y2Bn + 1Bn + 1> - <xBn + 1B;+ 1>]. (C13) n+1 n+1 The f i r s t i n t e g r a l on the r i g h t hand s i d e may be eva l u a t e d 2 2 0 by squaring the recurrence r e l a t i o n , multiplying by the weight function and integrating. The last integral may be col l e c t e d on the l e f t hand side to give 2<xBn+1B;+1> = [ y Cn + l f n + 1 - (p+1) + (C14) 2 (0n + 2 + "n+1 + < W]-However, the integral on the l e f t hand side has been evaluated in Eq.(C7), thus I have that ( 2 n+ P +l ) = yCnfn + 2(Bn + 2 + a2+ 1 + 0 ^ ) , (C15) which may be rewitten as = n + (P+ 1 ) / 2 " y Cn , n/ 2 " an " V ( C 1 6 ) Eq.(C12) may be rewritten as r n n+ 1 Z a, - \/B ^.yC k = 0 k n,n+1 /(2/3n+1) - an. (C17) Eqs.(C16) and (C17) together provide a recurrence relations for the c o e f f i c i e n t s a and B . The only difference in n n J these relations for the set of polynomials orthonormal on [y,°°] i s that the boundary term has the opposite sign. Before these relations may be used, i t i s necessary to find some recurrence r e l a t i o n for the boundary terms and to calculate the starting values of the recurrence c o e f f i c i e n t s . The boundary terms C and j / B _,_, C _,_. may 2 n,n n+1n,n+1 2 be calculated d i r e c t l y from the recurrence relation Eq.(C3), since only the values of the recurrence c o e f f i c i e n t s that have previously been calculated need to be used. 221 To s t a r t the recurrence i t i s enough to know the value of a0 s i n c e Bo = 0 and B, i s given by Eq.(C16) and a, i s then given by Eq.(Cl7) and so on. T h i s value may be o b t a i n e d i n the f o l l o w i n g manner. Since the polynomials are orthonormal the f i r s t p o l y n o m i a l , B0, a c o n s t a n t , must s a t i s f y the r e l a t ion J B g x p e x dx = 1, (C18) 0 or B0 = rY rt -x2 1"1/2 Txpe x dx . (C19) -0 S i m i l a r l y , the value of the second polynomial w i l l be given by /M, = ( x - a0) B0. (C20) If Eq.(C20) i s m u l t i p l i e d by xpe x B0 and i n t e g r a t e d , then I have, using the o r t h o n o r m a l i t y of the p o l y n o m i a l s , that y 2 a0 = B g / xp + 1e ~x dx, (C21) 0 or a 0 = / xP + 1e "x 2d x / j f xpe "x 2d x . (C22) 0 0 A s i m i l a r r e l a t i o n holds for the value of aQ f o r the upper range. In the s p e c i a l case for p=2, Eq.(C22) reduces to G o = 2[ i - e 'y 2( i H- y2) ]t ( C 2 3 ) v/7rerf (y) - 2ye y S i m i l a r l y , i n t h i s s p e c i a l c a s e , the i n i t i a l value f o r a0 f o r the upper range polynomials i s given by 222 a o . ^ * " y 2 " + y 2> _1 , (C24) i / 7 r e r f c ( y ) - 2ye y Eq.(C16), (C17) and (C23) together provide a recurrence formula for the recurrence coefficents for polynomials rj — x 2 defined with the weight function x*e and the interval [0,y]. The corresponding quadrature rule Eq.(Cl) may be found by diagonalizing the matrix defined by Eq.(Bl8). 

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