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

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OK THE QUANTUM STATISTICAL THEORY OF THERMAL CONDUCTIVITY.  fey PETER ALLAN GRIFFIN B. Sc., University of B r i t i s h Columbia, i 9 6 0 .  A THESIS PRESENTED IN PARTIAL FULFILMENT- OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS  We accept t h i s thesis as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA SEPTEMBER, I 9 6 I .  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 shall make i t freely available for reference and study.  I further agree that permission  for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allox^ed without my written permission.  Department of  PI-} VS  I C S  The University of British Columbia, Vancouver 8, Canada. Date  UJUyCuU^  £LO,\ M$\ c  ABSTRACT  A c r i t i c a l survey of the present state of the quantum s t a t i s t i c a l theory of thermal conductivity i s given. Recently several attempts have been made to extend Kubo's treatment of e l e c t r i c a l conduction t o other i r r e v e r s i b l e transport processes i n -which the interaction between the d r i v i n g system and the system of i n t e r e s t i s not p r e c i s e l y known. No completely s a t i s f a c t o r y solution of the problems involved i s contained i n the l i t e r a t u r e . In t h i s t h e s i s , a d e t a i l e d derivation of a Kubo-type formula f o r thermal conductivity i s given, using e s s e n t i a l l y the concepts and methods of Nakajima and Mori, with no pretense that i t s e t t l e s the problem completely. Some general remarks are made on the evaluation of a Kubo-type expression, i n p a r t i c u l a r , the use of Van Hove's master equations and the reduction of the usual N-particle formula t o a single p a r t i c l e formula, An e x p l i c i t c a l c u l a t i o n of thermal conduct i v i t y i s made f o r the simple model of e l a s t i c electron scattering by randoml y d i s t r i b u t e d , s p h e r i c a l l y symmetric impurities.  -iiiTABLE OF CONTENTS  ABSTRACT  i i  TABLE OF CONTENTS  i l l  ACKNOWLEDGEMENTS  iv  INTRODUCTION CHAPTER 1 .  1 NON EQUILIBRIUM PROCESSES IN ISOLATED MANY-PARTICLE  SYSTEMS CHAPTER 2 .  8 THE PROBLEM OF TRANSPORT PROCESSES IN OPEN SYSTEMS AND  THE STANDARD "BOLTZMANN EQUATION" SOLUTION  .23  SECTION 2.1 :TRANSPORT PHENOMENA IN OPEN SYSTEMS  .23  SECTION 2.2 :OUTLINE AND CRITICISM OF THE STANDARD. METHOD OF FINDING ELECTRICAL AND THERMAL CONDUCTIVITY CHAPTER 3 .  "  KUBO'S DERIVATION OF FORMAL EXPRESSIONS FOR THE TRANS-  PORT COEFFICIENTS DESCRIBING MECHANICAL DISTURBANCES CHAPTER k .  .28  33  ATTEMPTS TO FIND KUBO-TYPE FORMULAS FOR TRANSPORT  COEFFICIENTS CONNECTED WITH THERMAL DISTURBANCES (THERMAL CONDUCTIVITY) SECTION k.l  A?  :INTRODUCTION AND CHOICE OF OPERATORS  ^5  SECTION k.2 :DERIVATION OF A KUBO-TYPE FORMULA FOR THERMAL (X5NDUCTIVITY  5  SECTION U.3 : CRITICAL REVIEW OF OTHER METHODS CHAPTER 5 .  5  .65  EVALUATION OF KUBO-TYPE EXPRESSIONS  .755  SECTION 5.1 :RELATION TO VAN HOVE'S MASTER EQUATIONS AND THE REDUCTION TO SINGLE PARTICLE EXPRESSIONS 0  .75  SECTION 5.2 :EXPLICIT EVALUATION OF THERMAL CONDUCTIVITY FOR ELECTRONS SCATTERED BY IMPURITIES BIBLIOGRAPHY  £2 .5 8  -ivACKNOWLEDGEMENTS  I wish to express my gratitude t o Professor W. Opechowski f o r suggesting t h i s problem and f o r h i s continued i n t e r e s t and valuable advice throughout the performance of this research*  I n addition, I am indebted  to Mr. K. Nishikawa f o r many u s e f u l conversations as w e l l as f o r considerable assistance i n the preparation of the f i n a l d r a f t of t h i s t h e s i s .  I wish  also t o acknowledge the f i n a n c i a l assistance of the National Research  r Council of Canada.  (1) INTRODUCTION  Experimentally,  i t i s w e l l known that when a weak external d i s -  turbance i s a p p l i e d to an Isolated system, the r e s u l t i n g "currents" are proportional to the external "forces". The constants of p r o p o r t i o n a l i t y enable us to describe i r r e v e r s i b l e transport processes by a few experimental parameters. I t i s a fundamental problem to give a t h e o r e t i c a l basis f o r the existence of such " f i r s t order transport c o e f f i c i e n t s " (as they are c a l l e d by the t h e o r i s t ) and to show how  to c a l c u l a t e them i n terms of the micro-  scopic properties of the p h y s i c a l system involved. Examples of these c o e f f i cients are e l e c t r i c a l conductivity, thermal conductivity, v i s c o s i t y and  the  d i f f u s i o n constant. In t h i s t h e s i s , we s h a l l be mainly interested i n the f i r s t two  coefficients. U n t i l very recently, the only method of c a l c u l a t i n g these c o e f f i -  cients was by the use of Boltzmann-type k i n e t i c equations. These are assumed to be s a t i s f i e d by s i n g l e p a r t i c l e d i s t r i b u t i o n functions of the p a r t i c l e s and q u a s i - p a r t i c l e s involved. Except f o r c e r t a i n s p e c i a l cases, the v a l i d i t y of these transport equations i s highly doubtful. In addition, these complex i n t e g r o - d i f f e r e n t i a l equations are not e a s i l y solved and,  i n most s i t u a t i o n s ,  a u n i v e r s a l relaxation time must be introduced before c a l c u l a t i o n s are f e a s i b l e . The use of a d i s t r i b u t i o n function i s suspect from the point of view of quantum mechanics and a t best gives only p a r t i a l information, since i t corresponds to a diagonal matrix element of a reduced density matrix. Over and above any t h e o r e t i c a l objections, the whole transport equation method does not give too much p h y s i c a l i n s i g h t i n t o the processes involved. Neither does i t involve any consistent manner of making approximations. The extreme d i f f i c u l t y of extending the standard treatment i n order to f i n d corrections of higher  order  ( 2 ) i n the i n t e r a c t i o n between the p a r t i c l e s and quas i - p a r t i c l e s i s w e l l known. Recently there has been considerable success achieved i n the attempt to give a rigourous quantum s t a t i s t i c a l theory of f i r s t order i r r e v e r s i b l e transport processes. Same authors have put the Boltzmann-type equations on a sounder f o o t i n g . A more i n t e r e s t i n g step has been a whole reformulation of the problem, i n i t i a t e d mainly by the Japanese p h y s i c i s t s M o r i (1956), Nakano (1956), Nakajima (1956) and Kubo (1956). Following the p r a c t i c e of the current l i t e r a t u r e , we s h a l l c a l l t h i s new approach Kubo's method (or Kubo*8 formalism). I t i s based on the use of the density matrix whose equation of motion i s solved to the f i r s t order i n the "external disturbance". I t i s a quantum s t a t i s t i c a l theory applicable t o any transport process where the weak i n t e r a c t i o n between the i s o l a t e d system and the " d r i v i n g " system can be expressed by some appropriate quantum mechanical operator. The main r e s u l t of t h i s theory i s that we are able to express c e r t a i n transport c o e f f i c i e n t s by means o f a double i n t e g r a l involving a time dependent c o r r e l a t i o n function evaluated over an equilibrium ensemble. These formal expressions are derived without any important r e s t r i c t i o n s being introduced. The evaluation f o r s p e c i f i c p h y s i c a l models generally involves a f a i r l y complicate ed c a l c u l a t i o n , although the form o f these expressions enables us to develop general approximation methods. Now  '  there e x i s t s a whole class of i r r e v e r s i b l e transport processes  to which the above-mentioned Kubo's formalism i s not d i r e c t l y applicable,since, i n these processes, the i n t e r a c t i o n between the " d r i v i n g " system and the system of i n t e r e s t cannot e a s i l y be expressed by means of an unambiguous quantum mechanical operator. In these processes, the p r e c i s e mathematical descript i o n of the external disturbance  ( f o r example, a temperature  obvious, other than i n a macroscopic  gradient) i s not  sense. Several authors have attempted to  derive Kubo-type expressions f o r the transport c o e f f i c i e n t s occurring i n t h i s  ( 3 ) c l a s s of transport processes. The major aim of t h i s thesis i s to give a c r i t i c a l review of these derivations, taking thermal conductivity as a concrete example. In our opinion none of the discussions i n the l i t e r a t u r e i s completely successful. Chapter 1 i s e n t i r e l y devoted to a discussion of the dynamical evol u t i o n of a large system of i n t e r a c t i n g p a r t i c l e s , with p a r t i c u l a r reference to the work of Van Hove (1955, 1957)  on quantum mechanical transport  At f i r s t s i g h t , the study of the non-equilibrium  equations.  behaviour of an i s o l a t e d  system has l i t t l e to do with the main topic of t h i s t h e s i s , which i s the i r r e v e r s i b l e transport processes occurring i n open systems. However, one of the important c h a r a c t e r i s t i c s of Kubo-type expressions f o r transport c o e f f i c i e n t s i s that t h e i r e x p l i c i t evaluation involves only a knowledge of the dynamical evolution of an i s o l a t e d system. Indeed, we can look upon the derivation of closed, formally exact expressions  f o r transport c o e f f i c i e n t s as simply a  transformation of a complex problem to a more basic  one.  That i s , we are o r i g i n a l l y dealing with-an open system (an i s o l a t e d system under the influence of an external disturbance); but, i n the end,  the  transport properties are incorporated into a Kubo-type expression which involves only the properties of an i s o l a t e d system. In chapter 2, we come to transport processes i n open systems. A f t e r the problem has been stated mathematical 1y, the phienomenological description of i r r e v e r s i b l e processes i n open systems i s touched upon. In p a r t i c u l a r , the macroscopic description of e l e c t r i c and thermal conduction i s outlined. In the second h a l f of chapter 2, we give a b r i e f analysis of the disadvantages of the standard method of evaluating e l e c t r i c a l and thermal conductivity by means of Boltzmann-type transport  equations.  An account of Kubo's formalism (1957&) i s contained i n chapter 3. T h i s formalism deals successfully with mechanical disturbances  such as an  (h) external e l e c t r i c f i e l d . D e t a i l e d discussions of Kubo's work have been given by several authors - f o r example, Van Hove ( i 9 6 0 ) and Montroll ( 1 9 5 9 ) . This chapter contains nothing new,  except possibly a more c r i t i c a l treatment of the  p h y s i c a l assumptions necessary i n order that the response current be  independent  of when and how the external f i e l d i s a p p l i e d . There i s some ambiguity on t h i s point i n the l i t e r a t u r e , f o r example, i n the treatments given by Lax ( 1 9 5 8 ) * Kubo ( 1 9 5 7 a ) and Van Hove ( i 9 6 0 ) . One of the main reasons f o r including the material of chapter 5 i s to provide an example of what we mean by a s a t i s f a c tory quantum s t a t i s t i c a l theory of l i n e a r , i r r e v e r s i b l e transport processes. I t may be pointed out that i t has been the great success of Kubo's treatment of mechanical disturbances which has encouraged attempts to extend the t r e a t ment to external disturbances which are, p h y s i c a l l y , e n t i r e l y d i f f e r e n t . That we are able to c a l c u l a t e the e f f e c t on an i s o l a t e d system due to an e l e c t r i c f i e l d , f o r example, does not J u s t i f y the expectation, a p r i o r i , that we can do likewise f o r a temperature gradient. With chapter k, we come to the main part of t h i s t h e s i s , a c r i t i c a l survey of the attempts t o f i n d a closed, formally exact Kubo-type expression f o r thermal conductivity. The f i r s t problem to be considered i s the proper choice of a quantum mechanical operator whose ensemble average gives the energy current. The usual d e f i n i t i o n Involves a summation over the products of the energy and momentum operators of the p a r t i c l e s and q u a s i - p a r t i c l e s composing the system. Ideally the proper d e f i n i t i o n should f o l l o w from a rigourous theory of heat conduction. The manner i n which we define l o c a l density operators i s another problem often skipped over i n the l i t e r a t u r e . Both Mori ( I 9 5 8 ) and Nakajima ( 1 9 5 8 , 1 9 5 9 b ) simply take the quantum mechanical operators corresponding, v i a Weyl's r u l e , to the c l a s s i c a l functions involving delta s i n g u l a r i t i e s i n the positions of the p a r t i c l e s . As a consequence of these s i n g u l a r i t i e s , the content of these theories i s b a s i c a l l y s e m i - c l a s s i c a l .  ( 5 ) In the second section of chapter 4, we give a derivation of a Kubotype formula f o r thermal conductivity by making use of the concept of the l o c a l equilibrium density matrix. The f i n a l formula has been written down by several authors including Mori (1956) and Kubo, Yokota and Nakajima (1957b), but the concepts and methods used i n the l i t e r a t u r e db not as yet provide a s a t i s f a c t o r y proof. Our discussion follows the work of Mori (1958, 1959) and Nakajima (1959). I t i s based on the use of an equivalent i s o l a t e d system relaxing from a weak f l u c t u a t i o n . This f l u c t u a t i o n i s chosen so that, i n i t i a l l y , i t leads to the same energy current flow which exists i n the open system involving a small temperature gradient. E s s e n t i a l use i s made of the smallness of the temperature gradient. As a consequence, the analysis cannot be generalized to deal with quadratic and higher order e f f e c t s , i n contrast to Kubo's o r i g i n a l method. As i n chapter 3* we pay p a r t i c u l a r attention t o the importance of the d i s s i p a t i v e properties of the many-particle system. In our opinion, the l i t e r a ture does not place enough stress on the f a c t that any s a t i s f a c t o r y derivation of Kubo-type formulas f o r the transport c o e f f i c i e n t s presupposes that the dynamical evolution of the i s o l a t e d system has been investigated. This i s quite apart from the evaluation of these abstract formulas. In section 4.3, same b r i e f remarks are made on other methods of deriving Kubo-type formulas. Of s p e c i a l i n t e r e s t i s the work of Montroll (1959) who attempts to e x p r e s s l y , t r i c k " boundary conditions, the i n t e r a c t i o n between the w  d r i v i n g system (the heat reservoirs i n our case ) and the system of i n t e r e s t by means of some quantum mechanical operator. Lebovitz (1957, 1959) a l s o attempts to e x p l i c i t l y f i n d the i n t e r a c t i o n , although h i s analysis makes use of a k i n e t i c equation. Although we are mainly interested i n quantum mechanical systems, i t should be mentioned that M. S. Green (195*0 developed a theory f o r c l a s s i c a l  r systems which i n many ways i s the c l a s s i c a l analogue of the treatment i n section 4.2. A recent paper by Kirkwood and F i t t s (i960) r e s u l t s i n the thermal  ( 6 ) conduct!vity tensor (among other transport c o e f f i c i e n t s ) f o r c l a s s i c a l systems being expressed as a time i n t e g r a l over a c o r r e l a t i o n function. Indeed, i n a c e r t a i n sense, Kubo's method i s simply a quantum generalization of Kirkwood s pioneering work ( 1 9 4 6 ) . 1  In the f i n a l chapter, we consider the general problem of evaluating Kubo-type formulas. The r o l e of these abstract expressions i n l i n e a r transport processes Is s i m i l a r to that played by the formal expressions used i n e q u i l i brium s t a t i s t i c a l mechanics which give the average value of some operator by means of the trace over the product of the operator and the quantum mechanical p a r t i t i o n function. By means of the l a t t e r , we can express a l l thermodynamic variables, the only problem being that of e x p l i c i t evaluation. Except f o r c e r t a i n investigations into the foundations, most of the recent l i t e r a t u r e on equilibrium s t a t i s t i c a l mechanics i s a search f o r b e t t e r methods of approximat i o n of these formal expressions involving the p a r t i t i o n function. S i m i l a r l y , once we have derived Kubo-type formulas f o r transport c o e f f i c i e n t s , the only remaining problem i s t o evaluate the complex, N-particle, time-dependent correl a t i o n functions involving the p a r t i t i o n function. Only a beginning has been made i n the l i t e r a t u r e i n f i n d i n g general methods of approximate evaluation of Kubo-type expressions. The c l u s t e r i n t e g r a l method of Montroll and Ward (1959b) and the use of Green's functions (see, f o r example, the review a r t i c l e by Zubarev ( 1 9 6 l ) ) both enable us to evaluate these transport c o e f f i c i e n t s to any stage of approximation, i n p r i n c i p l e a t l e a s t . No c a l c u l a t i o n s are made using e i t h e r of these methods i n t h i s t h e s i s . In the second part of chapter 5* we discuss the r e l a t i o n of Van Hove's work on master equations t o the evaluation of the c o r r e l a t i o n functions involved i n the exact expressions f o r transport c o e f f i c i e n t s . As Chester and Thellung (1959) and Klinger ( 1 9 6 l ) have stressed, Van Hove's work i s one of the most f r u i t f u l methods of evaluation, i f not as general as those mentioned a t the end of the above paragraph,, A f t e r a few remarks on the assumptions  < 7 ) necessary t o reduce the usual N-particle Kubo-type formula t o a single p a r t i c l e formula (see, f o r example, Verboven (i960) ), we make use of the l a t t e r t o e x p l i c i t l y evaluate the thermal conductivity f o r a simple model of a metal i n which electrons are scattered e l a s t i c a l l y by s t a t i c , randomly d i s tributed,  s p h e r i c a l l y symmetric impurities. This c a l c u l a t i o n i s simply an  extension of the work of Chester and Thellung (1959) on e l e c t r i c a l  conductivity.  To conclude t h i s introduction, we should l i k e t o add two points. F i r s t , Dresden (I961) has recently published the f i r s t part of an exhaustive review a r t i c l e on the recent progress i n transport theory. While t h i s f i r s t part only emphasizes the d i f f i c u l t i e s o f the standard methods, the published summary of the second part indicates that i t w i l l cover s i m i l a r topics as are discussed i n t h i s thesis, although a t t e n t i o n w i l l not be r e s t r i c t e d to thermal conductivity. L a s t l y , though the work of many authors i s quoted i n t h i s thesis, no attempt a t completeness has been made. Only those papers which have been used d i r e c t l y i n w r i t i n g the text have been e x p l i c i t l y r e f e r r e d to.  ( ) 8  CHAPTER 1 NON  EQUILIBRIUM PROCESSES IH ISOLATED MANY-PARTICLE SYSTEMS.  C o n s i d e r an i s o l a t e d p h y s i c a l system composed o f a l a r g e number o f i n t e r a c t i n g p a r t i c l e s e n c l o s e d i n a volume _TL  . A fundamental problem o f  t h e o r e t i c a l p h y s i c s i s , g i v e n t h e k i n d s o f p a r t i c l e s and t h e v a r i o u s  inter-  a c t i o n s among them, t o d e r i v e t h e m a c r o s c o p i c p r o p e r t i e s o f t h e p h y s i c a l s y s tem.  I n r e c e n t y e a r s , c o n s i d e r a b l e p r o g r e s s has been made i n c a r r y i n g o u t  this  program f o r v a r i o u s n o n - t r i v i a l c a s e s . The advances i n t h i s so c a l l e d N-body problem have been l a r g e l y due t o t h e improved m a t h e m a t i c a l t e c h n i q u e s were o r i g i n a l l y developed diagram  which  i n quantum f i e l d t h e o r y , as f o r example, t h e use  of  techniques. An  i s o l a t e d p h y s i c a l system ( t h r o u g h out t h i s t h e s i s , t h i s  always r e f e r t o a quantum m e c h a n i c a l m a n y - p a r t i c l e s p e c i f i e d ) may  system u n l e s s  will  otherwise  be i n an " e q u i l i b r i u m " o r " n o n e q u i l i b r i u m " s t a t e a t any  partic-  u l a r time. Before g i v i n g a mathematical f o r m u l a t i o n of t h i s d i s t i n c t i o n , s h a l l make a s h o r t d i g r e s s i o n i n o r d e r t o i n t r o d u c e t h e d e n s i t y m a t r i x  we  formal-  ism u s e d i n quantum s t a t i s t i c a l mechanics. Let  t h e n o r m a l i z e d quantum m e c h a n i c a l s t a t e f u n c t i o n  t h e p h y s i c a l system be expanded i n t o a l i n e a r combination o r t h o n o r m a l f u n c t i o n s denoted by  f  (i.i)  We  u,t)  «  ^  |TJ.(^-)J  o f a complete s e t o f  * that i s ,  have u s e d a c o o r d i n a t e r e p r e s e n t a t i o n w i t h ft r e p r e s e n t i n g t h e t  d e n o t i n g t h e time dependence. F o r  c i t y , we have assumed t h a t t h e f u n c t i o n s n  of  Cj(t)Y.(/t X  c o o r d i n a t e degrees o f freedom, and  index " h «  (rv,t)  4 n ( A* )  Otherwise, we w o u l d have i n p l a c e o f  simpli-  a r e s p e c i f i e d by a d i s c r e t e  (1.1),  t h e f o l l o w i n g expan-  sion  (1.2)  ¥(yi,t) - J c j W t ( * > ) d * We  now  i n t r o d u c e an ensemble o f systems, each system i n t h e  ensemble  (9) being denoted by the set of functions ^ C ( t )} • The ensemble average of n  C ( t ) C * ( t ) v i l l be denoted by p m  n  U.3) p  m  n  p  m n  (t)  E<C (t) m  m n  ( t ) . That i s ,  C *(*)>,  ("t) being an element of the so called density matrix p(-t ) describing  the ensemble. The ensemble average, except when a l l the systems in the ensealble are identical ("pure case"), introduces an additional s t a t i s t i c a l averaging apart from any averaging inherent i n quantum mechanics. As i s well known, the expectation value at time "t of some dynamical observable A of the system i n the state ^ ( A j t ) i s given by  A (-fc) - f ¥ K t ) 4 ¥ u , t ) c U  (i.M  where the integration Is over the volume SL of the system. As a matter of notation, a cap A over a letter always denotes a quantum mechanical operator. The only exception to this convention w i l l be the density matrix p ( t ). Now, i t can be easily shown that the ensemble average of a dynamical operator at time "t i s  d.5) <AX=<A(t)> =ZA„ p (t), vith A =(y*U)AH^)c{/r m  mn  ,M  Mm  Since the choice of I^T^CA. )]  can be any complete orthonormal set  of wave functions describing the many-particle system, we can write (1.5) as (1.6)  <A> = t  Tji(Ap{t)) ;  where T / t stands for the trace. I t should be noted that  |  ( /i) ] must  satisfy any periodic boundary conditions imposed j. on the isolated system. The evolution i n time of the system with a time-independent HamilA  tonian f< i s described by the solution of Schrodinger s equation, 8  d.7) i u , t ) = e ~ * - i W o ) . L  (t  te)  Using the density matrix formalism, Schro dinger's equation i s transformed into Von Neumann's equation,  (10)  -r i [  (1/8)  A  , p (  where [ A , B] E A B - B A  i )] «  O  ,  and- u n i t s are chosen so  The c l a s s i c a l analogue of (1.8)  « j.  i s L i o u v i l l e ' s equation.  One can e a s i l y check by d i f f e r e n t i a t i o n that the solution o f  (1.8)  Is given formally by  W9) ( t ) - e-^-VVctoe* **-*-'. 1  P  Both (1.7)  and. (1.9)  express the determinate q u a l i t y of the dynamical evolution  of the i s o l a t e d p h y s i c a l system. I f a t time " t , the properties of the system 0  are known, then f o r a l l values o f "t > "t o , the state Is completely determined. We assume that the ensemble average given i n (1.6) observed value o f the dynamical observable A  0  i s t o evaluate the r i g h t hand side of p ("t) into  (1.6)  (1.9),  i s the average  In p r i n c i p l e , a l l that remains substitute t h i s expression f o r  and c a l c u l a t e the trace i n order t o f i n d the average value  of A a t time "t . The d i f f i c u l t problem l i e s I n evaluating (1.9)  since t h i s  i n d i r e c t l y involves the solution of a N-body problem. I t may be appropriate t o make a few remarks on the Heisenberg representation which w i l l be used i n l a t e r chapters. In .the discussion so .far, the Schrodinger "picture" o r representation has been used. The complete •dynamical properties of the system have been incorporated i n p ( " t ) , with the Hermitian operator A  constant i n time. On the other hand, i n Heisenberg representation,  the state functions (and^ consequently the density matrix) are constant i n time, }  while the dynamical observables are functions of time. The new operator A("fc ), associated with the o l d operator A ,  dAl*L  (l.io) where X  Ait)]  i s assumed t o have no e x p l i c i t , time-dependence. As before, the formal  solution of (i.oi)  -iftft.  s a t i s f i e s the following equation of motion  (1.10) A (t)  i s given by  =  e  l  A t . )  e^*""** 1  (11)  The obvious physical interpretation of ( l . l l ) is that i t describes the "natural" behavior of the operator A ( t ) .  If we define  then the relation between  A  (1.12)  Act)  =  and  A  e  = A  A(O)  7  (-fc ) is  lk±  Ae'  iHt  T/i(AB)  Using the formal trace identity  = 7/L (BA)  , where A  and B  are any two operators, we can rewrite (1.6) in the form  (ia ) 5  <A>t = T u ( / \ w p c o ) ) .  ..  The advantage of the Schrodinger "picture" is that i t puts the emphasis on the dynamical properties of the system, rather, than on the dynamical observable A . It separates the problem of the time evolution of the system and the problem of what the dynamical observables A  p (t )  are in any particular  case. For isolated systems, the main problem is the former one. For open systems resulting in transport processes, the latter problem is not so easy, as we shall see in later chapters. Let us return to the question of equilibrium and non equilibrium states of the system. If of  p i t ) 1 independent of 8  t , then &7 . is independent <  i  "t also and the system is said to be in equilibrium. The equilibrium density  matrix CJ  U.110  BQ  describing the ensemble of systems is a solution of the equation  C « , p  E  0  > o  We state, without any further discussion, that we can take for the equilibrium state a microcanonical ensemble i f the energy is fixed, a canonical ensemble i f the temperature is fixed and a grand canonical ensemble i f the temperature is fixed but the number of particles is not constant. For future reference, the canonical ensemble, in particular, has the following density matrix (1-15)  where k is Boltzmann*s constant and T is the absolute temperature. As is well  (12) known, the canonical ensemble describes a small system i n contact with a l a r g e r system which acts as a temperature bath. That i s , we s p l i t a very large system i n equilibrium into two unequal parts and concentrate our attenA  t i o n on the smaller one, whose Hamiltonian i s denoted by H . For the r e s t of t h i s t h e s i s , the equilibrium density matrix p w i l l be that given by ( 1 . 1 5 ) . M  The above paragraph deals with the quantum analogue of the e q u i l i brium s t a t i s t i c a l mechanics as developed by Gibbs and others a t the turn of the century. The p r i n c i p l e s are c l e a r l y formulated i n a manner which Is independent of any model or s p e c i a l postulates about the p a r t i c u l a r system being considered. A l l macroscopic "thermodynamic*  1  properties are d i r e c t l y r e l a t e d to  the i n t e r a c t i o n forces and i n t r i n s i c c h a r a c t e r i s t i c s of the p a r t i c l e s of system by the r e l a t i o n  (1.16)  <A> = RA  T/t (APBO)  = TA  (A  e-**)  = £  TA(e-0«) As mentioned before, In c e r t a i n cases the mlcrocanonical or grand canonical ensemble may be more appropriate. Once A and H are known f o r the p a r t i c u l a r p h y s i c a l problem a t hand, the evaluation of (1.16) i s e s s e n t i a l l y a mathematica l problem. Any e x p l i c i t c a l c u l a t i o n of A may become quite a complicated blend of approximations made on p h y s i c a l grounds, but t h i s does not a l t e r the u s e f u l ness of ( 1 . 1 6 ) , which i s a sort of half-way house between the general p r i n c i ples of quantum s t a t i s t i c a l mechanics and the e x p l i c i t formulas an experiment a l i s t uses. As w i l l be seen l a t e r , some transport c o e f f i c i e n t s can be expressed by formally exact expressions d i f f e r e n t from (1.16) but having the same usefulness and generality. In case p(-fc) i s not independent of the t i m e t , then obviously <A)  t  i s not, and the system i s s a i d t o be i n a non equilibrium state. The task of f i n d i n g the evolution of the system i n time involves evaluating,. (0(t)  7  -  (13)  p(t) =  (1.17;  e  ( pio) i s assumed to be known)  p(o)e  Now from physical experience, most isolated systems i n a non equilibrium state approach an equilibrium state after a "long enough" time. This time interval, of course, depends on the properties of the system, especially on the strength of the coupling or interaction forces. In terms of our formalism, this irreversible approach towards equilibrium can be expressed by the following relation,  t  ~»  OO  where for any particular system, the limiting process really means,  p (t  (1.19)  and T  R  ) *  p  e f t  , with  t  »  T  R  being same appropriate relaxation time. The fundamental conceptual problem i s to somehow arrive at irreversi-  b i l i t y , starting from a framework of reversible equations. Over the years, many discussions have dealt with this d i f f i c u l t problem without trying to describe in any detail the time-development of the system. We may briefly mention ergo die theory. Quoting from a recent a r t i c l e by Farquhar ( 1 9 6 1 ) , the purpose of this theory " i s to establish the necessary and sufficient conditions for a system to approach equilibrium from an arbitrary non equilibrium situation", that Is, "to derive conditions for the equality of time and ensemble averages...of expectation values of dynamical observables". A much more specialized, as well as more ambitious, line of attack l i e s i n trying to evaluate ( 1 . 1 7 ) for particular systems and thus follow their total time-development. Equation ( 1 . 1 8 ) would follow as an asymptotic result for long times. It may be noted that ergodie theory gives no estimate of Relaxation times, while this method definitely does. In any discussion along these  (lk)  l i l i e s , we must introduce i r r e v e r s i b i l i t y i n some natural manner as w e l l as successfully approximate (asymptotically) the equations of motion of a system with an enormous number of degrees of freedom (of the order 1 0 ^ ) . I t turns out, as we might expect, that the two problems are intimately related, i r r e v e r s i b i l i t y appearing as a c h a r a c t e r i s t i c property of macroscopic systems, . . . . . . Van Hove's f i r s t paper ( 1 9 5 5 ) on the lowest order Markovian master equation must be considered t o be the f i r s t rigourous r e s o l u t i o n of these d i f f i c u l t i e s , although Kirkwpod and Bogoliubov both made s i g n i f i c a n t contributions i n the case of c l a s s i c a l systems. To do j u s t i c e t o the considerable progress that has been made i n t h i s f i e l d i n the l a s t few years would require a large monograph. Indeed, i f we consider the important gains i n understanding, the time i s r i p e f o r a modern version of the Ehrenfests* femous Encyklopadie a r t i c l e . We s h a l l give a b r i e f account of Van Hove's work not only as an excellent example of recent work on non equilibrium processes but a l s o since i t w i l l be used l a t e r i n chapter 5 . The mathematical  techniques used by Van Hove enable us t o evaluate the time  dependent c o r r e l a t i o n functions occurring i n c e r t a i n formal, expressions f o r transport c o e f f i c i e n t s . Van Hove assumes that the system of i n t e r e s t i s such that I t s t o t a l Hamiltonian (1.20)  ii  A  =  can be s p l i t into two parts. More p r e c i s e l y , we have -Ho  +  XV  where the "free Hamiltonian"  ^  0  describes N/ f r e e excitations and the  A  "perturbation"  XV  represents the interactions among the e x c i t a t i o n s .  The term " e x c i t a t i o n " i s used by Van Hove to denote both p a r t i c l e s and quasip a r t i c l e s , while  A  i s a dimen3ionless parameter measuring the strength of  the interactions among the e x c i t a t i o n s . The eigenstates and eigenvalues of A  A  ~K  a  are assumed to be known and  AV  i s treated formally by perturbation  (15) theory even though i t i s not necessarily small.  The basic representation  used throughout Van Hove's work i s that i n which f<o i s diagonal, that i s  (1.21)  ft  0  /*> = €« lot)  where both the eigenstates and eigenvalues quantum numbers ot •  Some of these <X  are l a b e l l e d by a complete set of  quantum numbers are always d i s c r e t e ,  such as the ones denoting the spin and p o l a r i z a t i o n indices of the e x c i t a tions. vector.  Each elementary plane wave e x c i t a t i o n i s a l s o l a b e l l e d by a wave The components of t h i s wave vector* i n the l i m i t of a large system,  are regarded as continuously varying quantum numbers.  In general, we  may  adopt the normalization  (1.22)  <<*U-> - cf  where £(ot -<*')  i s . t h e product of Dirac delta functions f o r the continuous  quantum numbers and Kronecker symbols for the discrete ones. a n a l y s i s , Van Hove (1955, 1957, numbers are continuous.  1958)  To s i m p l i f y the  often assumes that a l l t h e c < quantum  In a recent a r t i c l e (I960), he always keeps i n mind  the f a c t that r e a l p h y s i c a l systems are f i n i t e i n s i z e and consequently a l l the  c< quantum numbers are d i s c r e t e .  I n our o u t l i n e , we s h a l l follow the  presentation o f t h i s a r t i c l e , and therefore we adopt the normalization  (1.23)  SKI**'  <<*lcx'> = -CO 1  oi 4 aL* I  oL^oH  and use summation notation throughout.  I t i s e s s e n t i a l to Van Hove's analysis  that the unperturbed energy spectrum formed by the at l e a s t , quasi-continuous).  £ « ^ i s continuous (or,  I f the system of i n t e r e s t i s large, the unper-  turbed eigenvalues have t h i s property (see, f o r example, Landau and  Lifshitz  (1958), Chapter 1, f o r an i n t e r e s t i n g discussion of t h i s p o i n t ) . The entire discussion i n t h i s chapter deals with the whole system and therefore i s i n T-space.''  Much of Van Hove's work i s v a l i d f o r a  (16)  s i n g l e system, but i n many other portions of t h i s and other chapters, we often speak of an i s o l a t e d system when we are r e a l l y dealing with an ensemb l e of such systems. The problem of making a t r a n s i t i o n t o a description involving a s i n g l e e x c i t a t i o n by integration (or summation) over the degrees of freedom o f the other excitations w i l l be commented on In chapter 2 . The l a t t e r description i s s a i d t o be i n ytx-space'* and w i l l be used i n chapter 5 . The aim of Van Hove's investigation i s t o f i n d the time-dependent A  expectation value o r ensemble average of an operator A diagonal i n the l<*>representation. The eigenvalues AW are given by (1.24)  AI ay = A(ot)l*> .  and, as we s h a l l see, Van Hove's a n a l y s i s requires that they be smooth functions of the a quantum numbers characterizing the state. I f the system Is In the quantum state |  d.25)  a t t - o , then a t time t , i t i s In the state  given by  »<K> = e - " ? - ^ * / * ^  = 0(t)i<P . The expectation value of A a t time t i s consequently o/  (1.26)  ^(t)=  <<p \A\<P+> t  Expanding the i n i t i a l state J&>in the lot)-representation, *  (1.27)  |4>.> = TLct4)\ot>  and using (1.24), we may rewrite Atfiin (1.28)  A(+) = £ H  cU)CU)  =2Z \ccoi )\ Zl i  t  (1.26)  A(oi ) }  as  <oMU(-tM  UL-t)lot > t  ko^/ucoio^r  V"r^cc^o^/4c^)<^ju(-t)u ><^/L)(t)Wz> J  (17) The prime' i n the double summation means that <*,#y  r  .  We introduce the  following definitions: (1.29)  Pt (<*>i  (1.30)  ItC^,,cf u )=,  (1.31)  P(^/t)=  (1.32)  I (^  (1.33)  PtM=  i;  3  =  \Q(-t)\*z><'*3lUHr)\ol,> . \\J(-t)\d ><^  3  ,t) =  3  Z  Pt (•«*.;a/*) Ic6x.)|*  jy I  t  P(* rt) 3  i  (Ute,)Mi7 • ( P ter-m)  (*.,^;«? ) C * G O c i < * J 3  +  (I  . term).  I(ot ,t) ?  Using these new quantities and (1.28), we have (1.34)  A(t)  = Z  AW,)Pf^i)  So far, everything we have done i s completely formal since no important assumptions have been made. ' Van Hove assumes that the perturbation operator V s a t i s f i e s the so called diagonal singularity condition. Physically, this analytic property of V i s connected with the fact that the perturbation extends over the whole system.  I t may be mentioned that another consequenoe of this character  of the perturbation i s that even though the interactions among the excitations may be quite small, the perturbation energy of the whole system i s quite large.  Therefore f i r s t order time-dependent perturbation theory may not be  valid.  The deep relationship between the diagonal singularity condition  and the dissipative properties of the system i s discussed i n the series of papers by Van Hove (1955, 1957, 1959).  I n the l i m i t of a large system,  the diagonal singularity condition can be expressed by the equation (1.35)  <C<X I v Vl<*'> - c f O -*') W(o() + <oilYloC>  where the second term i n the right hand side of (1.35) contains a weaker singularity than the one contained i n the f i r s t term.  (18)  Van Hove (1955, 1958) was able to evaluate  2Z A (ot ) p 3  t  by a.straightforward but lengthy method i n the weak coupling, long time He expanded the evolution operator , U u ;  approximation.  s  c  ,  by the use of the well known Schwinger formula for an ordered exponential, into a power series i n A • Substituting this series into the right hand side of (1.29), the formal expression for  At(<*-;°<j) was systematically  simplified through the use of the diagonal singularity condition (1.35) and asymptotic time integrations* Van Hove ended up with an i n f i n i t e series with terms contributing as  yy\ ~ o, \ 2 ... ) • In the long time, weak coupling t  approximation, i n which only terms with  h=0  }  A—)0 "t-*<x>  such that  ;  need be kept.  (A 1) 2  Van Hove found that  remains f i n i t e ,  P± (<*  1,0(3)  was given by iterated solution of the equation  (1.36)  ± d P,  (+  q.;-,) £  r  =  fort>0  p t(  J - |or  with the i n i t i a l conditions  _ ,  j  t<o)  Po (rf,;<rfj) =  and where  W/Grf,^)^ <2 7T A J(f«rf-f^)fC°<.lvU,>|. This latter quantity i s the same as 2  1  1  the transition probability per unit time as calculated by f i r s t order timedependent perturbation theory.  We should add that the solution of (1.36)  i s a valid approximation for  P-t (<*,', 1*3)  YL  A tol ) P+ 3  ; o^) 3  only when i t i s used i n  and /U^) i s a smooth function of o( . 3  3  In addition, Van Hove's analysis only gives us the various probabilities for times "t which are sufficiently large.  (19)  A very similar analysis concentrates on the so called P term P(c< ,t) =  (1.31)  ICCoOl*  3  P (o(,;oC ) t  3  where the C(oO specify the initial quantum state of the system. In using the previous method of finding  P(<*M°<3), t  Van Hove must assume that the initial  probability amplitudes C ( ° 0 are appropriately slowly varying. We refer to Van Hove's papers saying the  (1955,  1958)  for the precise meaning of what is meant by  C(d,) (or, the eigenvalues A(«(,) ) are slowly varying when consi-  dered as functions of the unperturbed energy £  0(i  . Van Hove has shown  (1955)  by direct calculation that for slowly varying C(o<.)ii,the I term d-32)  I(c(3,-t)  = Z'  I U,^« ) t  cU)C  3  M  is negligible when compared to the P term. This is, as in the whole discussion so far, for the long time, weak coupling approximation. As a result, the occupation probability ^(o^)of the system being in the state l ^ a t time t is Pt(* )  (1.37)  3  =  P(«*3,-0  and satisfies the equation (1.38)  ±  <LEg&=  Z  with the initial condition  WCo^foCoO-P-tCoO} p (o< ) 0  3  =  I  C(o( )| 3  2  . The mathematical  properties of the linear first order time differential equations given in (1.36)  and  (1.38)  are well known since a similar equation describes the  evolution of Markovian processes. It may be mentioned that the so called lowest order master equation in  (1.38)  for a "fine-grained" occupation  probability was first derived by W. Pauli in 1 9 2 8 . He was forced to make the so called "random phase assumption" repeatedly in his derivation.  (20)  To a v o i d t h i s u n s a t i s f a c t o r y assumption, Van Hove i r i t r b d u c e d a "coarse-grained" t r a n s i t i o n p r o b a b i l i t y  22 i  £  i  where t h e denominator i s t h e number o f <x - v a l u e s i n a s m a l l neighbourhood Ao(  around  o<  .  3  I f we use t h i s c o a r s e - g r a i n e d t r a n s i t i o n p r o b a b i l i t y ,  we c a n a p p l y Van Hove's m a t h e m a t i c a l method,based upon t h e use o f t h e diagonal s i n g u l a r i t y c o n d i t i o n t o the evaluation o f i t . This  "coarse-grained"  ?  t r a n s i t i o n p r o b a b i l i t y s a t i s f i e s f o r m a l l y t h e same e q u a t i o n a s t h e " f i n e grained" t r a n s i t i o n p r o b a b i l i t y Pt (<* i'>°* 3)  0  .  I n contrast to  Ptfai,'*^),  » a s c a l c u l a t e d by Van Hove,, has a meaning by i t s e l f s i n c e a  summation o v e r "coarse-grained"  (1.1*0)  P+Wi,' ^)  has a l r e a d y b e e n made. P  We may s i m i l a r l y i n t r o d u c e a  term  P(o<3,t) H  H  PU yt) 3  <<* '< * +4* 3  2  12 2  3  T  1  a  U s i n g t h e s e " c o a r s e - g r a i n e d " q u a n t i t i e s , we may d i s c u s s t h e d i s s i p a t i v e p r o p e r t i e s o f t h e system i n d e p e n d e n t l y eigenvalues  o f the diagonal operator  o f the i n i t i a l state  | <J>„> o r t h e  A*  As t o t h e q u e s t i o n o f when I t i s p o s s i b l e t o n e g l e c t t h e I term, i t i s q u i t e obvious  that t h i s i s not p o s s i b l e f o r a r b i t r a r y i n i t i a l s t a t e s .  If  s o , t h e n a l l systems ( f o r which t h e weak c o u p l i n g , l o n g time  approximation  is  v a l i d ) would e v o l v e a c c o r d i n g t o t h e i r r e v e r s i b l e master e q u a t i o n  given  (21) above and I t i s easy t o f i n d e x c e p t i o n s t o t h i s .  F o r example, l e t t h e  i n i t i a l s t a t e be a n e x a c t e i g e n s t a t e o f t h e t o t a l H a m i l t o n i a n o f the system. As w i t h some o t h e r t o p i c s d i s c u s s e d above, o n l y a b r i e f o u t l i n e o f Van  (1957) p e n e t r a t i n g remarks on t h i s s u b j e c t w i l l be g i v e n . ensemble and make t h e s o c a l l e d random phase assumption then t h e  I  term w i l l v a n i s h .  Hove's  I f we c o n s i d e r a n  (R.P.A.)  initially,  The r e p e a t e d use o f R.P.A., a s i n P a u l i ' s  d e r i v a t i o n , i m p l i e s t h a t t h e system i s i n e q u i l i b r i u m a t a l l times a n d consequently i s untenable.  T h i s r e s u l t f o l l o w s from the symmetry o f the  fundamental e q u a t i o n s o f m o t i o n under time In  reversal.  a l a t e r p a p e r (1957)» Van Hove succeeded  i n deriving a general-  i z e d non-Markovian m a s t e r e q u a t i o n d e s c r i b i n g t h e e v o l u t i o n o f c e r t a i n quantum m e c h a n i c a l m a n y - p a r t i c l e systems t o any o r d e r i n A • d e s c r i b e the h i g h l y s o p h i s t i c a t e d mathematical  No b r i e f note  technique  can  (resolvent operator  f o r m a l i s m ) used i n a r r i v i n g a t t h i s g e n e r a l i z e d e q u a t i o n o f d i s s i p a t i v e motion. of  The m a t h e m a t i c a l  i n v e s t i g a t i o n o f t h e p r o p e r t i e s and consequences  t h i s c o m p l i c a t e d e q u a t i o n i s no easy t a s k e i t h e r .  O t h e r than showing  t h a t i t d e s c r i b e s the approach t o e q u i l i b r i u m , few e x p l i c i t have been made u s i n g i t .  applications  For t h e weak c o u p l i n g c a s e , i t reduces  p r e v i o u s l o w e s t o r d e r m a s t e r e q u a t i o n and i n a r e c e n t p a p e r  t o the  (1961) w i t h  Verboven, Van Hove s o l v e d t h i s g e n e r a l i z e d m a s t e r e q u a t i o n f o r extremely crude models o f e l e c t r o n - i m p u r i t y and e l e c t r o n - p h o n o n  scattering i n a  metal. A l t h o u g h much more work o n s p e c i f i c models i s needed, t h e r e i s no doubt t h a t Van Hove's work i s the f i r s t  really successful investigation that  d e a l s w i t h t h e d e t a i l e d dynamics o f systems w i t h an enormous number o f degrees of  freedom.  I t shows how t h e l a t t e r p l a y a c e n t r a l r o l e i n the r e l a x a t i o n t o  e q u i l i b r i u m and e n a b l e s us t o d e s c r i b e t h i s i r r e v e r s i b l e b e h a v i o u r i n some detail.  (22) Prigogine and h i s co-workers have also developed t h e i r own approach to the rigourous study of i r r e v e r s i b l e processes  ( i n c l a s s i c a l and  quantal  systems) i n a s e r i e s of papers i n "Physica" and other journals i n the l a s t few years.  Although d i f f e r e n t i n method and d e t a i l , the main aim i s the same  as Van Hove's, that i s , to obtain the equations which describe the i r r e v e r s i ble behaviour of macroscopic systems from f i r s t p r i n c i p l e s .  In Prigogine's  case, t h i s again involves the s o l u t i o n of a N-body problem, not an exact one, but an asymptotic  s o l u t i o n f o r a large number of degrees of freedom and  long times compared to c h a r a c t e r i s t i c microscopic times.  He uses a complex,  diagram technique i n h i s a n a l y s i s . We may  note that Prigogine's group has mainly worked i n the  "Schrodlnger p i c t u r e " and, consequently,  attempts to describe the d i s s i p a t i v e  behavior of a system by means of the density matrix.  In p r i n c i p l e , i f we  could f i n d the solutions of t h e i r complicated equations, we could use r e s u l t s to f i n d the expectation or ensemble average values of any  these  operator.  In contrast, Van Hove's work uses the "Heisenberg p i c t u r e " and never loses sight of the p a r t i c u l a r dynamical observable he has i n mind.  Of course,  Van Hove's analysis only works f o r diagonal operators i n the  \ <*•> -representation  and therefore i s not as general as the work of Prigogine.  In a recent paper  (19$9) Van Hove has generalized h i s analysis to a c l a s s of operators which are "almost" diagonal, and states that t h i s class includes a l l the p h y s i c a l l y important  operators* In a recent paper (I960), Prigogine and h i s group have stated that  "non equilibrium s t a t i s t i c a l mechanics has been now brought to a stage s i m i l a r to that of equilibrium s t a t i s t i c a l mechanics."  This i s c e r t a i n l y over confident  and many a u t h o r i t i e s might disagree, but no one can doubt that considerable progress has been made i n t h i s f i e l d by Prigogine's group, Van Hove, and others•  Much remains on the computational  side, o f course*  CHAPTER 2 : THE PROBLEM OF TRANSPORT PROCESSES IN OPEN SYSTEMS AND THE STANDARD "BOLTZMANN EQUATION" SOLUTION.' SECTION 2.1:  TRANSPORT PHENOMENA IN OPEN SYSTEMS.  Throughout chapter 1, the many-particle system was assumed to be a p e r f e c t l y i s o l a t e d system, and described by some unique Hamiltonian vH  .As  a matter of fundamental physical p r i n c i p l e , no system can be rigourously i s o l a t e d from i t s surroundings. Consequently no system has a rigourously defined Hamiltonian. This f a i r l y subtle point w i l l be l a r g e l y disregarded i n the r e s t of t h i s t h e s i s , although t h i s weak i n t e r a c t i o n with the r e s t of the u n i verse must enter i n any complete t h e o r e t i c a l analysis of i r r e v e r s i b i l i t y . From p h y s i c a l experience, as we have mentioned, most systems i n a non equilibrium state approaches i r r e v e r s i b l y towards thermodynamic equilibrium a f t e r some time i n t e r v a l . In the l a s t decade, a d e t a i l e d description of t h i s process, s t a r t i n g from fundamental p r i n c i p l e s , has been attempted with considerable success. Of course, once the system i s i n equilibrium f o r some time, there i s always the p o s s i b i l i t y of spontaneous fluctuations r e s u l t i n g i n the system returning to same new non equilibrium state. The.equations derived by Van Hove and Prigogine t e l l us nothing about the time or frequency of these equilibrium f l u c t u a t i o n s . Now transport phenomena proper a r i s e when.an i s o l a t e d system i s placed under the influence.of external "forces" or "constraints". This prevents the system from reaching equilibrium and various transport or "flow" processes take place i n the system. Any dynamical observable JB( the tiJ,de denotes a vector) which i s defined i n terms of the microscopic variables of an i s o l a t e d system corresponds to a "current" i f the following c r i t e r i a . hold,  (24)  ( 2 . 1 ) <|> Q*T/t(p | ) = 0 EQ  r  ( 2 . 2 ) <|>  = T/t,(p<tV§ ) * O  {  where B - ( B , B, , B , ) and pit)  i f p(-t) *  pes  i s the density matrix of the i s o l a t e d  system., as t h i s system approaches equilibrium. Of course i f the system s t a r t < B >  ed from a non equilibrium state quite f a r from equilibrium,  t  would  i n i t i a l l y be quite large, but i t would decay to zero according to the equations of motion discussed i n chapter 1 . In the presence of external "constraints", the density matrix of the system must incorporate the e f f e c t s of these d i s t u r bances and consequently, so must < B>..  The t h e o r e t i c a l problem before us i s  to give a rigourous quantum s t a t i s t i c a l formulation of the e f f e c t s of these external "disturbances". We might attempt to include the " d r i v i n g systems" producing the external "disturbances" and the many-particle system of i n t e r e s t i n one  com-  A  posite, closed system. We assume that the t o t a l Hamiltonian K o f the compoT  s i t e "supersystem"  can be s p l i t as follows  (2.3 £  + £ . .+ V  T  -  D  S  A  A  where "ft i s the t o t a l Hamiltonian of the i s o l a t e d system of i n t e r e s t , "K o.s. A  i s the t o t a l Hamiltonian describing the " d r i v i n g systems" and  V  represents  the i n t e r a c t i o n between the system of i n t e r e s t and the " d r i v i n g systems". We s h a l l now reformulate the problem. We consider an open system c o n s i s t i n g of the system of i n t e r e s t acted upon by the external disturbances A A through the perturbation V . A c t u a l l y the Hamiltonian W  D  s  used i n describ-  ing the " d r i v i n g systems" i s a l i t t l e ambiguous. In transport theory, t h e "sources" are usually assumed t o be " i n f i n i t e l y large" (batteries produce a constant voltage,  heat reservoirs stay a t a constant temperature,  etc.).  Thus the "sources" have an open character and cannot be expressed i n the Hamiltonian formalism rigourously. I f the "sources" are large systems  (25) compared to the system of i n t e r e s t , the difference between a closed comp o s i t e system and a quasi-closed composite system i s n e g l i g i b l e and we neglect I t . As emphasised,  shall  what i s important i s t o be able to f i n d an appro-  > A  p r i a t e quantum mechanical operator expressing V .  For some " d r i v i n g systems",  t h i s Is a w e l l defined operator, while i n other cases, an unambiguous expression f o r V i s not so obvious. This d i s t i n c t i o n w i l l become basic i n the following chapters of t h i s t h e s i s . From an experimental or phenomenological  point of view, i n many cases  the behavior of Isolated systems under the influence of external "disturbances" can be simply described by means of a few parameters known as transport c o e f f i c i e n t s . Up to now everything has been f a i r l y general. For concreteness, we s h a l l consider a metal specimen under an a p p l i e d e l e c t r i c f i e l d and a "temperature gradient". This thesis w i l l concentrate on these two external "disturbances" and w i l l review recent attempts to f i n d t h e o r e t i c a l expressions f o r the transport c o e f f i c i e n t s experimentally connected with them ( i n p a r t i c u l a r , e l e c t r i c a l and thermal conductivity). In the remaining paragraphs of t h i s section, we s h a l l o u t l i n e the experimental or phenomenological  description of these trans-  port processes. Assume that we have a homogeneous metal specimen i n the shape of a c y l i n d e r of length L.  f  completely insulated from i t s surroundings except f o r  the ends. Apply a s u f f i c i e n t l y small e l e c t r i c f i e l d  E  x  along the a x i s  (X-direction) of the cylinder and assume that the temperature throughout. An e l e c t r i c "current" J  x  w i l l a r i s e along the length of the c y l i n d -  er, and i s found to be proportional to  (»3k) where  J  <T x * X  =  x  s  cr  x x  E  i s constant  £ * .  x  the e l e c t r i c a l conductivity, the transport c o e f f i c i e n t describ-  ing t h i s e l e c t r i c current flow. The equation (2.4) holds once the system has  (26)  reached a steady state with a constant current flow. When the external f i e l d i s f i r s t switched on, the r e s u l t i n g transport s i t u a t i o n i s more complex due to "transients" but these are soon damped out. We can immediately generalize (2.4)  (2.5) with  to a more general s i t u a t i o n i n three dimensions,  J » CT E CT  or J = E ( T . £ u v  K / i * 1* 2 , 3 or x, y, z)  the e l e c t r i c a l conductivity tensor (nine components). Macroscopically, the case of a "thermal gradient" i s s i m i l a r .  Assume that the two ends of the c y l i n d e r are i n contact with heat reservoirs (temperature baths) a t d i f f e r e n t constant temperatures, the difference AT being s u f f i c i e n t l y small. An energy "current" Q  w i l l flow along the c y l i n d e r .  We further make the r e s t r i c t i o n that the e l e c t r i c current  J  K  i s zero. Since  the energy current w i l l i n d i r e c t l y produce a charge flow, we must have e f f e c t i v e l y , i n addition to the temperature difference, an e l e c t r i c f i e l d  E whieh x  cancels out any e l e c t r i c current associated with the energy current. Mathematica l l y the experimental s i t u a t i o n involved i n thermoelectric processes can be expressed as follows  (2.6)  j  = cr  x  E  xx  •  x  B  *2  (1  L  (2.7)  Q  B  »  x  2  1  E  X  B  +  I  .  2  AT  L where B ,  2 ;  B  1  Now  and B  (  are experimental parameters.  1  i f we make the assumption that J  in (2.6)  In terms of  (2:8)  Q  =  2  AT  {B (-Bn)  - -  u  kl  M  sO  in (2.6),  express  , and i n s e r t the r e s u l t into ( 2 „ 7 ) , we have + B  I  2  | 4 I  AT  where k" x i s known as the thermal conductivity. Generalizing ( 2 . 8 ) X  analogous fashion to ( 2 . 4 ) , (2.9)  Q  =-K^T(A)or  a more general expression i s Q  y  =  -£  In an  Ex  C27)  where kl i s t h e t h e r m a l c o n d u c t i v i t y t e n s o r w i t h "current" Q coordinates. (2.10)  with  T  i s constant T(A)  TlA)  c  -  J= O  •  I f t h e energy  i n a steady, s t a t e and K. i s independent o f t h e s p a t i a l  must be o f t h e form T  some c o n s t a n t  +  c  -A.  a  temperature and  & =  ( .> *., 3) some c o n s t a n t Qi  a  vector.  a  S i n c e t h e temperature g r a d i e n t must be s m a l l , we must have i n a d d i t i o n (2.11)  |fl£Z£|<3U Tc  1  For the macroscopic f u n c t i o n  TCA)  t o have any p h y s i c a l meaning a s a tempera-  t u r e v a r i a t i o n i n t h e specimen, we must have some s o r t o f " l o c a l e q u i l i b r i u m " i n r e g i o n s o f t h e system m a c r o s c o p i c a l l y  s m a l l but m i c r o s c o p i c a l l y l a r g e .  c a s e , t h e c o n c e p t o f a " l o c a l temperature" may b e i n t r o d u c e d . t h i s type o f m a c r o s c o p i c a l l y  constant i s l e f t  with  fairly  We s h a l l r e t u r n t o t h i s " l o c a l temperature" concept l a t e r i n c h a p t e r U. We emphasize t h a t t h e above a n a l y s i s i s c o m p l e t e l y  The  As i s u s u a l  varying function of the coordinates, the size o f  the r e g i o n i n which t h e f u n c t i o n i s a p p r o x i m a t e l y vague.  I n this  phenomenological*  term " c u r r e n t " i s so f a r o n l y a name o f a' q u a n t i t y a p p e a r i n g  t i o n s which e x p e r i m e n t a l l y  i n t h e equa-  describe the various i r r e v e r s i b l e processes  As a l a s t remark, even though t h e p r e s e n t d i s c u s s i o n has no p r e t e n s e  discussed.  o f any  t h e o r e t i c a l background, t h e fundamental d i f f e r e n c e between t h e two e x t e r n a l "forces" i s noticeable. electromagnetic  The e l e c t r i c f i e l d i s p r e c i s e l y d e f i n e d i n terms o f  t h e o r y and has a d e f i n i t e c a l c u l a b l e e f f e c t o n a charge c a r r i e r  such a s an e l e c t r o n .  The s o c a l l e d temperature g r a d i e n t i s e a s i l y d e f i n e d macro-  s c o p i c a l l y b u t on a n a t o n i c s c a l e , i t i s q u i t e d i f f i c u l t meaning, l e t a l o n e Before equation  t o g i v e i t any p r e c i s e  c a l c u l a t e i t s e f f e c t s o n t h e atomic c o n s t i t u e n t s o f a m e t a l *  d i s c u s s i n g t h e method o f u s i n g a t r a n s p o r t o r Boltzmann  i n t h e t r a n s p o r t processes  mentioned above, a b r i e f mention may be  made o f t h e semi-macroscopic t h e o r y o f i r r e v e r s i b l e p r o c e s s e s  developed by  (28)  Onsager (1931). A very c l e a r discussion of t h i s type of t h e o r e t i c a l analysis i s given i n several monographs (for example, see S.-R.  de Groot (1952) ). In  t h i s theory, other than the standard procedures of s t a t i s t i c a l mechanics,  two  assumptions are used. The f i r s t i s the fundamental microscopic r e v e r s i b i l i t y of the dynamical equations of motion ( the laws are invariant under a change of the time variable  "t-*-•£«). Because of t h i s use of some microscopic prop-  e r t i e s of the system, i r r e v e r s i b l e thermodynamics i s not e n t i r e l y phenomenolog i c a l i n content. The second important assumption i s r e a l l y a hypothesis, as emphasized by Casimir (19^5). On the average, the decay of a f l u c t u a t i o n (which has caused the deviation from an equilibrium state i n an "aged" system) i s assumed to follow the ordinary phenomenological l i n e a r laws governing i r r e v e r s i b l e processes. By introducing appropriately defined "forces" and "fluxes", we can express the "fluxes" as a l i n e a r combination of the "forces". The constant coe f f i c i e n t s play the r o l e of transport c o e f f i c i e n t s and are known as " k i n e t i c c o e f f i c i e n t s " . Onsager*s theorem i s b a s i c a l l y a c r i t e r i o n of when same of these k i n e t i c c o e f f i c i e n t s are equal t o one another. I t should be  emphasized  that no information i s given as t o the evaluation of these k i n e t i c c o e f f i cients i n terms of the microscopic properties of the system.  SECTION 2 . 2 : OUTLINE AND CRITICISM OF THE STANDARD METHOD OF FINDING ELECTRICAL AND THERMAL CONDUCTIVITY. ~ A w e l l known "recipe", using Boltzmann type equations for jU.-space d i s t r i b u t i o n functions, i s used almost exclusively i n any discussion of transport phenomena i n s o l i d s . D e t a i l e d accounts may be found i n Wilson. (1953) or, f o r a s l i g h t l y more modern version, Ziman (i960). Although the t h e o r e t i c a l structure of t h i s "recipe" was never too rigourous, i t d i d and does provide a reasonable f i r s t approximation t o the s o l u t i o n of some f a i r l y complex physical  (29)  problems i n the theory of metals. The basis of t h i s method l i e s i n the use of  -space d i s t r i b u t i o n  functions. In the theory of metals, i t i s almost always assumed that i t i s possible to define a electron d i s t r i b u t i o n ^(vk)  which gives the probable  number of electrons (conduction) i n a s i n g l e p a r t i c l e state with wave vector k,  and a phonon d i s t r i b u t i o n n ( ^ ) which gives the probable number of phonons  i n a s i n g l e p a r t i c l e state with wave vector (jj>. The interactions between the f r e e electrons and phonons, the interactions with impurities, and the presence of the external e l e c t r i c and thermal gradients are a l l considered as causing " t r a n s i t i o n s " between the unperturbed states l a b e l l e d by Jk and <j . That t h i s procedure enables us to e f f e c t i v e l y describe a whole system of i n t e r a c t i n g p a r t i c l e s i s not a t a l l obvious u n t i l a f u l l scale many-body treatment, taking i n t o account the c o l l e c t i v e behavior of the p a r t i c l e s , has been undertaken. We must show that the A and ^ s i n g l e p a r t i c l e states define the appropriate "good" quantum numbers. Recent work by Bohm and Pines on the plasma theory of metals i s an example of what we have i n mind. By guessing appropriate c o l l e c t i v e coordinates, the system's t o t a l Hamiltonian may be s p l i t i n t o a part depending on the c o l l e c t i v e coordinates only and a part depending on the p a r t i c l e or quasip a r t i c l e coordinates only, the i n t e r a c t i o n being n e g l i g i b l e . In the case of metals, the q u a s i - p a r t i c l e s are fermions that behave as nearly f r e e fermions. That i s , they axe electrons "surrounded by a charge cloud" which produces a screened Coulomb f i e l d . This r e s u l t explains the success of the f r e e electron model. A rigourous development as sketched above i s e s p e c i a l l y important in. that i t may  l e a d to the occurrence of new  c a r r i e r s of energy. In the  standard  theory, i t i s simply assumed that only the electrons are charge c a r r i e r s , while only the phonons and electrons are energy c a r r i e r s . In terms of the d i s t r i b u t i o n functions ^ ( k ) a n d n ( <fy), the e l e c t r i c current density ^' and the energy current density <j> are assumed to be given by the "natural" expressions.  (50)  j  (2.i3)  j - j£CA)v^(A)^.^J^(^)cn(<j-)^^  ;  where e  =-/  e  v  f(A)d Jk,  (2.12)  3  i s the electronic charge, V and E(  ) are the electron's v e l o c i t y  and energy i n state k, and C and U/ (<fy) are phonon's v e l o c i t y and energy i n state  Even i f we could f i n d the values of £ ( X ) and rj ( <^), i t must be  remembered that these are the analogues of the diagonal elements of the reduced density matrix, and that the information contained i n the off-diagonal matrix elements i s completely ignored. The next problem i s t o set up gain-loss Boltzmann-type equations f o r both electron and phonon d i s t r i b u t i o n functions, using the t r a n s i t i o n p r o b a b i l i t i e s f o r the various processes. The manner i n which t h i s i s usually done i s highly i n t u i t i v e and r e a l l y can only be c a l l e d a "recipe" which works, a t l e a s t to a f i r s t approximation. When there exists e l e c t r i c and temperature gradients, and the system i s assumed to be i n a steady state, the transport equation f o r each d i s t r i b u t i o n function may be formally given as  \ ax/ -fiei^ terms * o w scattering terms The f i e l d terms contain the external gradients while the scattering terms describe the complicated i n t e r a c t i o n mechanisms between the atomic constituents of the metal. Assuming, as i s usually done, that the r i g h t hand side of (2.14) i s given by a Boltzmann gain-loss term ( generalized t o take into account the s t a t i s t i c s obeyed by the p a r t i c l e s and q u a s i - p a r t i c l e s ) , we end up with a coupled set o f extremely complicated i n t e g r o - d i f f e r e n t i a l equations f o r £ ( J t ) and n ( c j ) . T h e o r e t i c a l l y , the only remaining problem i s the mathematical one of solving these equations, but before any usable s o l u t i o n may be obtained, many further physical approximations are necessary. Since our goal Is to evaluate jr and f u s i n g the previous d e f i n i t i o n s , it  some obvious s i m p l i f i c a t i o n s are p e r f e c t l y J u s t i f i e d . The l i n e a r i z a t i o n of the  c o l l i s i o n term i s acceptable since we are only interested i n non equilibrium states which are not too f a r from the equilibrium state. I f the solution i s expressed as a series i n the external gradients,  we need only keep the l i n e a r  term. Other assumptions are not e a s i l y j u s t i f i e d , except on the ground that otherwise the problem i s almost hopelessly d i f f i c u l t . The most important example of t h i s i s the almost u n i v e r s a l use of a r e l a x a t i o n time i n simplifying the scattering term. S t r i c t l y speaking, t h i s use of a universal relaxa- , t i o n time Is j u s t i f i e d only i n the e l a s t i c and i s o t r o p i c scattering case. A recent a r t i c l e by Dresden ( l 6 l ) gives a d e t a i l e d account of the Q  assumptions, methods, d i f f i c u l t i e s and shortcomings of the transport equat i o n approach very b r i e f l y o u t l i n e d above. We should l i k e to emphasize  two  points. . F i r s t , i n recent years, attempts to derive transport equations from the general p r i n c i p l e s of quantum s t a t i s t i c a l theory have met with considerab l e success. We mention the work of Kohn and Luttinger (1957) i n which the usual Boltzmann equation was f i r s t successfully derived f o r a simple model of a metal. The work of Van Hove, Prigogine and others mentioned i n chapter 1 «>  //  i s of i n t e r e s t . Although these discussions are often i n I -space, i n contrast *  //  to the^C-space transport equations mentioned above, by appropriate summations and integrations over the degrees of freedom connected with the r e s t of the system, a Boltzmann-type equation f o r a s i n g l e p a r t i c l e may be obtained. Van Hove, i n p a r t i c u l a r , has c a r r i e d t h i s out i n the case of an electron-phonon system (1958) and the case of an electron-ion system (I960). I t should be emphasized that when we speak of a d i s t r i b u t i o n function f o r the number of p a r t i c l e s i n a c e r t a i n single p a r t i c l e state, t h i s i s the same as a occupat i o n p r o b a b i l i t y f o r a s i n g l e p a r t i c l e , except f o r a normalization f a c t o r . The second point i s that even i f a sounder t h e o r e t i c a l j u s t i f i c a t i o n could be given, the whole transport equation procedure i s unsatisfactory as a  (32)  mean of f i n d i n g transport c o e f f i c i e n t s .  Except f o r the simplest examples,  any use of t h i s method ends up i n a hopeless tangle of p h y s i c a l and mathematical approximations.  There i s no consistent manner by which we  can  compare various approximations or extend the theory to the case of strongly interacting particles. a complicated  The study of metals (or more generally, s o l i d s ) i a  problem at the best of i t , and the introduction of transport  equations and d i s t r i b u t i o n functions, as an intermediate transport c o e f f i c i e n t s , simply adds to the confusion.  We  step i n f i n d i n g s h a l l see that  another method i s possible which enables us to give formally correct N-particle expressions  f o r c e r t a i n transport c o e f f i c i e n t s by a d i r e c t quantum-statistical  arguement.  The problem of evaluation of these Kubo-type formulas i s of course  d i f f i c u l t , but they have the same advantage as the p a r t i t i o n function has ia equilibrium s t a t i s t i c a l mechanics i n that consistent procedures of approximation may  be found.  I t may  be mentioned that the p h y s i c a l i n s i g h t i n t o  i r r e v e r s i b l e processes and transport phenomena gained by the standard methods i s almost n i l .  Thus even though an e l e c t r i c and a temperature gradient  are r e a l l y e n t i r e l y d i f f e r e n t from a physical point o f view, almost no d i s t i n c t i o n i s made i n the way  they are treated.  i n the next three chapters removes seme of t h i s  The method to be hypocrisy.  described  (33) CHAPTER 3. KUBO'S DERIVATION OF FORMAL EXPRESSIONS FOR THE TRANSPORT COEFFICIENTS DESCRIBING MECHANICAL DISTURBANCES,  In t h i s chapter, we s h a l l deal with transport s i t u a t i o n s where the  A i s o l a t e d system has a w e l l defined time-independent Hamiltonian K including a l l interactions between the p a r t i c l e s and q u a s i - p a r t i c l e s composing the system and where the i n t e r a c t i o n operator  v  ,  describing the e f f e c t of the  'Striving systems" on the system of i n t e r e s t , i s same known quantum mechanical operator. We  s h a l l , f o r generality, assume that V  may have some e x p l i c i t time  A  dependence and therefore denote i t as V^. Any external disturbance which can be A  expressed unambiguously by means of a perturbation a "mechanical disturbance", a terminology  V  t  w i l l be r e f e r r e d to as  introduced by Kubo (195Tb).  example which we have i n mind i s an e l e c t r i c f i e l d  The  E ( " t ) . For "external A  disturbances" such as a temperature gradient, where the form of V or undefinable, Kubo uses the term "thermal disturbance". We these l a t t e r "disturbances" i n chapter  t  i s unknown  s h a l l discuss  4.  A  The e f f e c t of w i l l be now  V  on the density matrix  t  of the i s o l a t e d system  investigated. We assume that the mechanical disturbance i s turned  on a t time "t = 1 ,  and that before t h i s , the system of p a r t i c l e s was  0  i n equi-  l i b r i u m . Therefore, we must solve  (3.D  --iCK^Vt  t*t  f o r  0  with the i n i t i a l condition (3.2)  p^m p  E Q  ?  <±  t  0  The equation (3.2) can be transformed i n t o the following equivalent i n t e g r a l equation, as can be v e r i f i e d by d i r e c t d i f f e r e n t i a t i o n ,  (3.3) p t  P^-if - - '" Cv .,p Qe "- "ctt' t  i w  e  to  t  i  + ;  t  t  v >  O ) We can solve ( 3 . 3 ) formally by an i t e r a t i o n procedure but since we are interested i n weak external f i e l d s , the following l i n e a r approximation w i l l be. s u f f i c i e n t ^  to  where we s h a l l denote the second term on the r i g h t hand side by equation ( 3 . 4 ) gives  . The  P^^o the f i r s t order i n V V , and i t i s s u f f i c i e n t f o r  c a l c u l a t i n g f i r s t order transport c o e f f i c i e n t s such as e l e c t r i c a l conductivity, as we s h a l l see. I f we were interested i n second or higher order transport c o e f f i c i e n t s (such as the Joule heat), we should have t o use the neglected higher order terms i n ( 3 . 3 ) . More information on the l a t t e r subject may be found i n a paper by Bernard and Callen Let us denote by B  (1959).  some quantum mechanical operator describing A  some sort of "current". We have given the conditions which (2.1)  8  must s a t i s f y i n  and ( 2 . 2 ) . As i n our discussions of i s o l a t e d systems, we assume that the  ensemble average  (3.5)  < §>  T  -T/v(&B )  i s equal t o the actual observed value i f the system o f i n t e r e s t i s large enough. We s h a l l omit any reference t o the e f f e c t o f the observations on the quantum mechanical system, although t h i s i s a d e l i c a t e point worthy of attent i o n . Using ( 3 . 4 ) ,  (3.6)  <B>  ( 3 . 5 ) and ( 2 . 1 ) , we "find t  =~MAp  t  B ) .  Kubo's response method ( 1 9 5 7 a ) can be summarized as follows. Assuming that both B  and  are p r e c i s e l y defined operators, we evaluate A p^. and c a l c u l a t e the  trace i n the r i g h t hand side of ( 3 . 6 ) . The r e s u l t i n g expression i s then assumed to give the observable value o f the macroscopic current corresponding operator  t o the  B . We then t r y t o separate the external f i e l d (incorporated i n Ap«)  from t h i s expression, leaving a f a c t o r which w i l l be I d e n t i f i e d as a transport  (35) coefficient. Although i t is not the most general possibility, we are usually A  concerned with mechanical disturbances where the perturbing operator can be expressed as a scalar product of the form (3-7) where  •  = — M •  (- sign chosen for convenience)  F Ct) is a c-number and  A  is a, Hermitian operator, both of them  being vectors in the general case. . F C t ) may be an explicit function of the time and is related to the external disturbance while  i s not explicitly  A  dependent on the time and describes some property of the system of interacting particles. What A  and F(t) are in any particular example" is the result  of a separate investigation. The well known answers for an external electric f i e l d denoted by  (3.8) (3.9)  E Ct) are  ^=2  a-A;  p(t)= g(-t)  where  and  are the charge and position operator. , respectively, of the  th ^  charge carrier, and the summation is over the whole system of N charge  carriers.  (-3.4), using (3.7), as  We may write Ap in t  (3.10) &  e'^'^C&Petle^-^ott'.FU )  = L(*  e ±  J  1  ~to  .  ~  -~  Using (3.6) and (3.10), the value of the current, produced by the external  "t is given by  f i e l d , at time  a n ) <§> = T/i[i^e- ^' *rA,p„]e* i(t  1  £lt  t  ' ' *-F(.t') tt'g} 1:  ,  e  =LP<tf£tt )-TA.{e- «-* >*ca, ,  r  l  C  F(f)cU'-  l  TA,{ [AjPe*]  ,  (3.i5)  J  e*«*-  t - M 5  B(i) is in the Heisenberg  }  B W ^ e ' ^ S e ^ '  g}  B C t - f ) }  where the cyclic property of the trace was used and representation  Pl  with  s^)  =  |  be)  The time dependence of F ("t) should not be confused v i t h the Heisenberg representation notation used f o r operators. P a r t i c u l a r attention should be paid to the dot • appearing i n these and future equations. I t denotes the  scalar product of the tiro vectors which are separated by the dot. The response current  < § > I n the l a s t equation of (3.11) i s s t i l l t  not i n the desired form. As a further step, we introduce a new integration v a r i a b l e X = t - 1 ' • By means of t h i s transformation, we f i n d  (3.13)  <!>.= I { " dx x  fU-Z) . T / v ( [ A , p ^ ] B ( T ) } .  Xo  In order t o simplify £ A  ;  pt^J  * we s h a l l make a short digression. A l i t t l e  b i t of manipulation shows that  Now the Heisenberg  A = t[K,A]  (3.15)  where A  equation of motion i s  ct 6  i s not e x p l i c i t l y dependent on the time  t  . Remembering that  p  E < a  i s the canonical d i s t r i b u t i o n  (3.16)  p  e a  =^. -^  with  A  e  we may combine  z ~Ta(e"  p M  ).  (3.1 *), (3»15) and (3.l6) and a r r i v e a t the u s e f u l r e l a t i o n 1  The l a s t r e s u l t Is known as Kubo's Identity, and i s a very u s e f u l formula. Inserting (3.18) into (3*13), we have  (3.19) < B >  t=  r  f  '0  TA(PECI ^ C - i ^ g U ) )  (37) where i n the last equation, the cyclic property of the trace and the faett p  that  E ( 5  commutes with  G  were used.  .Summing up the results of the previous paragraph, the response of a dynaiaical observable perturbation  (3.20) where  < B >  V t  t  B , due to a mechanical force which introduces a  J*"Sr  =  ,  = - A • F(-fc) F(t-r).  = <--->  T/L-{PBQ  I  is given by <C(o) BCC + i A ) ^  and  e q  / j = iC^jA]  C  S  .This  result lias been derived by several authors, the f i r s t being Kubo, by the same method as used above. Following Kubo, we introduce the response or after-effect f  c g  (t)  , so named since i t gives, the response at a time  X  function  after the  application of a unit pulse of a mechanical disturbance. The response function (a tensor) is defined as ( 5  *  T^eCt) = \[d\ < c C o ) g C r + i ' A ) > ^  2 1 )  In terms of the response function, the response current is <B\=  (3.22)  (^'dr F(t-t)4c Ct) . B  By a change of integration variable,  T —> t'-(t-t)  , we have in place  of (3.22), (3.23)  <B> -r ca f(t').$ t  t  ,  c e  (t-t')  in which the physical meaning of the response function as the response to a unit pulse is more evident. If and  U.  (3.24)  F("t7 — 6(t'-t,)U.  , where to^ t, <t  is a unit vector, then  < i> = r "W ^ ( t H ) t  Where ( - f r - t ,  ^  -^CB  c-t-f)  ) is the time elapsed since .the pulse xra.s applied.  Making use of the assumption that the response current is linear in the external f i e l d , the most general equation for the response current  (38)  at time t" Is (3.25)  <§>  « = r°°c(r  Fa-T).fC(Y)  •where K ( T ) (a tensor) i s some function of the time which depends on the properties of the system. We have written ( 3 . 2 5 ) in a form compatibles with the "principle of causality" which, in this case, states that depend on F ( t )  <B> can only +  where -00<T< "t . On comparing ( 3 . 2 5 ) and the equation  ( 3 . 2 2 ) previously derived using Kubo's formalism, we see that the only difference l i e s i n that Kubo's method enables us to find an explicit formula for K ( r ) , namely that given by ( 3 . 2 1 ) . If we assume that p (-t) = F  e "  c  i  w  t  that i s , the external f i e l d has a harmonic time dependence, we can rewrite ( 3 . 2 5 ) as (3.26)  <|>  t  =  £(f).{nc(X KCT)e  =  F ( t ) • OC(UJ)  i w r  ]  where (X (LU ) i s known as the generalized •susceptibility tensor. The well known fluctuation-dissipation theorem uses C<(u)) to connect the equilibrium fluctuations of physical quantities and the dlssipative properties of the system when external disturbances act on i t (Callen and Welton (1951) )• A glance at ( 3 . 2 2 ) w i l l show that the response current  < 8 ) Is ~  dependent on "to,  t  the time at which the external f i e l d was turned on. We  shall now show how to get r i d of this i n i t i a l time dependence, following Van Hove'3 discussion ( i 9 6 0 ) . Consider <^ C(0) §\tjy , an expression usually known as a "time-relaxed correlation function". It gives a measure of the "correlation" between one operator  C  at time  Z = O and another operator  B at time T « T , the value of the latter being governed by the "natural" motion of the system of particles independently of the external f i e l d . Now i t w i l l be assumed that, due to the dlssipative coupling between the particles of the system, (3.27) Aw, < C(0) 0 ( t--»oo ~  T ) > _  =  Q  (39) The validity of (3.27) can be checked i n certain cases by means of calculations very similar to those used by .Van Hove i n his derivation of the master equation.  The problem of evaluating correlation functions w i l l be discussed  i n chapter 5. The limiting relation given i n (3.27) means that < i s negligible whenever t" » Now we shall take B '= A  }  where T  , where A -  c'(o) B'(x)y  e(k  i s appropriate relaxation time.  N  LCH.AJS  § , and  c'~  c  : we  may rewrite (3*27) as  (3.27)  =0  < C(o)A(?)>  Mm  eei  a result, we conclude that  (3.28)  £<*n, ^ K C ( o ) 0 ^ f U ) >  f  f  = O  t  T-*GO  since we can easily show that (3.29)  r'cfx < c ( o > f i C r + t A » -/ O  e a  «  t <C(o>A(T)>^-L<A(t)C(o)>^  -Ay  With (3*28) i n mind and assuming that  (3.30)  (t-to)»Tn  we can effectively replace the upper limit < f > i s independent of the time t t  0  ( t - t ) i n (3.22) by oo . Thus 0  at which the external disturbance was  switched on, as long as the time elapsed  Is large enough so that  (3.30) holds, and i s given by  (3.31)  § Ct)  < 0 > = i*F(t'Z)'  ce  t  dt  Thee physical meaning of this result i s clear, for we would intuitively expect that, even though the behaviour of the perturbed system might be complicated and highly dependent on the i n i t i a l conditions when the f i e l d was f i r s t applied, as time passes, these "transients" would be. "damped" out. We shall now specialize the problem to the case of an electric f i e l d Bit) (3.32)  • More particularly, we assume the following time dependence Bit)-  Ec  e~  iu,i:  where the subscript c denotes a time independent quantity. current operator J" i s given by (3.33) J~£e±Pi 1  /YY\ L  The electrical  (ao) where €y, pi  and  of the  charge c a r r i e r , and the summation i s made over the whole system.  are the charge, momentum operator and mass, respectively  For s i m p l i c i t y , we assume that we are dealing with electrons i n a metal, and therefore  (3*33)  s i m p l i f i e s to  (3-3U)  J" -  ® £1 p-  The  X -A  dipole moment operator  C = A  was given i n  (3.8).  I n order to f i n d  , we take the time d e r i v a t i v e o f  -A = S eAi  (3.35)  '  2. p  t  :  = J and consequently c - J and  (3.32)  (3.36)  (3.31)  into  i n t h i s s p e c i a l case. and (3.28), we have  <J\^?dt^e " '-* -$,r(T) i  (3.37)  L  )  with  -tfd\< £<•> J ( r + t > ) > «  Rewriting  (3.36),  (3.38)  C J> = t  F ( t ) . { > c  (3.39)  < J>  i  (3.U0)  < J y = 21 0~*Uw)g U;)  - E(±) • c r ( u J )  w  t„W  t  M  J = (J,, J  j,)  .  V (UJ) U)  , we may rewrite  (3»38) as  - 1, 2, 3 or x, y, a)  y  M  outside the i n t e g r a l  or i n component form  0,  t  Fit)  we take the e l e c t r i c f i e l d  Introducing the complex conductivity tensor  where  (3»35)» (3«3lO  Substituting  . The complex conductivity tensor i s e x p l i c i t l y  defined as  (3.UD  CT" (w)=  (3.U2)  CT^>^)=;*^te' w r X^ X  ,  J7VT e'  u , T  $,.r(t) (  <J,fo)£<T*fA)>  I t should be noticed that the indices ^u, V hand side t o the r i g h t hand side of  o r i n component form e  f  t  are permuted i n going from the  left  (3<>U2).  A w e l l known equation i n electromagnetism gives the t o t a l "miGroscopic current" i n terms of the obmic conductivity tensor  ° 0~ (w) and the medium O)  (41)  d i e l e c t r i c tensor A £ ( u> ), that i s ,  (3.^3)  <5>  -  J ( t )  where the macroscopic  •  P(-fr)  <>  current  j  (f ) i s defined by  (3.44) j ( t ) . ^ C t ) . ' c r ( ^ ) w  and the p o l a r i z a t i o n vector  OM)  Ct)-  P  P ( -fc ) i s defined by  E ( t ) * AE(  By combining  co)  (5.43) and (3.39), we have, assuming £ ( t ) = £  e~ * i W  c  (3.46) CT (u,)= [ °cr (w). - i w A £ ( o d ) ] c,)  0,  where, i n p a r t i c u l a r , the ohmic conductivity i s given by  (3.47)  i Co)^(.-r+iX)> ctX..  Coa.uiTf<  y  p  Generalizing once again, we s h a l l summarize the r e s u l t s of Kubo's A  a n a l y s i s . Assume that we have a "current" operator B a perturbation, due to a mechanical disturbance, with < 8>^  Then the current "response"  (3.48) < B> (3.^9)  L*  t  8  =  F( -t) • L  -  $™dz e  A S  iu,X  rigourously defined and V  t  = - A * £c  e  l  W  l i n e a r i n the external f i e l d i s given by  with the tensor given by  ^d\  Tn{pe*  A(0)  BU+<X)}  or i n component form  (3.50)  < §J> = ^  (3.5D  Li* -  t  F (~t)  with the tensor elements given by  "i'di^tfdxT^lpci.to  B^CT-JAI] .  u  The formally exact expressions given i n  (3.**2), (3.47) and (3.51)  are known as Kubo-type formulas f o r the components of f i r s t order transport c o e f f i c i e n t s . They are admittedly quite abstract i n appearance. As we mentioned i n the introduction, the main r e s u l t of  KUDO'S  analysis,. 4S that the study of  l i n e a r transport processes involving open systems (with mechanical disturbances) can be reduced to the study of non equilibrium processes i n i s o l a t e d systems. The l i n k i s the response function the time-relaxed c o r r e l a t i o n function,  <^ ( c8  t ), which i s an i n t e g r a l over  < c(o)B(z-^ixf)  (3.52)  = T/uj e  £<°>e.  gcoe  J-  A  where  -is the Hamiltonian of the i s o l a t e d system. As we s h a l l see i n more  d e t a i l i n chapter 5, the evaluation of the r i g h t hand side of (3.52) i s very c l o s e l y connected with the study of the d l s s i p a t i v e properties of an i s o l a t e d system as discussed i n chapter 1.  I t w i l l be seen that one of the most f r u i t -  f u l methods of c a l c u l a t i n g transport c o e f f i c i e n t s i s a combination of Kubo's general formalism as discussed i n t h i s chapter and the work of Van Hove on the master equations which describe the i r r e v e r s i b l e approach towards equilibrium. I t may be noted that i n passing from the f i n i t e upper l i m i t ("t — ~t ) D  i n (3.22) t o tiie e f f e c t i v e l y i n f i n i t e upper l i m i t i n ( 3 . 3 1 ) , the j u s t i f i c a t i o n given was that ^  c  8  (  ) i s n e g l i g i b l e as long as f » tn.  , where f  e  Is  some appropriate "relaxation time" r e l a t e d to the d l s s i p a t i v e properties of the system. In doing t h i s , we have followed Van Hove (I960); but, i n the l i t e r a t u r e , other methods of J u s t i f i c a t i o n have been used, A few remarks w i l l be made below on the r e l a t i o n s h i p between these various procedures. Lax (1958) assumes that the system of i n t e r e s t j undergoing an external perturbation represented by  V  t  , i s not completely i s o l a t e d but may weakly i n t e r A  act with the r e s t of the "universe" i n such a way that when Vt density matrix  p  t  approaches the equilibrium density matrix  p  = O eG)  .  , the The weaker  the i n t e r a c t i o n with the surroundings, the longer the "relaxation time" (denoted by T ) fi  i s . To describe t h i s state of a f f a i r s , Lax assumes p s a t i s f i e s t  the following equation of motion ^ y •••  (3.53) ijh^  +[  fe,  - H + V * ] + j(p^-pBQ)  _  Q  Karplus and Schwinger (1948) and others have used t h i s s o r t of equation p r e v i ously. Carrying through a s i m i l a r analysis as was given i n the f i r s t part of t h i s chapter f o r ( 3 . 1 ) , we are l e d to the r e s u l t • T h i s r e l a x a t i o n time T the r e l a x a t i o n time t  R  R  has, of course, a d i f f e r e n t p h y s i c a l meaning than introduced before.  If T »  Ti  , then  #  c  e  ( t )e  _  i s n e g l i g i b l e . Consequently  T / t c  f  if  t - t » Te  then the upper l i m i t i n  0  by OO . L a s t l y ve take the l i m i t  <g>  (3.55) or,  =  t  with the r e s u l t  (°° F(-t-T)  e'^dt  i f we introduce a new parameter £ ±  (3.56)  <8>,  (3.5*4 can be e f f e c t i v e l y replaced  re-  . J*n  '  E(i-TrV$cB^)^r  £ t  o o  where (3.56) defines the l i m i t i n g procedure used to evaluate the i n t e g r a l denoted by  j>  . I n contrast to our o r i g i n a l method (following Van Hove) i n deriving  (3.31), Lax's analysis makes no use of the vanishing of  § ra. it)  as a  r e s u l t of the i n t e r n a l d l s s i p a t i v e f o r c e s . On the other hand, Lax i s forced to introduce i r r e v e r s i b i l i t y through a weak i n t e r a c t i o n with the "universe". This r e s u l t s i n the convergence f a c t o r Q  appearing i n the integrand of (3»55).  However, introducing a convergence f a c t o r becomes redundant whenever the conditions (3.27) or (3.28) are v a l i d . In a d d i t i o n , although Lax's procedure i s based on a d e f i n i t e p h y s i c a l assumption, I t has the appearance of a mathematical " t r i c k " , e s p e c i a l l y when the l i m i t £ -»o  i s taken.  Since (t - ~t )» t , equa0  t i o n (3.52) i s only v a l i d f o r i n f i n i t e times, i n contrast to (3.31) whichholds f o r (t  -"t )>> t 0  e  where V  K  i s some s p e c i f i c , * calculable r e l a x a t i o n time  depending on the properties of the p a r t i c u l a r system i n question. In contrast to the methods used by Van Hove and Lax, Kubo (1957a) and others use a procedure which depends on how the external mechanical disturbance was "turned on". Here we assume that a t ' X  - - 00  >  the system was i n  thermal, equilibrium and the external f i e l d i s introduced a d i a b a t i c a l l y , that Is, extremely slowly. This can be done by Introducing an extra time dependence factor 6 * 6  i n the external f i e l d with  £  always being a very small p o s i t i v e  (kk) number. In the end, the l i m i t 6-»o i s taken. Again we f i n d , a f t e r the usual calculations,  (3.57) <|>-  i U ( ; ' ^  where " t = — CO  .' The l a s t equation i s formally i d e n t i c a l t o that obtained  0  e  - " F ( t . t ) . e  +  ^  M  l t )  by Lax i n (3.51). Kubo's r e s u l t i s d e f i n i t e l y dependent on how and when the f i e l d was switched on, i n contrast t o the procedure used by Lax. On the other hand, no p h y s i c a l assumption such as a weak i n t e r a c t i o n with the r e s t of the "universe" was necessary. As we have seen, neither Kubo nor Lax e x p l i c i t l y use any vanishing property o f the response function; but both introduce, rather a r t i f i c i a l l y , a convergence f a c t o r i n t o the integrand o f the i n f i n i t e i n t e g r a l i n (3.31), making the l a t t e r i n t e g r a l convergent and w e l l defined. By considering the longtime behavior of the response function, Van Hove succeeds i n giving a more d i r e c t j u s t i f i c a t i o n of the  (3.31) s t a r t i n g from (3.22). To v e r i f y the v a l i d i t y of  important c r i t e r i o n (3.27), o f course, involves some approximate c a l c u l a -  t i o n of the c o r r e l a t i o n function. This i s not easy task i n general, but t h i s i s no c r i t i c i s m since the whole of Kubo's formalism i s not too much help unl e s s we can evaluate the c o r r e l a t i o n function (or equivalently, the response function) i n some approximate manner.  (45) CHAPTER 4 ATTEMPTS TO FHiD KUBO-TYPE FORMULAS FOR  TRANSPORT COEFFICIENTS  CONNECTED WITH THERMAL DISTURBANCES (THERMAL COITOUCTIVITY) SECTION 4.1:  INTRODUCTION AND  I n c h a p t e r 3,  we  CHOICE OF OPERATORS.  have o u t l i n e d Kubo.'s method o f f i n d i n g  exact  general expressions f o r c e r t a i n transport c o e f f i c i e n t s . In the f i r s t place, i t was  assumed t h a t t h e quantum m e c h a n i c a l  current operator B  ambiguously d e f i n e d . The p a r t i c u l a r example we u s e d was  J  c o u l d be  , the  un-  electric  c u r r e n t o p e r a t o r o f a system o f N e l e c t r o n s i n a m e t a l undergoing  various  i n t e r a c t i o n s among themselves, w i t h i m p u r i t i e s , w i t h t h e l a t t i c e , e t c . F o l l o w i n g Kubo, we  next c a l c u l a t e d t h e change ( l i n e a r i n t h e e x t e r n a l f i e l d )  i n t h e d e n s i t y m a t r i x d e s c r i b i n g t h e system, t h e change b e i n g due t o an e x t e r n a l mechanical  d i s t u r b a n c e . I n t h e c a s e o f an e l e c t r i c f i e l d  p e r t u r b a t i o n energy  V  is  t  = - X • E(t)  moment o p e r a t o r o f t h e system. We  where  f  X  J=(t) , t h e  i s the e l e c t r i c d i p o l e p±  found the d e n s i t y matrix  t o the  first  order i n the e x t e r n a l f i e l d , Now  t h e r e i s a whole range o f i r r e v e r s i b l e p r o c e s s e s f o r w h i c h t h e  above p r o c e d u r e  cannot be e a s i l y c a r r i e d out, i f a t a l l . T h e r e a r e two  f o r t h i s . F i r s t of a l l ,  t h e quantum m e c h a n i c a l  reasons  operator corresponding t o the  c u r r e n t f l o w i n t h e s e t r a n s p o r t p r o c e s s e s i s n o t known as easily found.  Secondly, A  t h e e x t e r n a l d i s t u r b a n c e may  but the operator V  be g i v e n p h e n o m e n o l o g i c a l l y ,  w h i c h g i v e s t h e i n t e r a c t i o n between t h e system o f i n t e r e s t and t h e  t  "driving  system", nay n o t be e a s i l y d e f i n e d i n any p r e c i s e m i c r o s c o p i c manner. F o r t h e r e s t o f t h i s c h a p t e r , \re s h a l l c o n c e n t r a t e on i r r e v e r s i b l e t r a n s p o r t p r o c e s s e s i n which t h e s e d i f f i c u l t i e s a r e p r e s e n t . F o r c o n c r e t e n e s s , we  s h a l l consider  t h e s p e c i f i c example o f t h e r m o e l e c t r i c phenomena. A f t e r making a few i n t h i s s e c t i o n on t h e p r o p e r c r i b e t h e energy f l o w , we  d e f i n i t i o n  o f an o p e r a t o r  s h a l l t h e n review,  Q  i n s e c t i o n 4.2  remarks  which w o u l d desand 4.3,  recent  ,  attempts to generalize Kubo's results to thermal disturbances . . In particular, we shall work out the details for a macroscopic temperature gradient and attempt to find a Kubo-type formula for the thermal conductivity K. I t may be pointed out that the literature on this subject i s distinctly unsatisfactory from a rigourous quantum s t a t i s t i c a l standpoint* The physical problem should be stated precisely, and emphasis be given to the d i f f i c u l t i e s which must be surmounted. Instead, much of the literature boils down to an intuitive Justification of results written i n analogy to Kubo's formula for electrical conductivity given i n chapter 3« In the literature, there i s considerable variety i n the choice of an observable corresponding to the energy current. If a satisfactory quantum s t a t i s t i c a l theory of heat conduction was available, the appropriate dynamica l observable would be given as a result of the theory. In the absence of this, we are forced to make a reasonable "guess". Suppose the total Hamiltonian can be written as the sum of the energy operators of free noninteracting excitations (particles or quasiparticles) and the velocity of each excitation i s unambiguously defined. Then we may write the energy flow operator Q as  (4.1)  0 =  Eu  Yn th  where Em and V i , are the energy and the velocity operator of the L  type of  excitation i n the state labelled by £(there being M kinds of excitations), and n  lje  i s the occupation number of the state specified by t and X . The  operator i n (4.1) i s a well defined quantum mechanical operator and completely describes the energy flow i n the system. In the absence of any interaction between these excitations, though, the energy flow Q i s a constant of the motion. Its expectation value depends entirely on the i n i t i a l conditions . As soon as we introduce the interactions between the excitations,  the very concept of the energy carried by a particular excitation loses i t s precise meaning. In terms of an electron-phonon system, for example, we can speak about the total energy of the system but not of the. energy of the electrons or phonons separately. It Is the old question of which "part" of the interaction energy belongs to the electrons and which part to the phonons. What i s often done i s to consider separately the energy flow of the type of excitation which makes the dominant contribution to the transport of energy. In ordinary metals, for example, the conduction electrons, as carriers of energy, usually completely overshadow the other .'excitations. Therefore we may, to some approximation, only consider the energy flow-due to the electrons; the phonons, impurities, etc., enter.:  i n so far as" they  effect the dynamical behaviour of the electrons. On the other hand, i n an insulator, the heat conduction by phonons i s dominant as long as the temperature (and temperature gradient) i s not too large. Even for semimetals, where the heat conduction by electrons and phonons i s of comparable order, we may treat them separately to a f i r s t approximation. In this case, • the effects of the interactions with each other are taken i n account through scattering processes and the total energy flow is simply taken as the sum of the electron and phonon energy flows. The sort of physical approximation discussed in the above paragraph A  A  '  consists of concentrating on the part H,, of the total Hamiltonian H which t  contains the degrees of freedom associated with the "dominant" type of excitation. It i s usually assumed, i n addition, that H, may be s p l i t as follows  H, = H + V,  (4.2)  (0  where H represents the energy operator of the dominant excitations considered lo  as a separate system, while V, represents a l l the interactions which involve at least one excitation of the dominant type and at least one other type of  (48) e x c i t a t i o n . In a c t u a l c a l c u l a t i o n s , V, l a usually treated as a perturbation on  Ho . Since the dominant excitations may be i n t e r a c t i n g among themselves,  i n general, we s t i l l have the fundamental problem of how the i n t e r a c t i o n energy i s to be "shared" among these excitations. Fortunately, i n the important case of an ordinary metal, the dominant excitations are usually electrons and these can be assumed to be noninteracting to a good f i r s t approximation. A"reasonable" choice of the e l e c t r o n i c energy flow operator i n t h i s example i s  J  rrr\  ~' » m\  J  where "H^ , pj. and <™ are the energy operator, momentum operator and mass of the 3  electron. A s i m i l a r sort of expression i s often used f o r the phonon  energy flow operator (see, f o r example, Carruthers  ( 1 9 6 1 )  ) .  feny authors b e l i e v e that the concept of an energy flow i s b a s i c a l l y a macroscopic concept and hence the corresponding quantum mechanical observable may not be necessarily an operator which describes the behaviour of the system i n microscopic d e t a i l . More p r e c i s e l y , I t may be that the energy observed by macroscopic flow measurements i s c a r r i e d by a large number of i n t e r a c t i n g excitations, and may not be the sum of the energies c a r r i e d by i n d i v i d u a l e x c i t a t i o n s . As an example of t h i s type of treatment, we r e f e r to Mori  Here the whole system i s s p l i t into many subregions each of  ( 1 9 5 8 ) .  which i s macroscopically small but microscopically large. In other words, each subregion has a large number of excitations but the dimension i s smaller than the experimental accuracy  0  To a s u f f i c i e n t approximation, we may  now  neglect the interactions between the subregions as surface e f f e c t s and assign " th a d e f i n i t e energy E-and v e l o c i t y operator Vi ^° *  n e  <•  subregion. In terms  of t h i s p i c t u r e , the energy operator i s often taken as (4.4)  •  Q  =  £  p  L  y i  where we sum over a l l the subregions. This type of coarse-graining has been used i n many theories of i r r e v e r s i b l e processes and i n the derivation of  0*9 > hydrodynamical equations. From both an experimental and intuitive point of view, i t i s good as a f i r s t approximation. The major weakness of this type of approach i s that i t i s very d i f f i c u l t to present It rigourously or precisely. In addition, i t takes our attention from one of the basic problems of theoretical physics, the precise definition., of an energy flow. In principle, there does not seem to be any reason why the energy current cannot be given by some precisely defined operator, although the discussion in the last few pages seems to indicate that a reformulation of the problem may be necessary before any further progress i s made. As we mentioned before, the electric current operator for a ordinary metal may be taken as  (3.33)  J  £  =  e  p.  where Q i s the charge on the electron. Here the choice i s unique, even in the presence of interactions among the electrons and other excitations. This is because the charge carried by each electron has a definite meaning to a very good degree of approximation. Another problem arises when we consider the appropriate definition of local densities. When we are discussing the classical N-partiele problem, the conventional definitions of the local number density n ( M and the local energy density £ ( / 0 a r e in terms of delta functions , f  j  ^  r u n ) = J~[ <$(n _ •) " j=, ~ ~ =  Z!  i  n(/v) dn, - N  such that  3  -n-  J  such that  f  ~  £(n)  and  ~  d/i- =  E  In this- equation, .n. and E are the total volume and energy respectively, of the }  system; N i s the total number of particles, a l l of which are identical;  /t i s  "fell  the position vector; n,^ i s the position vector of the ^  particle; and  Ej  th is the kinetic energy plus the potential energy-of the j  particle, assuming  that the potential energy i s equally shared among the interacting particles.  (50)  The other two quantities of i n t e r e s t ,  .  the l o c a l e l e c t r i c current density /A.  and the l o c a l energy current density nM  '  V  are usually defined i n terms of  £6v) i n such a way that the equation of continuity  and  e  (4.6)  dm£)  +  v  A  .£c&) = o  d t and the equation of conservation of energy  ^  (4.7)  + V  f t  -^)=Q  dt are s a t i s f i e d .  Of course, the equations (4.6) and (4.7) do not uniquely  specify  and  c ^ M j . To extend t h i s c l a s s i c a l treatment to a quantum  mechanical N - p a r t i c l e system, we may make use of Weyl's correspondence r u l e . The l a t t e r enables us t o transform c l a s s i c a l phase space functions of the coordinates and momenta of the p a r t i c l e s i n t o quantum mechanical operators by means of a consistent procedure. operators such as  (4.8)  By means of t h i s , Me f i n d l o c a l density  ^  n(/L) - £  £(&-&i)  for the l o c a l number density, where of the # *  particle.  n  , now denotes the p o s i t i o n operator  I t must be emphasized that quantum mechanical operators  such as (4.8) are e s s e n t i a l l y  c l a s s i c a l i n content.  This may be seen from the  method of f i n d i n g these l o c a l density operators from the c l a s s i c a l l o c a l density functions.  Another nay of seeing t h i s i s by noting that the operator  A  r\(/i) has a physical meaning only when i t i s used i n a very r e s t r i c t e d  way.  To make, the l a s t point more precise, we s h a l l c a l c u l a t e the matrix elements of H(/t)  (4.9)  with respect t o the N - p a r t i c l e state functions  £  ^ K ^ I ^ I ,  •• •  it)  j  <^|n(A)|^,> L  E  "(*)  $u>(4j,-,lL*;t)  cU.-.. eU»  " x dsi, ...~OIA.J.-I dbi $.+<••• CIA*  =  W  J  A  ^^.A  L R  ..,^;<)^ (A A )  ;  l r  .^-t)^ ...^A»; L  (51) where i n the l a s t step, use was made of the f a c t that a l l the p a r t i c l e s are assumed to be i d e n t i c a l .  I f we put cy* =. u' , the l a s t expression turns  out to be, by d e f i n i t i o n , the diagonal element ( i n the 5p<* (-Hr*, \i • - ^«/;^) t  representation) of the single p a r t i c l e reduced density matrix. d e f i n i t e p h y s i c a l meaning as the p r o b a b i l i t y at / t at time t  , i f the state functions  eigenfunctions of the t o t a l hamiltonian. elements  < ^  ifo<'/*  I t has a  density of finding a p a r t i c l e 6£>, ...,Av;-r)  are the  In contrast, the off-diagonal  $ where  , are not equivalent to  the off-diagonal elements of the single p a r t i c l e reduced density matrix. Indeed, these off-diagonal elements of meaning.  YM/J)  have no d i r e c t p h y s i c a l  For t h i s reason, i n dealing with the operator  /1(A) , we have  always to keep i n mind that only the diagonal elements of t h i s operator may be used i n a c a l c u l a t i o n .  Although s e v e r a l authors use t h i s operator  as the d e f i n i t i o n of the number density operator, the above point i s an e s s e n t i a l drawback of such a d e f i n i t i o n and hence we w i l l not use t h i s operator i n the following argument. In t h i s t h e s i s , we s h a l l define our operators using a quantized theory, i n contrast to unquantized theory used up to now.  In p a r t i c u l a r  the various l o c a l densities w i l l be assumed to be expressed i n second quantized form.  The density of the number of conserved p a r t i c l e s M ( A )  i s given by the usual formula, (4.10)  h(/t)  =  YU)  with the t o t a l number operator  N =  n (A.) dA-  denotes the quantized p a r t i c l e f i e l d f u n c t i o n and conjugate of ^(^L) . we have  and where ^ ( A . ) YVA)  the Hermitian  For the p e r t i n e n t example of electrons i n a metal,  (52)  (lull)  A.)  II  —  n  (4oi2) where  OLKV$KV(A)  (X^v  and  are the usual c r e a t i o n and a n n i h i l a t i o n operators,  respectively, o f the conduction electrons with the reduced wave vector k and i n e l e c t r o n i c energy band V . trons.  We have ignored the s p i n o f the e l e c -  ^PKUCA.)} i s some complete orthogonal s e t of s i n g l e p a r t i c l e  states, properly normalized i n the volume Sl of the whole system.  (PKVCA.)  i s the Bloch wave f u n c t i o n describing an electron of reduced wave  vector k i n the y * for  Typically,  n  band.  The usual fermion anti-commutation  r u l e s hold  and fl^v . For the purpose of argument, we s h a l l assume that the l o c a l  density operators are known so that we can concentrate on the next problem, namely t o evaluate &P± operator  .  In p a r t i c u l a r , we assume that the l o c a l density  to ( A ) corresponding to the energy flow i s known.  requires only the existence of such an operator.  The next section  In chapter 5, where the  problem o f evaluating the formal expressions discussed i n chapter 3 and k i s t a c k l e d , we must be more e x p l i c i t . as the operator defined i n  (U.3),  with  Q  As an example, we could take  u)(A) defined i n the usual manner, ****  S\r  I f -H- i s the t o t a l Hamiltonian of the i s o l a t e d system, then  (U.1U) where  ^  r  L  i s the l o c a l energy density. I f we apply the general r e s u l t s of chapter 3, we may now  f i n d the energy flow due to an e l e c t r i c f i e l d  (3»U8) and (3*1*9), the  §[tt) £/• € ~ s  "response" energy flow i s given by  t u /  ^  easily .  Using  (53)  We may introduce the tensor  0~ (LO) SO that  (4,16)  <T >(co)  <$\=  <=rt)•  u)  u  where  (4.1?)  cr">cu>) = $?dT  ^  TA[  z'^JfcUi  Ctt  i«» 3^T+a>}  In component form, we have as usual  (4.18)  <(^> = X t  ry^t^EUt)  y-i and  (4.19)  ^(cp).r  J^te "^ 1  f?d\TA.fp a e  JyMQ^Ct+O)]  SECTION 4.2 : DERIVATION OF A KUBO-TYPE FORMULA. FOR THERMAL CONDUCTIVITY. Although a thermal gradient i s physically very different from an external electric f i e l d , i t may be hoped that we can derive a formal expression for thermal conductivity similar i n form to that obtained for electrical conductivity.  The derivation outlined i n this section i s based on the approach  used i n the papers of Mori (1956, 1958, 1959) and Nakajima (1958, I960). Although formally different i n development, both these authors make use of local distribution functions and study transport processes involving open systems i n terms of the regression of an appropriately chosen fluctuation in the isolated system. Further comments on the literature, which i s fairly extensive, w i l l be given in section 4*3* Our f i r s t object i s to find the equation of motion of the density matrix of the many-particle system subject to a "sufficiently small" tempera—  ture gradient  ^^.TitC) (see section 2.1) which constrains the system from  reaching thermodynamic equilibrium.  It i s assumed that i f the system has been  <5U)  i s o l a t e d long enough ("aged system"), o r equivalently, i f i t has been i n contact with a constant temperature heat bath, i t w i l l be i n thermal equilibrium.  As mentioned before, the equilibrium density matrix  p^ e  i s taken  f o r a canonical ensemble and therefore Is  (U.20)  p  e<k  =.  e  P*-* * 3  with the normalization constant  chosen so TA(p^)  =/•  To even define macroscppically a temperature gradient, we must assume the existence of a " l o c a l temperature," as was noted i n section 2 . 1 . The specimen can then be e f f e c t i v e l y s p l i t up i n t o small volume elements each with an approximately constant temperature and each being i n thermal e q u i l i brium.  The v a l i d i t y and meaning of t h i s " l o c a l equilibrium assumption" has  been discussed extensively by Kirkwood and hie co-workers, f o r the case o f c l a s s i c a l mechanics, from 1°U6  Mori (1958) puts considerable emphasis  onward.  on the importance o f the existence of macroscopically small mass elements i n e f f e c t i v e thermal equilibrium.  I f the " i n t e r a c t i o n s " between these small  elements are assumed to vanish, then each volume element i s assumed to a t t a i n i t s own thermal equilibrium i n a time t  0  , the l a t t e r time being quite small  since i t i s a r e s u l t of the i n t e r n a l microscopic processes.  The other relaxa-  t i o n process i s the hydrodynamical process of a t t a i n i n g s p a t i a l uniformity throughout the whole system, the r e l a x a t i o n time  Te. being much larger than  Much of Mori's analysis (1958) i s b u i l t upon the q u a l i t a t i v e d i s t i n c t i o n between these two relaxation mechanisms and t h e i r coupling but, as one would expect, the discussion r e l i e s on i n t u i t i v e arguments i n many places.  In our opinion,  I t i s d i f f i c u l t enough to deal with a macroscopic temperature gradient within the framework of a quantum s t a t i s t i c a l theory without bringing i n a d d i t i o n a l macroscopic concepts and arguments.  Mori, i n t r y i n g to give some p h y s i c a l  (55)  i n s i g h t i n t o transport and relaxation processes, succeeds i n complicating the  theoretical analysis.  As we s h a l l see. c e r t a i n i n t u i t i v e assumptions  are  necessary but by stating them as simply as possible and only when  needed, we never forget that these are assumptions and'may be eliminated i n future work. An open system, which i s kept under a constant small temperature }  gradient by means of heat reservoirs, w i l l approach to a steady state a f t e r a s u f f i c i e n t l y long time.  We s h a l l denote by p  which describes t h i s steady state.  the density matrix  The quantity which we want to calculate  i s the s t a t i s t i c a l average of the energy flow with respect to the density matrix p • the  However, the d i r e c t evaluation of t h i s quantity i s not easy f o r  following reasons.  F i r s t , the e x p l i c i t form of the density matrix  i s not known; i t i s obviously d i f f e r e n t from the l o c a l equilibrium density matrix since t h i s l a t t e r density matrix neglects the i n t e r a c t i o n s between the  d i f f e r e n t mass elements which give r i s e to the d l s s i p a t i v e heat flow  (without these i n t e r a c t i o n s , the hydrodynamical processes w i l l follow the i d e a l f l u i d equations).  Secondly* the time evolution o f the density matrix  i s d i f f i c u l t t o describe because i t depends on the i n t e r a c t i o n between the system and the heat r e s e r v o i r s .  Hence, instead o f studying the open system  d i r e c t l y , many authors consider an a u x i l i a r y i s o l a t e d system which a t the i n i t i a l moment i s i n the same macroscopic state as the r e a l open system, and then t r y to i d e n t i f y the p h y s i c a l s i t u a t i o n taking place i n t h i s a u x i l i a r y system with that i n the r e a l system under c e r t a i n conditions. We s h a l l denote the density matrix of the a u x i l i a r y i s o l a t e d system by p the  t  •  I f i t i s to describe the same macroscopic state as p a t  i n i t i a l moment, the following condition must be s a t i s f i e d  (56)  (4.21)  T/u(p  t  =  0  i(&)=  T^CpciA))  A  i s the l o c a l energy density operator defined by (4»14).  where  Equation ( 4 . 2 1 ) , however, does not determine i s dictated by convenience.  Pt*-o  uniquely and the choice  Mori has pointed out that the choice of Pt^o  >  with the r e s t r i c t i o n (a.21), i s rather immaterial i f we assume the d i s t i n c t separability  of the two r e l a x a t i o n mechanisms discussed above (for more  d e t a i l s , see the discussion a f t e r (4*51))•  Accepting Mori's argument, we  choose as the i n i t i a l density matrix the " l o c a l equilibrium" density matrix defined by  (4.22)  p  L  ^  e  (3(A) £ U)  P o * - S ^  cU  with the a u x i l i a r y condition (4.21) and the normalization condition  (4.23)  T/u(/9 ) = f L  In (4.22),  which determines $ •  (4.24)  T = - ^  -  '  T  i s the average temperature defined by  | — ^  ]  ,  Using the Fourier transform defined as follows,  (4.25)  g ( A ) r; ± £  2«  e  , , £  '*  i  we can rewrite the argument of the exponential i n (4*22) as follows  (57)  where i n the last equation we have used the relation (4.27)  i.  = J  c U - n  lu)  A  In the Fourier expansion of /3<A) = J-0. +  (4.28)  i ^ ' / j ,  -it-  -it.  »  K  £ e  * *  ' ~  the coordinate-independent f i r s t term can be interpreted, for the case of no temperature gradient, as /S - 7=. system. n  » where T i s the temperature of the  I f we l e t the temperature gradient approach zero, this i s  equivalent to assuming that the values of K present i n the summation n  i n (4.28). approach zero. gradient, only the /3*  In other words, for a small temperature with small values of K. (say K. < *r ) are c  nonnegligible. As a result of this analysis, we may rewrite (4.26) as (4.29)  0(A) CC/i)cU  =  fi*L  To the f i r s t order i n the  ±1'fi<£->c  BtcC^-o)  , which are assumed  to be small since the temperature gradient i s small, we find from (4.22)  (4.30)  p =-pev-pj™ L  f^dxle™  r where  £ -K  i s i n the Heisenberg representation generalized to  imaginary time variables.  More explicitly, we have  (58)  Thus a t t=o , the deviation of the density matrix from p  r4  is ( p - p L  e Q  ) * or  more e x p l i c i t l y ,  where A p = p - p L  L  E Q  As the i s o l a t e d system "relaxes to equilibrium" f o r t > o , i t i s governed s o l e l y by the i n t e r n a l Hamiltonian. We are dealing with the "regression of a f l u c t u a t i o n which had reached i t s peak a t t = O **. We are not concerned with an open system under an external temperature  gradient any more but  with a f l u c t u a t i o n i n i t i a l l y s p e c i f i e d so that, a t t" = O , the r e s u l t i n g s t a t i s t i c a l average of the energy density operator i s equal to the macroscopic energy v a r i a t i o n found i n the open system. As before, we must solve the usual equation of motion f o r an i s o l a t e d dynamical system  dt  p  where  t =  (MY)  and with the i n i t i a l condition  p -*e<s  /^p -/\p 0  L  The formal solution of  (4.33)  and  (4.34)  ^ p ^ e - ^ ^ e ^ * 4  (4.35)  is - t > o .  I t i s convenient to introduce the L i o u v l l l e operator L defined by the r e l a t i o n (M6)  L A  where A  LA,ii]  -  i s any operator. Using (4.J6) and the series expansion of an exponen-  t i a l , we can derive the u s e f u l r e l a t i o n (4.37)  e  L  A  =  e  "A  e~*  quite e a s i l y ( f o r example, see Kubo (1957a)). Using (4.35)  more compactly as  (4.37),  we may rewrite  (59)  From (4.37) we s e e t h a t t h e L i o u v i l l e o p e r a t o r i s , i n 'the quantum  mechanical  c a s e , s i m p l y a n o t h e r way o f d e n o t i n g t h e H e i s e n b e r g r e p r e s e n t a t i o n . A s i m i l a r self-adjoint  o p e r a t o r i s o f t e n u s e d i n c l a s s i c a l mechanics t o d e s c r i b e t h e .  dynamical e v o l u t i o n o f a n i s o l a t e d  system, composed o f a n enormous number o f  degrees o f freedom. We s h a l l now make t h e r i g h t hand s i d e o f (4.37) more e x p l i c i t a s follows  (4.38)  Ap  ±  = 4 p  u  t  °  1  =  +C^p -4pJ'.  J  di'  Ap +(-i){ye"  £ i : t ,  L  L*Ap " t  ft  where i n t h e l a s t two e q u a t i o n s we have i n s e r t e d t h e e x p l i c i t e x p r e s s i o n f o r  Ap  L  g i v e n i n (4.32-) a n d have made u s e o f  C Pets > "H ~\ O =  . Using t h e  energy c o n s e r v a t i o n e q u a t i o n w i t h t h e o p e r a t o r s i n t h e H e i s e n b e r g  representa-  t i o n ( t h a t i s , t h e d y n a m i c a l e v o l u t i o n o f t h e system i s i n c o r p o r a t e d d i r e c t l y into the operators)  (4..*0)  c)  V^.d)(A,t)  =Q • w^ere  i t follows t h a t t h e F o u r i e r components s a t i s f y  (Ml)  - lZi-At)iii]-  ix-A-Klt)  =o  6 = fuD(/t)c6t  (60)  As a f i n a l remark on (4.41), have no e x p l i c i t time dependence.  [  (4.42)  L^C-iA),*]  Now  and  i t follows  a r e assumed t o that  =• - ittz-tZ-Kt-iA)) =  ^ ^ - K C - C X )  (h»3$)  and t h e r e f o r e , from  (4.43)  £(%)  Ap  ±  or  (4.44)  Ap  Ap  ±  u  +  '  where  (4*37) was  wanted t o do,  2T J«-  Q  It  used i n the l a s t s t e p .  t h a t i s , we  pea u>-£  f*ott'L We  have succeeded i n d o i n g what  have found an e x p l i c i t e x p r e s s i o n  f o r the  density  m a t r i x d e s c r i b i n g the i s o l a t e d system whose d e n s i t y m a t r i x i n i t i a l l y  p. i n  g i v e n by  (4.22), t h e l a t t e r b e i n g chosen so t h a t  L  (4.44),  Using and  < T) .  <  s h a l l now  was  holds.  c a l c u l a t e the ensemble a v e r a g e s  These a r e the c u r r e n t s a r i s i n g as the I s o l a t e d system  T  from i t s i n i t i a l non for  we  (4.21)  equilibrium state.  , since the c a l c u l a t i o n o f  We  <J/.  we  <  ffi>  ±  evolves  s h a l l o n l y work out the d e t a i l s f o l l o w s i n an analogous manner.  A  More p r e c i s e l y , we  (4.U5*)  <.uJ(/L)y - T/L ±  Rewriting  (4.46)  v.^(£)'t  first find  Up  t  , which i s g i v e n  UJIA,))  t h e f i r s t term more e x p l i c i t l y by means o f  TA,C*PU  by  =  (4.32),  tegd\TA{p *i-« e  and p e r f o r m i n g the o p e r a t i o n o f time r e v e r s a l , we  see t h a t i t  vanishes.  A  Assuming t h e r e i s no magnetic f i e l d s p r e s e n t ,  £  and  K  fi  K  are i n v a r i a n t  A  in  sign while  (4.46)  tOt&)  switches i n s i g n .  As a r e s u l t , the l e f t  must be i d e n t i c a l l y zero s i n c e i t i s i n v a r i a n t .  the l o c a l d i s t r i b u t i o n d e n s i t y m a t r i x  P {~ L  ^IS  + ^PL)  We  hand s i d e o f  have thus shown t h a t  does not  contribute  (61)  d i r e c t l y t c the response energy current*  Expanding OJ(^) i n K -space  we may rewrite (iuU5) as  J-n  (U.U7)  < ^ A e  1  ' -  =  J. iT'E  e *[*vU [ dxT*\(>»&iw& itK  ,  fi  or  <. £> * . > =  (4.48)  t  ^•^J c6t'| c<ATAf/Prs £-*<r'A-t') t  / 3  6  As i n chapter 3 , we introduce the response function  (4.4?)  (+) = ffdxTA.  {peo &-*Co) u> c'C*'+C\)} t  which, by using the c y c l i c invariance property o f the trace and the d e f i n i t i o n of pg-$ , i s equal to the f a c t o r enclosed i n the brackets C...7  i n (4.48).  As before and for s i m i l a r reasons (see chapter 3,where mechanical external  eft disturbances are considered), i t s h a l l be assumed that  T  vJ^'ft \ C r y dies  o f f within some d e f i n i t e time T « (as a r e s u l t of the inherently d l s s i p a t i v e character of the many-particle system under consideration)*  On t h i s assumption,  equation (4.48) can be replaced for  (U.5Da)  t  »  with the i m p l i c i t r e s t r i c t i o n that only small values of f£ and *r' are present* Now, the question a r i s e s under what conditions equation (4*£l) which we have derived f o r an i s o l a t e d system with the i n i t i a l condition p - f> 0  u  can be used to describe the physical s i t u a t i o n i n the case of  an open system under a temperature gradient*  To answer t h i s question,  l e t us examine the p h y s i c a l s i t u a t i o n i n the case of an i s o l a t e d system  (62)  i n more d e t a i l *  (4.49)  We  vanishes f o r  have assumed above t h a t t h e response f u n c t i o n  X  satisfying  (4«S>0a).  This i m p l i e s t h a t the  F o u r i e r t r a n s f o r m o f t h e energy f l o w g i v e n by  (4-U8)  s t a t e v a l u e a f t e r a time l o n g compared w i t h T& •  attains a  steady  On t h e o t h e r h a n d  r  we assume, as u s u a l , t h a t i f t h e system i s l e f t t o I t s e l f the energy f l o w w i l l d i s a p p e a r a f t e r a c e r t a i n r e l a x a t i o n time w i l l come t o e q u i l i b r i u m . as f o l l o w s :  we  We  may  assume t h a t t h e r e e x i s t two  d i s t i n c t processes,  T  K  and  two  the l o n g e r  s a t i s f y i n g the c o n d i t i o n  "decay" o f t h e response  process.  (4.48)  one  which a r e c h a r a c t e r i z e d by  d i f f e r e n t r e l a x a t i o n t i m e s , t h e s h o r t e r one b e i n g  The  and t h e system  s t a t e t h i s assumption more c l e a r l y  m i c r o s c o p i c and t h e o t h e r macroscopic,  one T / / ,  T&  function  (U.49)  to a microscopic  On t h e o t h e r hand, the f a c t t h a t t h e energy flow g i v e n i n  d i e s o f f i s due t o a m a c r o s c o p i c  be extremely  p r o c e s s which i s assumed t o  slow f o r an i n f i n i t e l y l a r g e system.  assumption, t h e n t h e macroscopic region  i s due  I f we  accept  p r o c e s s t a k i n g p l a c e i n the  this  time  satisfying  (14.50b)  «  ± «  i s approximately  a steady s t a t e p r o c e s s under a temperature g r a d i e n t  which i s approximately L e t us now  Ti  constant.  c o n s i d e r an open system i n c o n t a c t w i t h  heat  r e s e r v o i r s which keep t h e system a t a c o n s t a n t temperature g r a d i e n t .  (63)  I f the system i s s u f f i c i e n t l y large, then the i n t e r a c t i o n between the system and the r e s e r v o i r may be regarded as a boundary condition and be omitted i n studying the process taking place i n the i n t e r i o r of t h e system.  Then, f o r the time s a t i s f y i n g (U.50b), we may assume that the  macroscopic physical s i t u a t i o n i n the i n t e r i o r of an open system i s the same as that taking place i n an i s o l a t e d system with the same, but "approximate l y  n  constant, temperature gradient, provided that t>he i n i t i a l  conditions o f the two systems are the same. There s t i l l remains a difference i n the i n i t i a l  conditions  since we have assumed that the i n i t i a l condition o f the i s o l a t e d system i s given by  p  0  - PL  > while the i n i t i a l density matrix of the  r e a l open system i s d i f f e r e n t from p  L  • The l a t t e r i s the density matrix  which describes the steady state under the influence of heat r e s e r v o i r s . However, since both i n i t i a l density matrices  s a t i s f y the same boundary  condition (U.21), they describe the same macroscopic state with respect to the energy d i s t r i b u t i o n .  Their difference i s of microscopic  scale and  w i l l change rapidly i n a time of order •£<> due t o the i n t e r n a l i n t e r a c t i o n s of the system.  Since, on the other hand, the macroscopic energy d i s t r i b u -  t i o n varies slowly i n time ( i n time of order  TR ), the i s o l a t e d system  which started from the i n i t i a l condition given i n (U.3U) w i l l a t t a i n the same steady state as the open system i n time of order  T  0  . Hence the  e f f e c t of the difference between the two i n i t i a l conditions w i l l die o f f i n time T  0  •  Thus one may disregard the difference of the i n i t i a l  conditions f o r time s u f f i c i e n t l y l a r g e r than To •  This whole argument,  which follows M o r l j W i l l give a q u a l i t a t i v e answer to our o r i g i n a l  (64)  question, although i t s v a l i d i t y must be checked f o r any p a r t i c u l a r case. Returning t o the discussion (4.51)> consider the operator uJ * when  K-0 j from ( 4 . 2 0 ) , we have  (4.52)  o =  where W (o)  J^^C^dA VJ(0)  i s the net constant energy flow operator independent of the  s p a t i a l coordinates.  Thus we may rewrite (4.51), f o r the case  £' ~0  as  (4.53)  < $<o)> = t  J -£  t£j3«  J" °°oU: J fclxTA.{p« «5-!S^) W ( t 4 c > ) } ,  0  i  The l a s t step makes use o f the following transformation, s i m i l a r to that given i n ( 4 . 2 6 ) , (4.54)  -i- H Sl  tc  U'0*&-Kto)  -  1 1 ^ f i ^ / 3 * e ' * * . A«<o>e^"*ck St K. ST. Js\ ~ ~ ~1  V/v Bin)  cU  (65)  where ^^cLk-Sl*  The l e f t hand side i s independent of the s p a t i a l  coordinates and; i s the response energy current due to the presence of a weak external temperature gradient*  We assume that the tempera-  ture gradient i s constant or equivalently, the l o c a l temperature varies l i n e a r l y with A as given i n  (2.10).  I f the system i s homogeneous and o f  regular cross section, a constant temperature gradient w i l l lead to a constant energy current.  Thus i t i s i n the steady state, that i s ,  a f t e r the i n i t i a l "transients," present when the external i s f i r s t applied, are "damped" out.  Assuming that  disturbance  T(A ) —  T  c  ••• a • A  with  (U.55)  then  (U.56)  S- V,v ftU) = T  Vn\  c  V,,  '  TCn)  Tc  L  ' c  ~  where the system i s "small" and therefore A  i s not too large.  From  —+  (U.56), i f  so Is  V/\ TCA)  Vn (3<4)  l a independent o f A then, to a good approximation,  * Coming back to (1**53)»  0  ft  (66)  where the last relation defines the tensor  f j ^  . In terms of  components, we have (iu*8>  - T cr^ djW  < vCU)> = t  with/*, V - 1, 2, 3» or x, y, z. If we make no restriction that the total electric current must be zero in the definition of the thermal conductivity (see section 2.1), then  (T ^ 14  is the thermal conductivity  tensor. By an exactly similar discussion, we can show that the electric current produced by the temperature gradient is (4.60)  < £ ( 0 ) >  with the tensor (4.61)  =  t  CT ^ (  -  T(A)  0-<» ^ $™ol*  JCO) =  The elements of the (4.63)  a-^l =  cr< > 3  given by J , f ih  T^{p  T(o)  and the net electric current (U.62)  .  e a  W  ro)  J  (t+tA)}  is  |  CT ^tensor are (3  J^d?  J*  Tr, {pe*  WV(O)J^CUL\)]  If we consider our system under both a constant electric field Ec and a constant temperature gradient V*. TCA.)  F  w e  c a n  ±te  VT  (67)  down the resultant electric and energy current, linear in the external disturbances, by combining (J+.60), (4.57)» (3.U3) and (4.16) (4.64)  <Jfo)> =  (4.65)  < W ( o »  E  +  +  -  .o-  (o)  —  f c • CT»>^o)  -  c  where CT (o) , <T (0) <T U)  (0  ( , )  %  *cr<3)  V*rc*).<T  and < T  t3)  ;  TU)  K )  w )  are explicitly given in  (3.42), (4.17), (4.63) and (4.58), respectively.- Remembering the usual definition of the thermal conductivity tensor, (4.66)  CW(0)>  +  =  -  %TU).  IC  with  u-^co)  <r ]  < J C  0  ) )  +  =  O  we can easily show that (4.67)  S  x=  -  L3)  or, in component form,  where  £ Cr (o)]~' = l,)  CT">(o)  We have thus "derived" a formally exact expression for the thermal conductivity tensor which is analogous to the expression for the electrical conductivity tensor, although slightly more complex since i t involves four tensors, each given by a Kubo-type formula.  (68) SECTION U.3 : CRITICAL REVIEW OF OTHER METHODS The e s s e n t i a l c r i t e r i o n used i n f i n d i n g the change  due to  a small temperature gradient was that a c e r t a i n response function i n v o l v i n g a time-dependent c o r r e l a t i o n f u n c t i o n must be n e g l i g i b l e a f t e r a time  r  £  while the f l u c t u a t i o n should have hardly changed, i . e . , i t s relaxation time is  „ i  it  In » where 7 R » T  k  .  have a "steady flow."  .  i  t  Thus when the time t i s such that Tn<z ~t <•<• TR. , we  This argument i s modeled a f t e r Nakajima's work (1958,  I960), though he i s s p e c i f i c a l l y interested i n f i n d i n g a Kubo-type formula f o r the d i f f u s i o n constant.  As mentioned before, Mori (1956, 1958, 1959)  has also given an analysis of i r r e v e r s i b l e processes caused by thermal disturbances which i s b a s i c a l l y the same as the one given i n section U.2* The discussion given by Kubo, lokota and Nakajima (1957b) i s also based on the study of the time evolution of an i s o l a t e d system which was i n i t i a l l y i n a non equilibrium state.  The system i s considered as macro-  s c o p i c a l l y defined by a set of variables (U.2).  In a d d i t i o n , Onsager's hypothesis  a priori.  (1931) i s assumed to be v a l i d -  We r e f e r to the assumption that the average regression o f a f l u c -  tuation follows the usual phenomenological laws.  Although formal  expressions  for the k i n e t i c c o e f f i c i e n t s involved i n electron transport phenomena are e x p l i c i t l y written down by Kubo e t a l i i , t h e i r d e r i v a t i o n can only be described as phenomenological.  The analysis makes e s s e n t i a l use of the  concepts and usual r e s u l t s of i r r e v e r s i b l e thermodynamics. Montroll has not yet published anything on thermal conductivity, but recently he has derived Kubo-type expressions f o r both the d i f f u s i o n constant and v i s c o s i t y .  Both of these c o e f f i c i e n t s are related to thermal  disturbances, as defined by Kubo, but Montroll (1959a) managed to apply Kubo's method of dealing with mechanical disturbances.  What he did was  to  (69) f i n d an appropriate boundary oondition which enabled him to express the i n t e r a c t i o n operator v interesting.  explicitly.  t  His discussion of v i s c o s i t y i s p a r t i c u l a r l y  S t a r t i n g with an i s o l a t e d system In the shape o f a cube with a  known t o t a l Hamiltonian  tl  a l t e r a t i o n o f the volume.  , Montroll a r t i f i c i a l l y introduced a time-dependent Without going i n t o any o f the d e t a i l s , we state  that he was able to show that the open system, c o n s i s t i n g of the system of i n t e r e s t and the boundary condition i n so f a r as i t effected the system of i n t e r e s t , could be described by means of a p r e c i s e l y given perturbation V^. on the Hamiltonian.  Montroll then applied Kubo's formalism, as outlined i n  chapter 3, i n a straightforward manner.  Although there i s considerable  algebraic manipulation i n v o l v i n g several canonical transformations of the o r i g i n a l coordinates and momenta of the many-particle system, Montroll's discussion i s p e r f e c t l y rigourous.  I t would be very i n t e r e s t i n g to see i f  t h i s type of a n a l y s i s , i n v o l v i n g a " t r i c k " boundary condition leading to an e x p l i c i t form f o r the perturbation operator  , could be extended to the  case of a temperature gradient and thermal conductivity. Another i n t e r e s t i n g approach follows from the work of Lebowitz and others set  (1957, 1958:  f u r t h e r references are given i n these papers) who have  themselves the task of e x p l i c i t l y constructing the stationary Oibbsian  ensembles which describe open systems i n the steady s t a t e .  In order to do t h i s ,  Lebowitz makes c e r t a i n assumptions as to the microscopic constituents of the d r i v i n g systems (heat reservoirs, p a r t i c l e reservoirs, e t c . ) .  Using these, he  i s able to f i n d the i n t e r a c t i o n between the many-particle system of i n t e r e s t and the d r i v i n g systems. for  Although most of the d e t a i l e d discussions so f a r are  c l a s s i c a l systems, the method can be generalized to deal with quantum  mechanical systems.  A very important r e s u l t of the d e t a i l e d c a l c u l a t i o n s given  so f a r i s that the stationary non equilibrium properties of the system do not depend c r i t i c a l l y  on the p a r t i c u l a r mechanism by which we assume the d r i v i n g  (70)  systems i n t e r a c t with the system of i n t e r e s t . For concreteness, we take the example of R heat reservoirs, the th  -r reservoir being a t the constant temperature  ' . A  Lebowitz assumes that  reservoir consists of an i n f i n i t e number of noninteracting components.  Prior  to the i n t e r a c t i o n with the many-particle system o f i n t e r e s t (henceforward to be denoted by S), the components of the / t *  n  r e s e r v o i r are assumed to have a T.  Maxwell-Boltzmann v e l o c i t y d i s t r i b u t i o n corresponding to the temperature  A  L a s t l y , each component i n t e r a c t s a t most once with S and the i n t e r a c t i o n i s purely impulsively. We s h a l l denote by yU(x/t) the c l a s s i c a l phase space d i s t r i b u t i o n function which describes an ensemble of S systems, *. standing f o r a l l the degrees of freedom of the ensemble.  In the absence of heat reservoirs and  other external driving systems, JULlx^-t) would be given by L i o u v i l l e ' s equation  £*  d/A  (4.69) where {...} of S.  l±)  = {  HCx^c^f)}  represents the Poisson bracket and HOx) i s the t o t a l Hamiltonian  Lebowitz has "shown" that, with the R reservoirs mentioned above, the  new equation of motion i s given by the stochastic modification of  In  (U.69)  (U.70) the function Kn($,X%8 assumed to be a d i f f e r e n t i a b l e function of TA.  but independent of T/i' A^/t. ;  I t i s defined i n such  a  way that lOiCjf.vOdxett:  i s the c o n d i t i o n a l p r o b a b i l i t y that S a t the phase space point *' w i l l i n t e r a c t with the  r e s e r v o i r i n the time i n t e r v a l  end at the point x., the change being given by c(.K . Shimony  di,andas a  result,  Recently, Lebowitz  and  (1961) have used Kubo's response method i n order to f i n d a Kubo-type  formula f o r thermal conductivity.  This approach has the merit of being  (71) direct i n that i t explicitly evaluates the interaction between S and the driving systems, but i t has the disadvantage that this evaluation involves many ad-hoc assumptions. Of course, Lebowltz's whole analysis i s b u i l t upon the hypothesis that, i n studying open systems in the steady state, the Interaction between the driving systems and S need not be specified In complete detail. In other words, several different interaction mechanisms may lead to the same stationary non equilibrium ensemble, at least to a f i r s t approximation. The last theory which we should like to discuss b r i e f l y i s the one given by Kllnger (196&C*) In his series of papers on Kubo-type expressions for kinetic coefficients and their evaluation. The generalized " s t a t i s t i c a l forces", —»  such as  V/y.  fi(A)  }  which Kllnger deals with are those used In irreversible  , thermodynamics. According to Kllnger, a l l attempts made so far In trying to f i n d formal expressions for the kinetic coefficients related to these statist i c a l forces lack generality. That Is,the results are only v a l i d for "a special type of relaxation process and time dependence of the response function". In contrast, Klinger's method i s supposed to be v a l i d i n a manner Independent of the frequency behaviour of the relaxation process. The great disadvantage of his analysis i s that i t Is based on the hypothesis that the s t a t i s t i c a l forces affect the density matrix p^of the system of Interest in exactly the same way as the mechanical disturbances do. For mechanical disturbances or forces, wa know the Hamiltonian of the system i n the mechanical force f i e l d . As a result, we need only solve for p In the, linear approximat  tion. Statistical forces cannot be described by a definite Hamiltonian. To' do so, we would have to deal with a larger system including the driving systems. Since this has not been successfully done so far (see the remarks on Lebowltz's work), Kllnger makes a "guess" as to the correct solution. These remarks should become clearer as Klinger's method i s outlined. We  do  (72) not discuss the problem In the same generality as Kllnger does. The density matrix p of the perturbed system o f Interest Is given t  by  f  (U.71)  ^«-=-f. + ?;  where p Is the equilibrium density matrix^ 7/v f>» - ' ) , o  (U.72) p = e ^ ' ^ ^ 4  -  0  /3JV + \ J n ( A ) o ^ _ (i J £ ( A ) o U  e  We are working with a grand canonical ensemble and  ju. - Sj^ Is the chemical  p o t e n t i a l of the system i n equilibrium. How we consider the s p e c i a l case of an electric field, (4.73)  E(A,t) = - % $ ( A , t )  where A and t stand f o r f u n c t i o n a l dependence on the p o s i t i o n and time. I t i s assumed that  (4.74) with  #(4,-0 = $•  l<5$|  +  s u f f i c i e n t l y small and <$ a constant. The i n t e r a c t i o n V, between D  the system and the external d r i v i n g system i s known i n t h i s case,  (4.75)  V  T  j ^ e " U ) 6§<A/OC6I  =  where €> , J i and nd) are the charge of the electron, the t o t a l volume o f the system o f i n t e r e s t and the l o c a l number density operator, r e s p e c t i v e l y . By a c a l c u l a t i o n s i m i l a r t o that given i n (4.22),  (4.76)  v  t  = J-  can be expressed i n K * s p a c e as  £en*8$- (t) K  In the Heisenberg representation (with respect t o the Internal Hamiltonian ft), we have  (H.77)  v w=iLen" (t)^a). t  "  K  Now the equation of motion which we must solve i s , i n the Heisenberg representation,  0.78)  ,  t£k*)  a  iCPk(t),V,W]  (73) The linear approximation of (4.78) Is (4.79)  ^ t )  dt  If we use  =  t  T p „ V  .  t  t t ) ]  .  ^ ~ ' n«(.t)= i CK, n^Ct)j  ,(^.77),  and Kubo's  identity as given i n (3.18), ve can rewrite (4.79) as  (4.80)  <¥±M dt  -£fEL^e^(t-iA)(5i,W.  =  *  We shall not go into details, but i f we solve (4.80) in a straightforward manner, we can use the solution to get the usual Kubo-type formula for elect r i c a l conductivity. It Is important to note that the result (4.80) depends on the knowledge of V* given in (4.76). We now make a small digression. We assume that the electric f i e l d  E(A t)  Is given by  f  (4.81)  where  B(A,t) = - %  yU(A,i:)  <$U,t) - £  and yU (A,+) e  are known as the generalized chemical  potential and electrochemical potential, respectively. In addition, we assume that (4.82)  = yU*  jU.*(A. t) t  +  SjU, (A,t) e  whereyClf i s a constant and /  i s sufficiently small. KLinger then  introduces the operator p by the following definition o  (4.83)  p  6  =  e  /SA'r^Jci^jofA  ^//t^A.t-)  We note that (4.84)  J fy*(*,f) in(A)oU_= ^ Z  ri* o > £ f t )  Now using (4.84) and substituting (4.82) into (4.83), we may expand respect to the Fourier components  with  S^U-Kft)hy means of the usual Schwinger  formula for an ordered exponential. The result i s , i n the linear approximation Po - po +• $p  }  where  (7»0  In the Heisenberg representation, we have  (4.86)  o>(+) =  ,  £ 5 1 e n \ ( t - a ) ^ ( t ) .  Now, d i f f e r e n t i a t i n g (4.86) with respect t o time and keeping  c£a-*(t) cons-  tant,  (4.87) \#  *P(lA  The r i g h t hand sidesof (4.87) and (4.80) are the same except f o r the signs, or  '  (  (4.88)  *r 6  C t )  ) ,  m  In the case of a s t a t i s t i c a l force  (3(Hjt)  (fc.89) where  p  =(3+ )  {3(A)  , we assume  tipC*,*)  i s the equilibrium temperature, and I  T  small. We can e a s i l y show that i f (1 i n p  0  l a replaced by  I i s sufficiently  ( 3 ( * , T ) a n d ^ Is kept  constant, we have  (4.90)  Spit) = -_g ^L^  X  &<*-i*> <5/3^«)(^)  Now Kllnger makes the fundamental assumption that (4.88) again holds, that Is  (4.91) (dtpW)  = _  Using (4.91) and (4.90), Kllnger was able t o f i n d a Kubo-type formula f o r thermal conductivity.  (75) CHAPTER 5 EVALUATION OF KUBO-TYPE EXPRESSIONS. SECTION 5.1: RELATION TO VAN HOVE'S MASTER EQUATIONS AND THE REDUCTION TO' SINGLE PARTICLE EXPRESSIONS. Consider a many-particle system under the influence of several weak external disturbances; experimentally, i t i s w e l l known that various transport processes w i l l a r i s e . Some of the disturbances may involve "mechanical'* forces and, as we have seen i n chapter 3, there Is no fundamental d i f f i c u l t y i n f i n d i n g the l i n e a r "response" currents due t o t h i s "type of force i n terms of transport c o e f f i c i e n t s given by Kubo-type expressions. Others may be "thermal" o r " s t a t i s t i c a l " disturbances and f o r these, the above problem i s not so easy. In chapter 4, we discussed the e f f e c t of a constant temperature gradient i n some d e t a i l , and derived a Kubo-type formula f o r thermal conduct i v i t y . Although there are s t i l l some d i f f i c u l t i e s In the proof, t h i s and other discussions i n the l i t e r a t u r e seem to give a good Indication that we can express thermal conductivity i n the form of a Kubo-type expression. In t h i s chapter, we s h a l l leave a l l doubts aside and assume that the r e s u l t given i n (4.68),  OM), (4.1.9), (4.63)  and (4.58) i s c o r r e c t .  Speaking generally, the r e s u l t s of the work over the l a s t few years seem to Indicate the f o l l o w i n g : I f a weak external "force",, Pit)  ~ Fc e"^*"  , acts upon an ensemble of systems characterized by  the temperature T, then the. l i n e a r "response" o f current operator 8 i s given by  (5.1)  < | \ =  %-  tU  ^  8  ^  )  where the transport c o e f f i c i e n t i s formally expressed as  (5.2)  cr  C 8  M = ff  e ^ f ^ x T ^ f e - ^  ceo  ejoe^^}  ( 76 ) *  = Tolt  J?^  e  iu)ir  A  ^ £  Co)  |(-t  +  ^<a  a)  The operator £ depends on the particular process we have in mind but i n general i t i s the observable corresponding to the macroscopic flow most directly related to the force Rf)(for example, i f c  i if  tr 7  (itnjt)  R*> =  , then  FJ- >= +  C = §  )  , then . The next problem  i s to evaluate the right hand side of (5.2), the basic element being the correlation function  < C(o) B(<-+<A»  r(S  . This part of the subject  is at present i n a rapid stage of development and only a few remarks w i l l be made i n this section. In general, the transport coefficients given by (5.2) (or "kinetic" coefficients i f the generalized "forces" r(t)  have been chosen so as to lead  to Onsager's reciprocity'relations) are complex quantities. Following KUnger (I961), we may show that (5.3)  ReaKCTcoM) =  e >o  where 7l,-r\*-=  E-iio  #  y  e  refer to the end of chapter 3 for the significance - £+  of the convergence factor  G  .It lends generality to our discussion, but  i t i s somewhat a r t i f i c i a l and in certain cases, redundant. It i s sufficient to find  Real  ((Tce(UJ))  , since we can easily derive the following disper-  sion relation r  (5.4)  Re«Tc,<u>))=  -Lj  00  O^U)' ( J ^ (cTca^))}  (77) where we have taken the p r i n c i p a l part o f the Integral. K e CO"colw))ls often r e f e r r e d t o as the d i s s i p a t i v e part of OCBCU))  the transport c o e f f i c i e n t  (5.5)  since  Jbsvrx Re C<r (u>)) ca  while  (5.6)  A^rt  I/^((T (u))) = 0 C8  I(rvi( °ca )) (u)  *  8 o  f  t  e  n  r e f e r r e d t o as the dispersive "part o f the trans-  port c o e f f i c i e n t . As we mentioned i n chapter 3$ i t i s often convenient t o introduce a quantity  (5.7)  £ca (•<*>)  C<r 8tu>)) C  = -  w  *  e  such that U c e C ^ ) )  I f we had external magnetic f i e l d s , i t would also be advantageous t o s p l i t the transport c o e f f i c i e n t , as given i n (5.2), into .symmetric and antisymmet r i c parts (with respect t o a r e v e r s a l of the direction.of the magnetic f i e l d } . We refer' t o the a r t i c l e s by Kubo  (1957a) and KLinger (196l) f o r  f u r t h e r formal r e l a t i o n s between the r e a l and Imaginary, symmetric and a n t i symmetric, components o f the formal Kubo-type expressions.  I t may be mention-  ed that Onsager's r e c i p r o c i t y r e l a t i o n s can be derived d i r e c t l y . The e x p l i c i t evaluation o f the r i g h t hand side o f (5.2) involves a trace over the product of an equilibrium density matrix and two operators, and a time i n t e g r a l . As we mentioned before, Van Hove's master equations provide one o f the most u s e f u l methods o f evaluation, e s p e c i a l l y f o r the case of weak s c a t t e r i n g . I t should be emphasized that Kubo-type formulas are, i n p r i n c i p l e , v a l i d f o r strong as w e l l as weak s c a t t e r i n g o f the "current c a r r i e r s " . Indeed, t h i s i s one o f t h e i r main advantages over the standard transport equation method which i s , s t r i c t l y speaking, only v a l i d f o r weakly interacting excitations.  (78) We shall restrict ourselves to the type of system which Van Hove worked with (see chapter 1 for details). For convenience we use the notation in Van Hove's f i r s t paper (1955). Our basic representation i s denoted by  lev*>  where  (5.8)  ft,  =  ElEuy  and we choose the normalization  (5.9)  <Fo(/ek'> =  ^foc-oC)  &LE-E  As (5.9) implies, we assume that E andoC act as continuous variables i n the limit of a large system ( N -*>oo> si^oo E  such that the density  i s constant). A l l our operators are written i n the second quantized  representation. In the long time, weak coupling approximation, the diagonal singularity can be written as  vAvlewy  (5.10)  = 6(e-E')j(o(-ee)  where A i s any diagonal operator in the ( £ o ( | v f f o(> r O  (5.11)  A  •+  <E*  I X*  IE'°<'> .  IE»-representation and  . Lastly, we assume =  N(E°0  <^EU  I NlEUy  <  l cje'oiy = c(rot)  EU  w (eoe)  6(E-E')f(ol-iX')  6(E-  e')S(ot-of)  where C(eor) and Btew)are "smooth" functions of E , the unperturbed energy. This last condition on the eigenvalues i s essential i f we are to use Van Hove's mathematical technique based on the use of the diagonal singularity. Now,  in the weak coupling, long time approximation, Van Hove has  shown that,  (5.12)  '  C(t)/Fi6= =  < F ^ | U C - i ) C J ( t ) / £F'<*• >  §c(E.ot,)  <e«Iu(--t)iE\O{,><E,O<,Iu(VIE'u'>d°t,ctE>  (79)  where  (5.13)  P+ ( £"  Eoi)  ^ £ ^ )  ±  ±  =  2  the s o l u t i o n of the "master equation**,  8  ^ , L [  w  (  £  ^  f  o  /  i  )  p  e*;  <  oft"  WYEc^Fo/i) =  Eoi) = 6(o(,-ot)  P (Eot,;  with the i n i t i a l condition  0  wrE^jFoi,)^  l<E°f-1 v l F o O l  .in  (5.13),  , t h e .second - t  1  equality following from the Hermitian character o f V . A s i m i l a r r e s u l t holds f o r  ^ l B ( t ) l £ W ' > I f we expand  Re  given i n (5.5)  C °cs/u>))  i n the /e<>0-repre-  sentation and make use of (5*15), we have  (5.1*0  £ e ( O-  c0  <u>»  •  * [ 8(£V) c f ( f ' - e > o W - o / ) J c t « " C ( ^ ^ " )  P L£oi"-Eu)  r  t  + cce« c ) ce '-£)d"c^ -rt)Joto<" gee*-) P c  : < 0  r  ,  HEp(u)  X j B(E<X) Jclo(»C(E*") -f C f E * ) J " G ^ » 6 ^ " )  P+tBd';?<*) P (EOC;EOL) ±  j  t  ( ^ " , E o < )  j  (80)  where  E C u O = ^ <J^rJa  Z = T/i { e  and  f l  ft/U  ".  I t should be noted that ve are using a grand p a r t i t i o n f u n c t i o n here... '  ;  -  Klinger  J:  (1961) formally  s i m p l i f i e s the r i g h t hand side of '(5.1*0 by introducing a resolvent operat o r j the r e s u l t i n g expression i s very compact and q u i t e u s e f u l i n making c a l c u l a t i o n s f o r s p e c i f i c models. As v e noted before, the dispersive part of  0~c eCvo)  can be found from the d l s s i p a t i v e part by means o f the disper-  sion r e l a t i o n  (5.15)  Lm((TcB<uO) = ~{°° KeCCTcsCu;)) . " '-co co-to' We return t o the s p e c i f i c case of thermoelectric transport c o e f f i -  c i e n t s . In chapters 3 and k, ve have derived the following Kubo-type formulas  ( s t a t i c ca.se,u~>=o) (5.16)  cr^>  (5.17)  a-JV -  =  J7 dt^dX  (5.18)  ^ = { *ot±  (5.19)  0>iN ["ctt^dX  o  TA{PEQ Jy(o) w^u db. Trv [ p  £Q  TrulpBQ  VV„(o) J^iUi  + n)}  A)j  Wvfo)W (t+tA)l A  As we mentioned before, a l l of these involve a double i n t e g r a t i o n . Verboven  (1961) has shown that  can be reduced so as t o involve a s i n g l e time  i n t e g r a t i o n by making use of the d l s s i p a t i v e properties of the many-particle system. His proof can be e a s i l y generalized t o  0^/ ° j 0^l' y  y  a n d cr^i^  As a r e s u l t , we may rewrite (5.8) - (5.H) i n the f o l l o w i n g form,  (5.16) '  <rj» = (3 j ^ c t f  (5.17) '  (7^=  (5.18) '  ft  TA { p  f o ) 5^  Jy  ft)}  T A { fa S (o) VV^f-O}  J°°ca T A { ^ o  ESl  v  V\Uo)£ft>}  (81) (5.19) '  J"ca T/tfpec W (o)vv>(:t)}  cj>^ = }  y  A l l of these formal expressions are many-particle formulas i n T  -space".  As Chester and Thellung (1959) have pointed out and Van Hove (see Verboven A  (196l) ) has proved, i f the total Hamiltonian # i s the sum of single particle Hamlltonians Wj (where <y refers to t h e ^ (5.20)  1  particle), then  -  where H Is the single particle Hamiltonian,  £ (+•)  i s the single particle  electric current operator In the Heisenberg representation (with respect to H) (H) i s the s t a t i s t i c a l distribution function for the particles when they are i n equilibrium. In the theory of metals, we are usually dealing with electrons which, to a sufficient approximation, are Independent. Electrons obey Ferml-Dlrac statistics and consequently (5.21)  J(fj) =  '  -  where// i s the chemical potential. As a last remark, "  stands for the trace  over any complete set of one particle wave functions. In section 5.2, we assume that the energy i s carried only by the electrons, and take as our energy current operator  (5.22) where  w /vn  f 0 )  £  -  W(o)  CPiJVjv)  , /y>i* , and  are the mass, "effective" mass and the momen-  •th turn operator of the ^ ;  CT y >t  :J)  =  (5.23) V*  electron. With this definition of W,  (TMV^  and  ^°  t n e  we can reduce  following single particle formulas  - M ^ J > ' 4 l ^  (5.24)  0>»> = _ t \ {  (5.25)  <£„ «  '4L^(0|v(6)+  -1* \ $JP£<& '/zOpit)  ' ^ ( » j v W ] .  6  w)  j (0) + y  j C0) M  fy(t)l(B.  (82)  by a calculation similar to that given by Van Hove for  G^tv i n  (5.20).  In  Klinger (I96I) points out that the reduction to single particle formulas i s valid for elastic and inelastic scattering when there i s no Fermi degeneracy. If the electron gas Is degenerate, then the reduction i s possible only for the case when the electrons are scattered elastically by the phonons, Impurities, etc.  SECTION 5.21 EXPLICIT EVALUATION OF THERMAL CORDICTIVITY FOR ELECTRONS SCATTERED BY IMPURITIES. The papers of Chester and Thellung have evaluated  (fjj^  given in  (5.20)  (1959)  and Verboven  (I961)  for the simple model of electrons  elastically scattered by a large number of randomly distributed static Impurit i e s . The single particle Hamiltonian with which they deal i s given by (5.26)  H = W«+AV  where Wo i s the Hamiltonian of the conduction electrons moving i n the periodic potential of the lattice and. V i s the scattering potential for one electron due to the static impurities. The basic representation i s denoted by Ut >  where < i s the wave vector of the electron. £  i s assumed to be diagonal  in this representation, i.e., we limit ourselves to a single energy band. The case of impurity scattering i s very special i n that ve may simple consider one electron interacting with a large number of static impurities which are assumed never to act as "carriers". Thus Van Hove's work in V -space" immen  diately reduces to equivalent results in "/i -space". In contrast, although we may deal with an electron-phonon system composed of a single electron weakly interacting with a large number of phonons, a complicated summation  (83) ' must be made In order that the master equation f o r the e n t i r e system i s reduced to a s t a t i s t i c a l equation involving the degrees of freedom of the single electron. The above authors both found correction terms to the s t a t i c e l e c t r i c a l conductivity tensor evaluated by means of Van Hove's lowest order master equation f o r a s i n g l e electron. From the order of magnitude of these corrections, they concluded that the r e s u l t s were correct as long as  £ <cr(  where t i s some c h a r a c t e r i s t i c " c o l l i s i o n time" and Y[ i s of the order of magnitude of the Fermi energy. Chester and Thellung imply that they have managed to v e r i f y Landau's conjecture because the single p a r t i c l e Kubo formula (3*20) separates the s t a t i s t i c a l factors such as a l factors such as of  $ (+)  It should also be  evaluation i m p l i c i t l y uses ,the f a c t that  Jj£ and  the dynamic*  mentioned tha[t t h e i r method  '  I^(o)jE  in (5.28)  <A'lj fr)lA> M  =  <e'«'lj (t)\Eot> A  i s an appropriately slowly varying function of H , Using (5.22) f o r our energy current operator, we may derive the following lowest order r e s u l t s f o r spherical symmetric impurities,  (5.29) cr^p  =  _ £  K (5.30) c r ^ c -pZI  (5.31)  dl t K l f r i K X t c i f a t c y v t E j dE  0  w K  ^<.KlpK><xij tlc>T(eK) v  yJZ U «lf\\KXKlf lK> v  t(e )  using (5.23), (5.24) and (5.25). In these formulas ,  K  .  m  (5.32)  where  ~  =  *  ¥7T  J*'(l-CoQsO) W(E,a&&)  (>(E) i a some appropriate density function and  dCcao^e)  W ( E , coa-e) i s  proportional to the t r a n s i t i o n p r o b a b i l i t y between the single p a r t i c l e , states (energy E") l a b e l l e d by the wave vectors and  0 the angle between the two vectors.  £  and K' , with  =  For further d e t a i l s oh the  c a l c u l a t i o n and notation, we r e f e r t o page 753 of Chester and Thellung  (1959). In this section, we have simply applied the c a l c u l a t i o n s made by Chester and Thellung to the lowest order evaluation of the thermal conductivity tensor.  That we may do so i s a r e s u l t of our p a r t i c u l a r  choice of an energy current operator, i . e . , the one defined i n (5.22). . I f we chose the energy current operator given i n ( H given by  (4.3), with  = &  (5.26)), then the c a l c u l a t i o n would not carry over so simply.  (85) BIBLIOGRAPHY  BERNARD,. W. and CALLEN, H.B., 1959, Rev. Mod. Phys., 31, 1017. CALLEN, H.B. and WELTON, T., 1951, Phys. Rev., 83, 3^. CARRUTHERS, P., I 9 6 I , Revs. Mod. Phys., 33, 92. CAS3MIR, H.B.G., 19^5, Rev. Mod. Phys., 3J, 3^3. CHESTER, G.V. and THELLUNGi A., 1959, Proc. Phys. Soc., 73, 7^5. DE GROOT, S.R., 1952, "Thermodynamics of I r r e v e r s i b l e Processes", NorthHolland Publ. Comp., Amsterdam. DRESDEN, M., I96I, Revs. Mod. Phys., 33, 265. FARQUHAR, I.E., I 9 6 I , Nature, 190, 17. GREEN, M.S., 195 *, J . Chem. Phys., 22, 398. 1  KARPLUS, R. and SCHWlNGERi.;, J . , 19^8, Phys. Rev., 73, 1020. KIRKWOOD, J.G., 19^6, J . Chem. Phys., 14, 180. and FITTS, D., i960, Ibid, 33, 1317. ' KLINGER, V.I., 1960a, Soviet Physics - S o l i d State, 1, 613.  1960b, Ibid, 1, 782. 1960c, Ibid, 1, 1122. 1960d, Ibid, 1, 12 69. I961,  Ibid, 2, 2747.  KOHN, W. and LUTTINGER, J.M., 1957, Phys. Rev., 108,.590. KUBO, R., 1956, Can. J . Phys., 3^, 127^. t  1957a, J . Phys. Soc. Japan, 12, 570.  , YOKOTA, M. and NAKAJIMA, S., 1957b, Ibid, 32, 1203. LANDAU, L.D. and LIFSHITZ, E.M., " S t a t i s t i c a l Physics", Pergamon Press, London,1958. LAX, M., 1958, Phys. Rev., 109, 1921. LEBOWITZ, J.L. and BERGMANN, P.G., 1957, Ann. Phys., 1, f  1.  I959, Phys. Rev., Ilk, 1192. and SHBDNY,  ,  I96I,  Preprint.  MARTIN, P.C. and SCHWINGER, J . , 1959, Phys. Rev., 115, 131*2.  (86)  MATSUBAIRA, N., I 9 6 I , Prog. Theor. Phys., 25, 1 5 3 MONTROLL, E.W., 1959a,  P  Suppl. t o I I Nuovo Cimento,  "Thermodynamic a del Process! I r r e v e r s i b i l i " , Varenna Summer School (1959) Lectures. and W&KB, J.C., 1959b, Physica, 2 5 , 4 2 3 . MORI, H., —  I956, J . Phys. Soc. Japan, Lj., 1 0 2 9 .  , 1958, Phys. Rev., 112, 1 8 2 9 . , 1959, Ibid, 115, 2 9 8 .  NAKAJIMA, S., 1956, Proc. Phys. Soc. A, 69, 441.  , I958, Prog. Theor. Phys., 2 0 , 948. , 1959a,Ibid, 2 1 , 6 5 9 . _  f  1959b,Suppl. t o I I Nuovo Cimento, "Thermodynamica del Process! I r r e v e r s i b i l i " , Varenna Summer School (1959) Lectures,  NAKANO, S., 1 9 5 6 , Prog. Theor. Phys., 15, 77* ONSAGER, L, 1 9 3 1 , Phys. Rev., 3 7 , 4 0 5 . PRIGOGINE, I., BALESCU, R., RENIN, F. and RESIBOIS, P., i 9 6 0 , Suppl. t o Physica, "Proceedings of the International Congress on Many-Particle Problems", Utrecht. VAN HOVE, L., 1955, Physica, 2 1 , 517. , 1957, Ibid, 23, 441. , 1958, "Selected Topics i n the Quantum S t a t i s t i c s of Interacting P a r t i c l e s " , Lecture Notes, University of Washington, Seattle. , 1959, Physica, 2 5 , 2 6 8 . , i 9 6 0 , "The Theory of Neutral and Ionized Gases", Les Houehes Summer School ( i 9 6 0 ) Lectures; J . Wiley and Sons, New York. , and VffiffiOVEN, E., 1 9 6 l , Physica, 27, 4 l 8 . VERBOVEN, E., 1961, Physica, 26, 1 0 9 1 . WILSON, A.H., 1 9 5 3 , "The Theory of Metals", 2 n d Ed., University Press, Cambridge.  (8 ) 7  ZIMAN, J.M., I960, "Electrons and Phonons", Clarendon Press, Oxford. ZUBAREV, D.N.,  i960, Soviet Physics - Uspechi, 3,  

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