Variational principles of irreversible processes

VARIATIONAL PRINCIPLES OF IRREVERSIBLE PROCESSES

Masakazu ICHIYANAGI Department of Mathematical Science, Gifu University of Economics, Ohgaki, G~fU503, Japan

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Physics Reports 243 (1994) 125—182

Variational principles of irreversible processes Masakazu Ichiyanagi Department of Mathematical Science, Gill, University of Economics, Ohgaki, Gifu 503, Japan Received November 1993; editor 1. Procaccia Contents I. Introduction 2. The linear irreversible thermodynamics 2.1. The linear phenomenological laws 2.2. The Le Chatelier-Braun principle 3. Variational principles of irreversible processes 3.1. The variational principle of Onsager’s type 3.2. The variational principle of Prigogine’s type 3.3. The relation between Onsager’s and Prigogine’s principle 3.4. Hamilton’s principle 4. The general evolution criterion 5. Statistical significance of dissipation functions 6. Variational principle in Boltzmann equations 6.1. The Kohler—Sondheimer—Umeda variational principle 6.2. Conduction problems in a magnetic field 6.3. High-frequency transport problems 6.4. Upper and lower bounds on transport coefficients 6.5. Relation between the variational principles in the Boltzmann and the Onsager theories of transport processes 7. Variational principles in quantum theories of transport processes

128 132 132 134 135 135 137 139 139 142 144 148 148 151 152 154

156

7.1. Nakano’s variational principle 7.2. Reduction from the quantum to the classical variational principles 7.3. Contraction of information 8. The variational principles for dynamical susceptibilities 9. The variational principle for dynamical structure function 9.1. The variational principle in the Boltzmann equation 9.2. The variational principle for the Bethe Salpeter equation 9.3. Relation between the variational principles in the coupled electron phonon system 10. Concluding remarks Appendix A. A note on the Djukic—Vujanovic theory Appendix B. Analysis of the Onsager—Machlup theory Appendix C. Time-reversal problem Appendix D. The relative entropy formula References

159 163 164 166 167 168 169

172 174 175 176 177 179 180

159

Abstract This article reviews developments of variational principles in the study of irreversible processes during the past three decades or so. The variational principles we consider here are related to entropy production. The purpose of this article is to explicate that we can formulate a variational principle which relates the transport coefficients to microscopic dynamics of fluctuations. The quantum variational principle restricts the nonequilibrium density matrix to a class conforming to 0370-l573/94/$26.00 © 1994 Elsevier Science By. All rights reserved. SSDI 0370-1573(94)00005-N

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the requirement demanded by the second law of thermodynamics. There are various kinds of variational principles according to different stages of a macroscopic system. The three stages are known, which are dynamical, kinetic, and thermodynamical stages. The relationships among these variational principles are discussed from the point of view of the contraction of information about irrelevant components. Nakano’s variational principle has close similarity to the Lippmann—Schwinger theory of scattering, in which some incoming and outgoing disturbances have to be considered in a pair. It is also shown that the variational principle of Onsager’s type can be reformulated in the form of Hamilton’s principle if a generalization of Hamilton’s principle proposed by Djukic and Vujanovic is used. A variational principle in the diagrammatic method is also reviewed, which utilizes the generalized Ward—Takahashi relations.

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I. Introduction Variational principles have been used in many branches of theoretical physics as a tool for calculation and sometimes to elucidate the nature of a problem. The second law of thermodynamics is an example of a law expressed in the form of a variational principle. Another famous example is the principle of the least dissipation of energy proposed by Onsager [1]. In the present paper the variational principles, which are the most useful tools in the study of irreversible processes, will be treated and discussed from a thermodynamical and statistical mechanical point of view. The reason why the variational principles are reviewed here is as follows. In the first place, the laws expressed in the form of variational principles are themselves important, as can be seen from the importance of d’Alembert’s and Hamilton’s principles in analytical mechanics. With regard to this, we note that Onsager’s principle, in which one considers the two fundamental quantities, the complete variation of the dissipation function and a differential form, is analogous to d’Alembert’s principle. Here, the differential form is not reducible to the variation of a scalar function. It is well known that d’Alembert’s principle can be mathematically reformulated as Hamilton’s principle which requires that a definite time integral of the Lagrangian shall be stationary. Accordingly, the following question inevitably arises: Is there an analogous transformation within the realm of irreversible thermodynamics? This question has been the subject of a number of papers and at least one review article to which the reader is directed for further references [2]. The unifying quality of a variational principle is truly remarkable. The idea of deriving the basic equations of nature from the variational principle has never been abandoned. They are derivable from the “principle of least action”. It is only the Lagrangian which has to be defined in a different manner. Quite similarly, the fundamental equations of irreversible processes can be embraced with a single variational principle, the principle of the least dissipation of energy [1]. It is only the dissipation function which has to be defined in each case. The primary purpose there is to develop a variational principle, which would provide an intrinsic relationship between the two concepts, microscopic dynamics and transport coefficient. For this paper, three known variational principles are relevant. The first one is Onsager’s principle which belongs to the thermodynamical stage. This type of variational principle has been reviewed by Ono [3]. The second one is related to the Boltzmann equation of transport theory which was first used by Umeda and coworkers [4] to justify Kroll’s treatment of transport coefficients [5, 6]. This type of variational principle belongs to the kinetic stage. The last one, which concerns the dynamical stage of irreversible processes, is a principle in the quantum theory of irreversible processes proposed first by Nakano [7]. In these treatments, we have to consider separately the two equal and opposite contributions to the entropy production, from a drift and collisions, respectively. Then, it is not difficult to see that an intrinsic entropy production, which comes from collisions, can be expressed in terms of the dissipation function which is quadratic with respect to currents, while an extrinsic component is equal to the Joule heat generated in the system by the external perturbation [8, 9]. The dissipation functions are classified according to the stages of temporal change of entropy of the considered system, by which a dynamical basis of irreversibility may be elucidated. We stress the second reason for the present article. In developing variational principles of irreversible processes, one combines the use of the forward and backward equations of motion. One

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may summarize that statistical mechanics aims to relate the macroscopic properties of matter to microscopic dynamics. The microscopic dynamics in fact governs the motion of molecules constituting matter. However, it governs motion less straightforwardly in statistical mechanics than in classical mechanics. From this point of view, it seems that thermodynamics is only a phenomenological theory and cannot be derived from the first principles of molecular dynamics. As argued by Penrose [10], the microscopic dynamics by itself is not enough to characterize general nonequilibrium processes. Instead, we have to use an ensemble, in which reversed motions we consider have much lower probabilities than the original ones. This is the real distinction between past and future. In order to discuss this situation in more detail, we have to sharpen our notion of a macroscopic description of irreversible processes. We first note that there are two distinct ways of describing irreversible processes. One way is to focus our attention on the differential equations of motion for macroscopic variables, e.g., on the Langevin equations which can be taken as the basis of the macroscopic description. An alternative way of thought concerns the phenomenological laws which relate thermodynamical forces with macroscopic currents defining measurable transport coefficients. The phenomenological laws express the content of causality, in a sense that the thermodynamical forces are seen as causing the corresponding currents. It is the second way of thought which we are going to discuss in detail in this article. We will arrive at the typical variational principle in the dynamical stage, which concerns the microscopic equations of motion in the respective cases that some incoming and outgoing external fields are applied to the considered system. In consequence, we have to use two kinds of variational functions in the variational expression necessary in a pair [7]. This outstanding feature is similar to the occurrence of incoming and outgoing waves in the Lippmann—Schwinger theory [11, 12]. A little thought shows that this is connected, with the combined use of probability and microscopic dynamics which creates a difference between past and future. The present article is devoted to the review of variational principles of irreversible processes providing the relations between these principles. Although many of the variational principles have been studied extensively, the present study will introduce additional clarity, as well as provide some new insights. We will discuss their physical interpretation, in particular, their relation to entropy production. In Section 3, we will give a review of the variational principles for linear irreversible processes which admit the phenomenological laws. The variational functional will be related to entropy production. After having reviewed the two types of variational principle of irreversible thermodynamics, we shall concentrate in Sections 3 and 4 upon a particular form of variational principle. It will be shown that the variational principle of Onsager’s type can be reformulated if the generalization of Hamilton’s principle proposed by Djukic and Vujanovic [13] is used. In this type of variational principle, the Lagrangian functional is related to the entropy, and the linear phenomenological laws are equivalent to the Euler—Lagrange equations in mechanics [14]. In order to give a general description of irreversible processes, we shall divide the systems of interest into two classes: discontinuous and continuous. The name “discontinuous” is given to a system whose physical properties are not continuous functions of the spatial coordinates. In this paper, we direct our attention to discontinuous systems. An integral principle of irreversible processes taking place in a continuous system has been proposed by Gyarmati [15]. He introduces F variables which are intensive functions conjugate to extensive quantities. Thermodynamical forces are given by their spatial gradients. Then, a Lagrangian density is expressed in terms of T~,

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VT’,, J, (currents), etc., from which we obtain the local balance equations for the extensive quantities as well as the phenomenological laws as the Euler—Lagrange equations. Since we want to discuss Gyarmati’s principle separately, we shall not dwell on it here. After having established the relation between the two types of variational principle belonging to the thermodynamical stage, a short summary will be given in Section 4 of the general evolution criterion proposed by Glansdorff and Prigogine [16]. Despite its significance, this principle has not been stated precisely and generally nor has its validity been demonstrated in general. As a consequence, some confusion surrounds it, such as the doubt about whether it is a new law of nature or a consequence of known laws. It will be shown that this principle, when it applies, is a consequence of known laws. The major advance in recent years towards an understanding of irreversibility is the discovery that linear irreversible processes can be formulated in terms of fluctuations about a thermal equilibrium state. The formula of Boltzmann, S = kB log W, defined the macroscopic entropy in terms of microscopic probabilities. Einstein [17] proposed the interpretation of Boltzmann’s formula which allows the calculation of microscopic probability from thermodynamics. Inspired by this formula, Hashitsume [18] and Onsager and Machlup [19] have given a very suggestive reformulation of linear fluctuation theory about equilibrium. They have derived a formula for the joint probability density, which is expressed in terms of the entropy production as well as the dissipation functions. In consequence, their formula tells the probability of temporal succession of states in terms of the entropy production and dissipation functions, and provides a dynamical variational principle (principle of least dissipation) for the most probable states. In Section 5, the statistical significance of the dissipation functions will be discussed by solving the Fokker—Planck equation for the joint probability function. The relationship is analyzed between the two variational principles, the stationary principle of dynamics and the extremum principle of the irreversible thermodynamics. An analysis of the Onsager—Machlup theory is presented for extracting its characteristic of the Gaussian Markoff processes (see also Appendix A). By developing a variational principle for the Fokker—Planck equation, we will show that phenomenological coefficients are the same which appear in the Fokker—Planck equation. The next type of variational principle is concerned with a kinetic or Boltzmann equation, the solution of which usually requires approximations. A powerful approximation for obtaining transport coefficients from the Boltzmann equation is based upon a certain variational principle, which gives bounds on transport coefficients. Such an extremal principle enables us to obtain information about solutions of the Boltzmann equation without any explicit knowledge of the solutions themselves. A discussion of such a principle may be found in standard texts [20,2 1]. It is usual to consider the variational method merely as a convenient mathematical device. It was Ziman [22] who suggested that the variational principle is a basic physical principle. The main purpose of this paper thus is to rescue the variational principle from being a mathematical tool and to show that it has considerable physical meaning. After Ziman, we stress the link between the variational principle and entropy production. In fact there is a great variety of possible variational principles to maximize (or minimize). Even so, we obtain a positive-definite quadratic form which agrees with the entropy production. The presence of a magnetic field or time variation, however, has frustrated attempts to establish a variational principle in the (linearized) Boltzmann equation which is an extremal problem. In such a case, a non-Hermitian operator occurs in the linearized

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Boltzmann equation, and the variational principle is a stationary problem. This essentially reflects the fact that the invariance of the Hamiltonian under time reversal is destroyed by the presence of a magnetic field. We will show in Section 6 that if we eliminate some part of the variational functions, we get an extremal principle as to the remaining part [23]. This means that the information concerned in an extremal principle in the kinetic stage is much less than that in a stationary principle. The key to this interpretation is found in the ambitious papers by Nakano [7]. In Section 7, we will concern ourselves with the quantum mechanical version of the variational principle of Onsager’s type, which appears to have a fairly wide range of applicability in various areas of transport theory. The well-known time-correlation-function expression (the Kubo—Nakano formula) for transport coefficients [24,25] is derived from the quantum mechanical variational principle. It is shown that, even in the absence of a magnetic field, the quantum mechanical variational principle, which is connected with the von Neumann equation for the density matrix, is not an extremal but a stationary problem [26]. This arises from the bilinearity of the variational functional in terms of independent operators which is necessary in order to deduce the timecorrelation-function expression for transport coefficients. Again, by eliminating the irrelevant components even with respect to time reversal, the variational principle changes to an extremal problem. Accordingly, the irreversibility immediately results from the contraction of information about the system. Conceptually, the notion of the contraction of information is the natural starting point for a theory of irreversible processes and it therefore deserves a careful analysis. We know that most observable phenomena in condensed matter are macroscopic. Then, our immediate question has been to ask how microscopic systems can manifest themselves in macroscopic phenomena which are ruled by the laws of the macroscopic world. One way to answer this question is given by intuition: condensation of a large number of quanta is needed in order to manifest macroscopic behavior. The quanta are in general not the original constituent entities but instead quasi-particles. This is the phenomenon of phase transition, in which the contraction of information in the microscopic scale of the system, the renormalization transformation, is essentially involved. The second example of contraction is a quantum Langevin-equation description, in which dynamical variables of irrelevant degrees of freedom appear only in random-force terms and interactions of the relevant components with the irrelevant degrees of freedom are characterized by the transport coefficients involved. In the step leading the quantum Langevin equation, we break the time-reversal symmetry of original dynamical equations by choosing the retarded solutions of the inhomogeneous equations of motion for the irrelevant components. This is the general way the invariance with respect to time-reversal is broken in microscopic equations. These microscopic equations, which are ruled by the causality principle, describe only the time development of a class of solutions of the microscopic equations of motion. The last type of contraction operates with quantum variational principles of irreversible processes. This is the main topic which we will address in this article. Sauermann [12] has established that the dynamic susceptibilities can be obtained from a Schwinger type variational principle in a vector space of observables which is treated in Section 8. Sauermann’s aim has been to consider different physical contents of reasonable ansatz simultaneously. In Section 9, Davies’ approach is reviewed, which considers the Bethe—Salpeter equation for various vertex functions playing the role of the fundamental transport equations. The

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relation between Nakano’s and Davies’ approach is examined. In Appendix A, a note on the Djukic—Vujanovic theory is given. In Appendix B, an analysis is given of the Onsager—Machlup theory for a simple model. Appendix C contains a simple discussion on a time-reversal problem. In Appendix D, some useful mathematical properties of a relative entropy are given.

2. The linear irreversible thermodynamics It will be necessary to give a general description of the irreversible processes to be considered in this article. We prefer to consider irreversible processes that take place in discontinuous systems. The name “discontinuous” is given since we suppose that the thermodynamic functions are not continuous functions of the spatial coordinates. One of the examples of a discontinuous system is that consisting of two chambers connected by a narrow capillary and maintained at uniform temperatures [27]. In order to identify thermodynamic forces and currents, we adopt the following definition. Let the system be described by a set of extensive thermodynamic variables The entropy then will satisfy the fundamental relations S = S[cz1, y1~].We identify the currents in the system as ~,

...

...,~.

,

the time derivatives of these variables; J, = ~ (i = 1,2, ,f). On the other hand, the thermodynamic forces are identified as X, = aS/act,. If the currents are linear functions of the thermodynamic forces, we call these irreversible processes linear. ...

2.1. The linear phenomenological laws We shall summarize the main aspects of linear irreversible thermodynamic processes [27]. It should be emphasized that the nonequilibrium thermodynamics hinges on our possibility to distinguish thermodynamic forces from macroscopic currents in the system under study. This is physically appropriate for systems not far from equilibrium but might be difficult to do for systems far from equilibrium. Let us denote the currents and the forces by J1, ...,J1 and X1, ...,Xf, respectively. Then, the entropy production per unit time due to these currents is expressed as P=>~X,J,0.

(2.1)

This quantity is a positive definite one according to the second law of thermodynamics. It is known empirically that for a large class of irreversible processes the currents are linear functions of the thermodynamical forces. If we restrict ourselves to these linear irreversible processes we can write generally J,

=

~ ~

i, k

=

1, 2,

. . . ,

f

(2.2)

or equivalently Xk

=

>RkjJj

.

(2.3)

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Here L,~are the phenomenological coefficients and the matrix R (R,~)is the inverse of the matrix L (L,k). Relations (2.2) and (2.3) will be referred to as the phenomenological laws. For linear systems, in the absence of magnetic fields and rotations, the phenomenological coefficients are symmetric, as a consequence of the Onsager reciprocity relations: L,,,

=

Lk,

or Rk

=

Rk,

.

(2.4)

Here it should be remembered that the reciprocity relations for L,,, not Rk can be proved from the principle of microscopic reversibility [1]. In fact, in Section 7, we will see that the phenomenological coefficients L,, have an intimate relationship with the dynamics of fluctuations. The validity of the Onsager reciprocity relations is not influenced by the linear homogeneous dependence between the currents. Similarly, if the forces are linearly related by a constraint while the currents are independent, the Onsager reciprocity relations remain valid. If homogeneous linear relationships exist between the forces as well as the currents, the phenomenological coefficients are not uniquely defined and the reciprocity relations are not necessarily fulfilled. However, it is true that the coefficients can be always be chosen in such a mannerthat the reciprocity relations shall remain valid. In this case, if one introduces Eq. (2.3) into Eq. (2.1), one gets a quadratic expression in the currents of the form P = 2~b[J,J] b[J,J]

=

(2.5)

,

(l/2)~R~kJtJk,

(2.6)

where ~[J,J] is referred to as the dissipation function (of the first kind). This function acts as the potential for the thermodynamic forces: that is, Eq. (2.3) is written in the form Xk

=

.

(2.7)

It should be noted that condition (2.5) is not equivalent to the linear laws (2.3), although these satisfy Eq. (2.5) automatically. The phenomenological laws are sufficient but not necessary for condition (2.5). There is still another formulation in the form P

=

2~P[X, X]

~P[X,X]

=

,

(1/2)~,L,kXIXk,

(2.8) (2.9)

~I’[X,X] is the dissipation function (of the second kind). The third form is written as P= ~P[X,X] + ‘P[J,J].

(2.10)

The potentials ~Pand ~bare the functions in which the premises of the Onsager theory are accumulated, i.e., the linear phenomenological laws together with the Onsager relations valid for the coefficients. Before closing this section, it should be noted that in the derivation of the Onsager relations the thermodynamic forces X, and the currents J, are assumed to form separately linearly independent

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sets. Then, it is a simple matter to verify that, if we have derived the Onsager relations for one independent set of forces and currents, any acceptable transformation yields forces and currents satisfying the Onsager relations. A linear transformation is acceptable if it keeps the dissipation functions invariant. 2.2. The Le Chatelier—Braun principle The Le Chatelier—Braun principle of equilibrium thermodynamics [28] implies a damping of the changes of extensive variables characterizing the equilibrium state of the system. de Groot [29] and Prigogine [30] extended this principle to nonequilibrium steady states. They expressed it in terms of inequalities referring to intensive variables. Let us consider the steady state characterized by the thermodynamical forces, some of which are fixed while others are not. For simplicity, let us assume that X,=fixed,

(
i= l,...,m

(2.11)

Since not all the thermodynamical forces are unspecified, one has to prescribe an equivalent number of conditions. These will frequently appear as boundary conditions: J,.=0,

n=m+ 1,...,f.

(2.12)

The unspecified forces X,, (n = m + I, ...,f) can be obtained by solving Eq. (2.12). Let us now assume that one of the unspecified forces, say X1, has changed by a certain amount. Then, the entropy production of the perturbed state is given by P’

=

=

P + Jf5Xf + ~

(2.13)

L~1~5X~

(2.14)

.

The second term of the right-hand side of Eq. (2.10) is equal to zero by Eq. (2.12). It is readily verified that 2 > 0 (2.15) 5J~5Xf= Lff(5Xf) Hence, .

p’>p.

(2.16)

This implies that the entropy production is minimum in the (stable) steady state. Since we have dXf/dt ~x—5J~,. ,

(2.17)

we can write Eq. (2.15) in the form (dXf/dt)Xf < 0.

(2.18)

We can generalize the above argument to more general cases. Let us assume that the unspecified forces have changed by amounts 5X~(n = m + 1 f). In this case we have

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P’ = >Ljk(Xj + ~X~)(X~ + 5Xk), öX 1=0, i=1,2,...,m, 5X,~0,

135

(2.19) i=m+1,...,f.

(2.20)

We can write Eq. (2.19) as P’

=

P + 2>~Jk~Xk + ~L~k~X~5Xk.

(2.21)

The second sum of the right-hand side of Eq. (2.21) equals zero by Eqs. (2.12) and (2.20). Accordingly, the Le Chatelier—Braun principle can be written in the form ~ L,k oX, OX~> 0.

(2.22)

This form was postulated by Yomosa [28]. Discussions have been made by many authors, among whom we refer here to the work by Glansdorff and Prigogine [31] and Blount [32].

3. Variational principles of linear irreversible processes In the thermodynamical stage, the variational principles were developed by Onsager [1] with the study of the reciprocity relations as to the transport coefficients. This type of variational principle has been reviewed by Ono [3]. In the following four sections, the variational principles for linear processes will be treated and discussed from a thermodynamical and statistical mechanical point of view. In Sections 3.1 and 3.2, based on the theory of reciprocal variational theorem [33, 34], a general problem is considered for the determination of thermodynamical forces and phenomenological currents describing irreversible processes in nonequilibrium thermodynamics. A clear definition of forces and currents is necessary to establish certain symmetry relations, known as the Onsager reciprocity relations, in phenomenological laws which connect the forces and the currents, and to ascribe physical meaning. The reason for considering this problem is to clarify Prigogine’s principle of minimum entropy production. We want to show that this theorem, when it applies, is a consequence of known physical laws. Ono [3] was the first who clarified the relation between the principle of least dissipation of energy and the principle of minimum entropy production (see also Refs. [7, 15]). In Section 3.4, we propose a new variational principle of Hamilton’s type in studies of linear irreversible processes. Work of this kind is motivated by the attempt to find a quantity which plays the same pivotal role in irreversible thermodynamics as the Lagrangian does in analytical mechanics. 3.1. The variational principle of Onsager’s type As we have stated in Section 2, the problem we consider is that of finding an alternative characterization of the phenomenological laws by means of variational principles. We formulate this problem, which we shall call: Problem P: Find the forces X and fluxes c~such that =

~

L~X~ .

(2.3’)

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We shall show that, for linear systems, ~ and X are the solutions of two reciprocal variational principles. A simple problem involving (2,3’) is that of prescribing the forces and finding the fluxes. A more general one is that of prescribing some forces and the complementary fluxes and finding both the forces and fluxes. In this section, we first consider the simple problem. The reason for considering this problem is to discuss the basis of the Onsager reciprocity relations between irreversible processes. From macroscopic point of view, the state of a system will be described by a set of extensive variables A1 A1 with their equilibrium values . Since the entropy Sofa system is maximum for the equilibrium state, we have S

=

S~ (1/2)~S~kcc,c~k —

(3.1)

,

where S~is the equilibrium value of the entropy, S,~the positive-definite symmetric tensor, and ~t, are the deviations of the extensive variables from their equilibrium values: =

A,

—

,

i

=

1, 2,

...,f.

(3.2)

In an aged system, the time derivative of the extensive variables generally correspond to the dissipative currents. Hence, we may write J,

=

da~,/dt

(3.3)

.

The thermodynamical forces Xk conjugate to c~kis defined as Xk

=

(3.4)

—~Sk,~I.

The entropy production in the system equals the time derivative of the entropy given by Eq. (3.1) and therefore we have P[ot;c~] S = ~(aS/aC~k)~k= ~Xkc~k

.

(3.5)

Then, the variational principle P[~c~] i[c~c~]= maximum —

(3.6)

is formally equivalent to the phenomenological laws (2.3), including the Onsager reciprocity relations (2.4) for the phenomenological constants R,,,. Here we only take the variations with respect to the odd variables with respect to time-reversal. It is to be noted that the thermodynamical forces Xk are functions of the state of the system while the dissipation function ‘b[~&] depends upon its change in time. As shown by Onsager [1], 2Tb[~2;~]is equal to the rate of dissipation of free energy, and hence the variational principle expressed in the form of Eq. (3.6) is called the principle of least dissipation of energy. This is a generalization of a similar principle in hydrodynamics due to Rayleigh [35]. In Onsager’s principle of least dissipation of energy, the thermodynamical forces are prescribed and only the dissipative currents are varied. However, this variational principle is not restricted to stationary state. If we assume that the irreversible processes in macroscopic system are all of quite the same as that of aged systems, Onsager’s principle is also applicable to open systems. ~,

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The hypothesis of a generalized entropy depending upon the so-called fl-variables, which are odd under time reversal, was advocated by Machlup and Onsager [36]. Let us assume that the generalized entropy E(~,c~)has the form S~ (1/2)~S,k~,~k(1/2)~m,kc~~~k ,

=

—

(3.7)

—

where m,~are constants. The letter m has been chosen because of the analogy to mass. The ultimate aim of this hypothesis is to write phenomenological laws in the form =

(3.8)

,

where the effective thermodynamical forces are defined by =

Xk + (d/dt)[(~E/ac~k)]

(3.9)

.

The second member of this sum is analogous to a D’Alembert force in classical mechanics: any phenomenological laws may be written in a canonical form like Eq. (2.18) if we add the thermodynamic forces to the forces of inertia. Machlup and Onsager [36] showed that the variational principle E(c~)

~[c~~]

—

—

~I’[~~]

=

maximum

(3.10)

is formally equivalent to Eq. (3.8). Here in Eq. (3.10) =

(1/2)~L,k~~~k

(3.11)

.

It is to be noted that the effective forces ~k are not functions of the state alone and accordingly the dissipation function (of the second kind) W [~ ~]depends upon the rate of change of state in time. On the other hand, the dissipation function D[c~; ~], in this case, is a function of the state alone. 3.2. The variational principle of Prigogine ~ type The variational principle of Onsager’s type is that of prescribing the thermodynamical forces and finding dissipative currents. Another one is the variational principle complementary to Eq. (3.6), in which the currents are kept constant. The following variational principle is valid: P[~c~] ~P[X;X] —

=

maximum,

(3.12)

where the dissipation function ~P[X ; X] is given by Eq. (2.9). It is readily verified that this principle, which is referred to as the variational principle of Prigogine’s type, gives the phenomenological laws (2.2). It provides the proof of the Onsager reciprocity relations for the phenomenological coefficients L,3. We want to give an alternative form of the principle (3.12). We construct the statement of the variational principle: minimize ‘I’[X;X], subject to 2~I’[X;X] = P[c’;~]

.

(3.13)

The question, however, arises whether any deeper significance is to be attributed to it. The point to be noted is that, as we have mentioned in Section 3.1, ~t’[X; X] is a function of the state of the system and acts as a potential for the dissipative currents.

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In Section 5, we will discuss that the variational principle (3.12) has a significant relation to the variational principle which determines the Fokker—Planck equation for a conditional probability function for fluctuations. Let us now consider the steady state described by the thermodynamic forces, some of which are fixed while others are not (see Sections 2 and 3). If some of the unspecified forces, say X,,, n = m + 1, ,[~ have changed by OX~,the currents would change by ...

=

~

n

m+ 1

=

f.

(3.14)

The dissipation function of the perturbed state takes the form ~P’[X;X]

W[X;X] + (1/2)>~oX~oJ~

=

(3.15)

.

Hence, if the boundary conditions (2.12) apply, we have ~P’[X;X] < W[X;X]

(3.16)

.

This inequality is based on the Le Chatelier—Braun principle. In consequence, if the system is operating in a steady state whose dissipative function W[X; X] is minimum, this state is stable. Here the word “stable” means that there would be no spontaneous change of that state to another. Hence, the theorem expressed in the form (3.13) is referred to as the principle of minimum entropy production for linear irreversible processes. We shall present a simple example. Let us consider a transport process of matter and energy between systems which are kept at different temperatures. We assume linear laws. The heat current Jq and mass current im are related to the thermal gradient Xq and mass density gradient Xm by pq~=~Lut Li2~[Xq~ L21 =L12 LL2i

Urn]

(3.17)

L22]LXmJ’

The dissipation function ‘I’ [X ; X] takes the form W[X;X]

=

(1/2)L11X~+ L12XqXm + (l/2)L22X~

.

(3.18)

At a fixed temperature gradient, the state of minimum entropy production is attained when =

L21Xq + L22Xm = ~rn= 0.

(3,19)

The minimum value of W[X;X] is given by ~1’min =

(3.20)

JqXq.

This is the actual solution, only if a boundary condition of the form of Eq. (2.12) is imposed. If the mass density gradient has changed by an amount OXrn in a time interval zlt, the mass current would occur: OJm

=

L22OXm

.

(3.21)

Then, we have OJrnOXrn < 0.

(3.22)

M. Ichiyanagi/Phvsics Reports 243 (1994) 125—182

139

That is to say, the excess mass current would flow in such a way as to restore the original mass current and density gradient. This is the statement of the Le Chatelier—Braun principle. 3.3. The relation between Onsager ‘s and Prigogine ~ principle At first sight, the minimum principle of Prigogine’s type seems to have some intimate connection with the variational principle of Onsager’s type. The essential part of Onsager’s principle is the variation with respect to the fluxes, whereas the essential part of Prigogine’s type is the simultaneous variation with respect to the fluxes and forces. In connection to these characteristics, we shall pay attention to the article by Keller [37]. Following Keller, let us consider a dissipative thermodynamic system in which there are various forces and fluxes, represented by the vectors X and ~, respectively, in a real Hubert space H. We introduce a subspace ~ of H and its orthogonal component Q, so that H = ~ + Q. Then, for any 2r~= ~ + X given vectors X0 and &~,we define the sets and Q0 by 0 and Q0 = Q + ~o. ~

Let us denote by ~* and X* the solution of Problem P. Then, we can prove that X* is also the solution of the minimum problem and that c~*is the solution of a maximum problem. These are: Minimum problem: Among all Xe~J0find one which minimizes g(X)

=

2~P[X,X]

—

2~0(X X0). —

(3.23)

Maximum problem: Among all c~ e Q0 find one which maximizes f(c~)=

—

2~[&,c~]+ 2c’X0

.

(3.24)

The following theorem can be established: (1) Minimum problem has a unique solution X~. (2) Maximum problem has a unique solution cr’. (3) &, = ~ L,1X~,so that ~ and X constitute the unique solution of problem P. (4) f(~*)= max~00f(&)= min~~0g(X) = G(X*). (5) ~* and X* are the unique stationary point of f(~)and g(X), respectively. (6) When X0e2, then f(c~)= 2~I[c~,c~]+ 2c~0X0,and when ~ then g(X) = 2~P[X,X]. In these two cases, and only in these cases, the principle of Prigogine’s type is valid. (7) When X* = X0, then f(~*)= g(X*) = 2~P[X*, X*]. When ~* = ~, then f(~*)= g(X*) = 2~[c~*, ck*] + 2c~0X0. This theorem shows that the variational principle of Prigogine’s type is valid if and only if either X0 e ~ or ~e Q, and that this principle is not an independent principle of irreversible thermodynamics but rather only an alternative reformulation of the variational principle of Onsager’s type valid for stationary processes. —

—

3.4. Hamilton ‘s principle The analysis of a variety of problems can be greatly simplified if the phenomenological laws admit an analytic representation in terms of Hamilton’s equations which come from a variational principle. This is the main reason for which Hamilton’s principle of irreversible thermodynamics

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M. Ichivanagi/Physics Reports 243 (1994) 125- 182

may be interesting: for obtaining constants of motion and symmetry properties by using the Noether theorem. It is known that not every problem can be expressed in the form of the classical Hamilton principle. It is not possible to find a classical Lagrangian for equations of odd-order. However, there exist limited Lagrangians. For instance, there is no classical Lagrangian for ~ + x = 0, but the so-called limited Lagrangian L,, = (1/2)(p.~2 + x2)exp(—t/p) gives this equation. It is important to recognize that this variational principle is sound only for slightly dissipative systems since it requires putting the dissipative parameter p equal to zero in the exponential factor everywhere before the results are interpreted physically. This direction is meant to be highly suggestive. Djukic and Vujanovic [13] have constructed a new variational principle of Hamilton’s type by introducing into Lagrangian parameters which tend to zero after the process of variation is completed. If one uses Eqs. (3.3) and (3.4) one can write the phenomenological laws in the form ~(R, 1~1 + ~

0

=

(3.25)

.

We then consider the limit Lagrangian of the form L0(t)

=

(l/2)~{~J.i,1(t;p)~,~1 +~

Here the functions lim ~‘,1(t; p)

=

have the following properties:

~

0

(3.26)

.

lim ô~/i~1(t ; p)/~t= R.~,

,

(3.27)

otherwise they are arbitrary functions of their arguments. It is necessary that the limit Lagrangian remains invariant under arbitrary point transformations of the extensive variables; L~= LM. Let us consider a nonsingular transformation of the form x,

=

~

,

i

=

1,2

(3.28)

which are quite arbitrary functions of the old variables variables are given by x,

=

~(ax,/~x1)~1.

ot,.

Then, fluxes associated with the new

(3.29)

Since the entropy is a function of the state only and does not depend upon the chosen representation of the state, we have

,...,x1]

=

S[c~

~]

,

(3.30)

.

(3.31)

which is equivalent to S~=

~(aX,/a~k)

(ax1/~,)Ski

The matrix of S~is a contravariant tensor. Accordingly, the limit Lagrangian is an invariant of the transformation if it is true that

M. Ichiyanagi/Physics Reports 243 (1994) 125—182

~ (t; p)

~ (ax,/~c’k)(~x~/~,) 111k! (t ; p)

=

141

(3.32)

.

The matrix of i/i,~(t; p) is required to contravariant. The first derivatives of the limit Lagrangian are given by =

~,i1i,~(t;p)~~

(3.33)

and the second derivatives become in the limit p

—±

0

lim (d/dt)(aL~/~~) = ~

(3.34)

In consequence, the Euler—Lagrange equation in the limit p 0 yield the phenomenological laws (3.16). There are two features of the limit Lagrangian which strike us as interesting. The first one is that the time derivative of the limit Lagrangian becomes in the limit p —p 0 —~

limdL,2/dt

~

=

+~

=

—

{P[z,ck]

—

P[ci;c~]} ,

(3.35)

~L—’0

which exactly corresponds to the variational functional (3.6) to be minimized. Accordingly, the principle of least dissipation of energy can be expressed as urn dL~/dt= minimum.

(3.36)

p-.o

The second one is that if we assume that lim ~,1(t; p)

=

m1~,

lirn a~/~,~(t ; p)/~t= R.~,

(3.37)

instead of Eq. (3.27), the Euler—Lagrange equations become i= 1,2,...,f

(3.38)

in the limit p 0. These exactly correspond to Eq. (3.8). This hypotheses yields —~

(3.39) ~~—‘0

where ~E’(c~ c~)is the generalized entropy given by Eq. (3.7). Hence, it follows from no more than the definition of the limit Lagrangian that the generalized thermodynamical forces ~k are defined by Eq. (3.9). In the original paper by Machlup and Onsager [36], was it not found how the generalized forces can be defined from the standard theory of thermodynamics. The discussion given above might be of use in the unification of the generalized entropy with the standard thermodynamic entropy. The most interesting question still remains: Does there exist a canonical formalism? it is a simple matter to show that if one introduces the “canonical momenta” Pk

aL,L/~k=

~l/IkJ~J,

(3.40)

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M. Ichivanagi/Phvsics Reports 243 (1994) 125—182

the canonical equations of motion can be written in the form aH~/aPk

=

Pk

‘

=

—~H~/~k

(3.41)

.

Here the limit Hamiltonian H~is defined by H,,

>~pjc~, L0

=

—

P’[p,p]

=

b’[p,p]

=

—

S[ot] ,

(3.42)

(1/2)~(~i’),1p,p1

(3.43)

.

Hence, the entropy S is analogous to the work function (the negative potential) in analytical mechanics. By making use of Eq. (3.42) in Eq. (3.41) we have Pk

~S/~k

=

=

Xk ,

(3.44)

which are analogous to Newton’s equations of motion. It is easy to verify that in the limit of p lim Pk = 0 , lim f,,, IA -~

=

~

—~

0 (3.45)

R~,&,(= Xk).

(3.46)

0

In consequence, we conclude that the canonical equations of motion exactly reproduce the phenomenological laws (3.4). Before concluding this section, we discuss the nonlinear case. To this order, we generalize the limit Lagrangian as =

11’(t;p)~b(~i, ~

—

S[~] ,

(3.47)

where 1(’) is the generalized dissipation function which is not necessarily a bilinear form as the first term of Eq. (3.24). The arbitrary function ~‘(t ; p) is assumed to satisfy the conditions lim t~/(t;p)= 0,

lim ~i(t;p)/~it

=

1

.

(3.48)

I1_,0

Then, in the limit of p

—÷

0, the Euler—Lagrange equations read

c~)= X~(= (ô/~~) S[~]) .

(3.49)

These are the nonlinear phenomenological laws, which can be rewritten as J,=~,=J,(X1

X1),

(3.50)

X1) being the nonlinear currents.

4. The general evolution criterion To prove the principle of minimum entropy production, Glansdorff and Prigogine [38] made rather restrictive assumptions. They assumed that the system is described by the linear phenomenological laws (2.2) with constant coefficients satisfying the Onsager reciprocity relations (2.4) and

M. Ichiyanagi/Physics Reports 243 (1994) 125—182

143

subject to time-independent conditions of the form given by Eq. (2.12). Glansdorff and Prigogine [16] have generalized the principle of minimum entropy production to an evolution criterion that does not invoke the linear phenomenological laws. The general evolution criterion has been established for continuous systems under the assumption of local equilibrium. Hence, in order to prove the theorem, Glansdorff and Prigogine had to utilize the balance equations for densities. In this Section, we want to approach this problem in the general framework set by Onsager for discontinuous systems. Indeed, the principle expressed as Eq. (3.14) is valid for a class of nonlinear processes also, as far as Onsager reciprocity relations can be assumed. Generalized Onsager reciprocity relations may be characterized in terms of a dissipation function, ~I’(X1,..., X1), which is not necessarily bilinear such as given by Eq. (2.9) [39]. An essential point is that ~I’(X1,..., X1) is a function of the state and acts as a potential function for nonlinear currents: =

aw(X1,...,J~)/~X,,.

Then, the generalized transport coefficients

(4.1)

() are given by 2~P(X

LIk(X!, ...,X1)

[~J,,(X1, ...,X1)]/8X,

=

[~

1,...,X1)]/aX,~X,,, i, k = 1, ...,f. (4.2)

Accordingly, the conditions 21p(X a 1, ...,XJ-)/aX, aX,,

2,p(X

=

a

1, ...,X1)/aX,,aX~

(4.3)

imply the reciprocity relations L~,,(X1,...,X1)

=

L,,,(X1, ...,X1)

(4.4)

between the generalized transport coefficients [40] if one would assume W(~)to be sufficiently smooth. Let us consider 1 the steady state ofWe the shall system described by the currents J~O)and the thermodyn~J~O) = J~°~(X~°~). write amic forces X~° Xk = X~°~ + OXk, k = 1,2, . .,f, (4.5) .

OX,,

being the excess forces. Then, it is readily seen that W(X) = W(X~°’) + >J~°~OXk + (1/2)>0X,,OJk ,

(4.6)

up to the second order of OX,,. Here in Eq. (4.6), Of,,

=

~L,,~(X’°~)OXI.

(4.7)

The last term of the right-hand side of Eq. (4.6) is negative by the Le Chatelier—Braun principle. Accordingly, if these excess forces are not independent but satisfy the condition (4.8) we obtain from Eq. (4.6) W(X~°1 + OX) < ~P(X(0)).

(4.9)

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M. Ichiyanagi/Phvsics Reports 243 (1994) 125—182

The value of the dissipation function ~t’(X) would decrease in this case. Eqs. (4.8) and (4.9) can be written as = 0, >J~°1dX,,/dt d~P(X)/dt< 0,

(4.10) (4.11)

respectively. This is the statement of the evolution criterion due to Glansdorff and Prigogine [16], which asserts that during the evolution of the system the thermodynamical forces Xk will change and adjust themselves in such a way that the dissipation function !P(X) will be minimum in the steady state for which Eq. (4.10) applies. The evolution criterion could serve as a starting point for a nonlinear stability analysis if a neighbourhood of the steady state of the system, in which we can distinguish the thermodynamical forces from the dissipative currents, is examined. In the original derivation of the evolution criterion, Glansdorff and Prigogine considered the partial change in entropy production due to the changes of the thermodynarnical forces and based on the structure of the balance equations. Then, they expressed their theorem in the form

~

(4.12)

without making restrictive assumptions, for example, the linear laws and Onsager reciprocity relations. From the standpoint of physics, however, we must examine the relevance and applicability of such a general theorem. Are there real physical systems which may be described in the way proposed by Glansdorff and Prigogine? The notion of the partial change in entropy production is not well understood. It should be noted that the partial change in entropy production of the form of Eq. (4.12) is equal to d W(X)/dt, if the sum ~ J,, OX,, is totally differentiable. Remember that condition (4.3) guarantees the total differentiability of 41’(X) which is a function of the state of the system. The existence of !P(X) thus assumes the reciprocity relations between the generalized (or differential) transport coefficients [41]. In this sense, we are less ambitious in scope than Glansdorff and Prigogine. The evolution criterion written as Eq. (4.11) is formulated in order to treat the special class of nonlinear irreversible processes satisfying the reciprocity relations (4.4) and we have, thus far, made no attempt to generalize it to other classes of irreversible processes.

5. Statistical significance of dissipation functions Let us study the statistical significance of the two dissipation functions 1~[c~; o~]and ~P[X ; X]. Following Onsager [1], we assume that a departure from thermal equilibrium can be described in terms of a set of extensive variables ~ = (~ ~, .. ., ~) of a macroscopic system. Then, the equilibrium condition, S = maximum, characterizes the most probable state and the probability W for a state characterized by ~ is given by Boltzmann’s principle klog W(~c)= S[~] + constant ,

(5.1)

M. Ichiyanagi/Phvsics Reports 243 (1994) 125—182

145

where k denotes the Boltzmann constant. We now introduce the joint probability W[~,ot’ ; At] for the state of the system defined by the sets {ot~,...,ot~,.}and {cc’~,...,ot’~.} at the successive times t and t + At, respectively. This probability is defined by W[cx,ot’;At]

W(oc)P(otlot’;At)

=

(5.2)

,

where P(otlot’;At) denotes the conditional probability. Note that we have the so-called Smolukowski equation P(ot I ot’; t + t’)

JP(ot I ot”; t)P(ot” I ot’; t’) dot”.

=

(5.3)

The conditional probability possesses the property W(ot)P(ctlot’;At)

W(ot’)P(ct’Iot;At)

=

.

(5.4)

This is the important property of microscopic irreversibility. The conditional probability P(ot I ot’, t) and the initial distribution P(ot, t0) characterize the Markovian process of fluctuations completely. The distribution P(ot, t) at time t then is given by P(ot, t)

=

JP(ot I ot’, t

—

t0)P(ot’, t0) dot’.

(5.5)

We shall assume that in short time intervals lim (1/At) IAot,P(otlot’,At)dot’

=

—

I’, = EL,1aS(~)/a~~,

(5.6)

z110

lim (1/At) .1~Aot,AotjP(otIot’,At)dot’ = 2kBL,J. 1t~0

(5.7)

L

Then, it is known from the theory of stochastic processes that for a Markovian process with properties (5.6) and (5.7), the conditional probability P(ot ot’, t) fulfills the Fokker—Planck equation

(a/at) P(ot I ot’, t)

=

~(a/a~,)L,

1(v~+ k~(a/a~~))P(~ I ot’, t) .

(5.8)

The conditional probability P has the properties P(ot

ot’ ;

t) ~ O(ot

ot’)

—

P(otIot’;t)—~W(ot)

,

, as t as

—~

0,

t—~oc.

(5.9) (5.10)

Thus, the stationary distribution equals the probability W(ot) given by Eq. (5.1). Furthermore, the conditional probability P(ot I ot’, t) is the Green’s function of the Fokker—Planck equation. In the limit of kB 0, the conditional probability P is given by [18] —4

P(ot0 lot, t) = JN(ot~t)exp

{

—

(1/2kB)JI(ot~~)dt}d(Path)

(5.11)

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M. Ichivanagi/Physics Reports 243 (1994) 125—182

in the path-integral form, where N(ot, t) is a normalization factor and I(ot,o~)=b[ck,c~]—P[ot,&] .

(5.12)

Onsager and Machlup [19] have proposed a more complete formulation of the same problem by confining themselves to the case of Gaussian random processes. They have proved that for a given initial and final state there is one more probable path, which is determined by the variational principle

Jr,

I(ot, ck) dt

=

minimum

(5.13)

.

The value of I(ot, c~)becomes minimal for the thermodynamically realized path ot(t) starting from t~ and reaching ot (t = t ot (t = t 1)

=

2)

ot

=

This is to be contrasted with the fact that the classical equations of motion are derived from the principle (‘(2

2’dt

Jti

=

stationary ,

(5.14)

provided that the variables are not varied at the two end-points. Here, 22 is the Lagrangian of a dynamical system. If we take t2 close to t1 (t2 = At = small), the integral (5.12) becomes —

I(ot, ~k)At = minimum ,

(5.15)

This is just the principle of the least dissipation of energy (3.6). Let us now introduce the quantities ~, which are defined by =

~ ~

,

i

=

1,

...,f.

(5.16)

In terms of these, we can write I (ot, c~)in the form I(ot,c~)= ~

0.

(5.17)

where the sign is due to the positive definiteness of the matrix R, and the equality sign holds when and only when = 0. The equations ~, = 0 correspond to the phenomenological laws (2.2). We can also verify that [18] ~‘,

dI(ot,&)/dt = ~

0,

(5.18)

which is due to the properties of the matrix S [see Eq. (3.1)]. Hence, the value of I(ot, &) increases except for the path, for which the value of I(ot, c~)remains constant. Eqs. (5.17) and (5.18) show the analogy of I(ot,&) and the entropy S. We want to analyze the problem somewhat more deeply by asking how the minimum (or maximum) principle is related to the stationary principle of dynamics. This is the real distinction between thermodynamics and microscopic dynamics.

M. Ichiyanagi/Phvsics Reports 243 (1994) 125—182

147

As we see in Appendix A, the most important characteristic of the Onsager—Machlup theory is that the diffusion process can be described by a dynamics with a suitable Lagrangian for the most probable path. Thus, the deterministic equation of motion (A.6) is sufficient for a stochastic description of nonequilibrium (Gaussian Markoff) processes not far from equilibrium. It should not be overlooked that the variational principle of the form (5.13) is equivalent to the principle given by (3.6) as far as we exclusively take variations with respect to Furthermore, the phenomenological laws are exactly the Euler—Lagrange equations (A.8) of the variational principle (5.14), in which we consider the variations with respect to ot and Eq. asserts that if the relaxation times are sufficiently short the action integral becomes independent of ot(t) and that the variational principle of dynamics (5.13) changes into the variational principle of the irreversible thermodynamics, (3.6). Thus, if we eliminate the even components with respect to time reversal, the stationary variational principle can be transformed to the minimum (or maximum) variational principle. Before closing this section, we discuss the variational principle complementary to Eq. (5.12), the principle of Prigogine’s type. First of all, the following Lagrangian is relevant [42]: &.

~.

2’[O,P]

=

J{OaP/at

—

(5.15)

(k8/2)~L~~(aO/aot1)(aO/aot~)P} dot.

(5.19)

Here 6(ot, t) is the variational function while P(ot, t) is not. Immediately after the process of variation is finished, this fixed function becomes unfrozen, i.e., equal to the original function of the problem. For simplicity, and not of course as a matter of necessity, we have set V, = 0. Then, it is readily verified that the Euler—Lagrange equations are given by

aP/at + kB> L,~(a/a~1)(ao/a~~) P = 0.

(5.20)

The fact that the function P(ot, t) is held fixed during a variation of an integral means that the variation is restricted variation [43, 44]. This situation has been recognized in Onsager’s principle which holds as long as the thermodynamical forces are kept frozen. Eq. (5.20) is identical with the Fokker—Planck equation provided (5.7),

O(ot, t)

=

—

log P(ot, t)

(5.21)

.

It is remarkable that if Eq. (5.21) is utilized in Eq. (5.19) the Lagrangian [0, P] becomes =

—

logP,P]

=

JP11[X~X]

—

~[ot,~]}dot,

(5.22)

where ~P[X, X] is defined by Eq. (2.9) and the thermodynamical forces are defined by Eq. (3.4). The integrand is the variational functional of the principle of Prigogine’s type, which has appeared in Eq. (3.12). In consequence, the Lagrangian (5.19) plays a dual role: It determines the dynamics of fluctuations which are described in terms of the Fokker—Planck equation, and gives the phenomenological laws. The transport coefficients are the same L.~that determine the dynamics of fluctuations. This result is predicted by an expansion method [45].

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M. Ichiyanagi/Physics Reports 243 (1994) 125—182

6. Variational principles in Boltzmann equations

The Boltzmann equation has become a generally accepted and central part of classical transport theory. An approximation method for obtaining transport coefficients from the Boltzmann equation is based on a certain variational principle. Boltzmann’s argument is of course not mathematically rigorous, but for linear Boltzmann equations all mathematical qualms have been resolved. It also turned out to be very difficult to make significant generalizations, to go beyond its limitations. Hence, in this section, we consider the variational principles of linearized Boltzmann equations. One motivation for the variational theory is the possibility that it might be of use in the unification of the kinetic theory with irreversible thermodynamics to go beyond Boltzmann. 6.1. The Kohler—Sondheimer—Umeda variational principle The variational principle, which we will review in what follows, is related to the Boltzmann equation of transport theory. This was first used by Umeda [4] to justify Kroll’s treatment [5] of transport coefficients. Kohler [6] also derived the same expression for the kinetic theory of gases. Sondheimer [46] developed this principle for the transport coefficients of metals, yielding corrections to the Bloch theory of electrical conductivity. Boltzmann’s equation can be deduced from the balancing of the rates at which electrons are induced to change from one quasi-stationary state (labeled by the wave number k) to another. The so-called collision rate of change of the distribution function in state k, .1k’ is given by

[af,,/at]~= Jd3k’{f~(l —f,,.)T(k,k’) —f,,.(1 —f,,)F(k’,k)},

(6.1)

where F(k, k’) is the probability of an electron, known to be in state k, being scattered into state k’. In general, it is assumed that r[k, k’] is independent of the external fields. According to the principle of microscopic reversibility, we have T’(k’, k)

=

F(k, k’),

(6.2)

which is an essential point in what follows. On the other hand, the electric field E causes change at the rate —eEV,,f,,+(k/m)Vf,,.

[afk/at]d=

It is convenient to define the new function,

f,,

to

(6.3) ~,,,

by

f,, =f0(~,,) ~,,(af~/a~,,), —

(6.4)

where ~,,is the energy in state k. Then, by substituting Eq. (6.4) into Eq. (6.1), one gets

[af,,/a~]~= (af0/&,,) (L~),,,

(6.5)

where L is the linear operator satisfying =

j’d3k’r(k~k’)(~,,

—

~,,.).

(6.6)

M. lchiyanagi/Physics Reports 243

(1994) 125—182

149

Making use of Eq. (6.4), Eq. (6.3) can be written as (afk/at)d

=

—

(af0/ae,,)(X1j,, + X2q~),

(6.7)

where the components of the electric and the heat currents are given by

j,,

(e/m)k ,

(6.8)

q~= (ck~/m)k,

(6.9)

=

respectively. Here p denotes the Fermi energy. The conjugate thermodynamical forces are chosen as X1

E

=

—

(1/e) Vp,

(6.10)

X2= —(1/T)Vp.

(6.11)

The part of the change of the distribution function in time due to the drift process can be expressed in bilinear form with respect to the forces and the currents. Consequently, we obtain the linearized Boltzmann equation from Eq. (6.3): L4.

=

(6.12)

it,

ir=X1j+X2q.

(6.13)

Let us define the inner product of two functions j’d3k(

(~, ~)

—

4’

and

~‘

as (6.14)

afO/a8k)cbk~k.

Then, by the symmetry of the kernel F(k, k’), it is readily verified that the linear operator L is self-adjoint in this function space: (~,L~)= (1/2)Jd3kd3k’ T(k,k’)(~k

—

~k’)(~k

—

~k’)

=

~

(6.15)

We also see that L is positive definite:

(11’~LçIi)

0.

(6.16) by Eq. (6.4) and the definition of the inner It shouldbybe Eq. emphasized introduction product (6.14) arethat not the independent butoftightly correlated. It would certainly be more desirable to define an inner product determined in a more fundamental way, e.g., by properties of the theory of transport processes itself. We will recognize this point in Section 7. The phenomenological currents are given by 4k

J = Jd3kf~f~=

Q

=

Jd3kfkqk

=

(~,j),

(6.17)

(~,q),

(6.18)

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M. Ichivanagi/Physics Reports 243 (1994) 125—182

respectively. They are expressed in terms of the inner products. Accordingly, if we write =

L/1

X1~1+ X242,

(6.19)

=1

(6.20)

and

L~2= q,

we obtain from Eqs. (6.18) and (6.19)

[.11 [Q]

[L1,

=

[~,

L121 [x11 L2j [X2j’

(6.21)

where the transport coefficients are expressed as L,3

=

(~,,L4~)

=

L1,,

(6.22)

which are symmetric by the microscopic reversibility expressed by Eq. (6.2) or Eq. (6.15). The variational principle can now be formulated. There are a number of different forms: (I) Find the function 4 which makes the functional 2(4,

it)

—

(4,

L~)

(6.23)

a maximum. (II) Maximize (4, L4i)2/(~, subject L4) = (4, L4) to or (~, minimize (~,it). L~)/(çb,it)2. (III) proof Maximize (q~, it) The is quite simple. Let 4’ be the solution of Eq. (6.12). Then, by making use of the fact that = (4, L~)and Eq. (6.16), we obtain E(~)—S(i!i)=(4,Lq~)—2(~,,Lq~)+(i/i,L~i)=(~— i!’,L(~—ifr)) 0,

(6.24)

which we are required to prove for principle (I). To prove principle (II), we note that (ifr, L4) = (ifr, it) = (i!” Li,li) by the condition. Then, Eq. (6.24) implies

(4, L~)

(i!’,

Li/i).

(6.25)

In the same way, we can prove principle (III). These variational principles are used in a following way [4—6,45]. Let us express ~ in the form = ~ (6.26) where & are known functions of k, while the variables a~are to be chosen latter to make a solution of the linearized Boltzmann equation (6.12). By substituting Eq. (6.26) into Eq. (6.23) we obtain formulae containing (&~L~m)and (~, it) which are coefficients of quadratic and linear forms in a~.The variational principles are then applied to give the best choice of a~.This yields a set of linear equations which can be solved by simple algebra. For instance, we assume the form =

>~a~(k)P~(cos0),

(6.27)

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151

where P~(cos0) are Legendre’s polynomials and 0 is the angle between k and E. This form has been applied by Adawi [47] to calculate deviations from Ohm’s law. 6.2. Conduction problems in a magnetic .field The variational principle discussed in the previous section can readily be applied to the Boltzmann equation, provided it has the form of a linear inhomogeneous integro-differential equation with a Hermitian operator acting on the deviation from equilibrium value of the distribution function. This principle, however, has some intrinsic difficulty that prevents a true maximum principle in the presence of a magnetic field or an alternating electric field [23,48,49]. For instance, nothing guarantees that the variational solution for a given class of trial functions is in any way better than some other member of the class. The Boltzmann equation in the presence of a magnetic field H can be setup as (L + M)4

=

(6.28)

it,

where 4 is defined by Eq. (6.4), L is the linear operator defined by Eq. (6.6) and Eq. (6.13). The operator M is the “magnetic scattering operator” (M4),,

=

—(.IeI/m)(kxH)~V,,4~,,.

it

is given by (6.29)

The general effect of the magnetic field thus appears similar to that of the scattering centers. Note that the electrons are scattered as they come into contact with the magnetic lines of force. There is, however, an important difference: this scattering process does not contribute to the entropy production. This is the reason why we have included the magnetic scattering operator in the left-hand side of Eq. (6.28). The linearized Boltzmann equation (6.28) can be solved formally in the form =

X~4~(H) + X2~2(H),

4’1(H) = (L + M)’j,

q’2(H)

(6.30) =

(L + M)~q.

(6.31a,b)

Hence, the transport coefficients are expressed as L,~(H)= (4,(H), (L + M)4~(H)).

(6.32)

It is readily verified that the magnetic scattering operator M satisfies the antisymmetric relation (i,1~’,Mçb)

=

—

(~,

Mi!’),

(6.33)

whichcorresponds to changing the sign of the magnetic field on the right-hand side of the equation. Accordingly, we have L,~(H)= L~,(—H)

.

(6.34)

The antisymmetric property (6.33) prevents a true maximum principle in the form studied in Section 5, since the operator L + M is no longer positive definite nor is it symmetric. However,

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M. Ichivanagi/Physics Reports 243 (1994) 125—182

Baylin [23] has demonstrated that a maximum principle does hold in the magnetic field (see also Ref. [50]). Following Baylin, let us consider 4, to be separated into an even part 4~and an odd part 4,(~)in the magnetic field: 4~= 4,(e) + Then, the Boltzmann equation (6.28) separates into 4,(O).

M4,~°=1 it, L4,~°~ + M4,(e) = 0,

L4,(e) +

(6.35) (6.36)

By solving Eq. (6.36) in the form 4,(0)

=

—

L_~M4,(e)

(6.37)

and substituting it into Eq. (3.32), we obtain A4,(e)

=

~,

A=L—ML1M.

(6.38) (6.39)

It is quite simple to prove that the new operator A is positive definite and symmetric (since L 1 is symmetric). Accordingly, Eq. (6.38) is identical in form with the linear Boltzmann equation (6.12) without a magnetic field, except that L has become A. The operator A has all the virtues of L. Eq. (6.38), thus, leads to a true extremal principle in the usual way as was developed in the previous section. However, the definition of A involves L 1, which may be evaluated by using the ordinary variational principle. These “internal” variational calculations should be carried out to the same order of approximation as the “external” calculations. It is emphasized that such an internal variational calculation in general breaks the time-reversal invariance of the original equations of motion by making use of their retarded (or advanced) solutions with an appropriate initial (or final) condition. For our purpose, the real question concerns the physical meaning of the variational principle. The important point here is that we have separated the variational function into two parts. It should be noted that the even part 4,~is just the time reversal of the odd part çb~°~. It is a part of a variational function for which there is an extremal principle. Eliminating the odd (or even) part might be understood as a kind of contraction of information about the microscopic dynamics of the system. 6.3. High-frequency transport problems The variational treatment of solving the linearized Boltzmann equation in a magnetic field can be extended to high-frequency conduction problems in a Boltzmann gas [34]. In the treatment of transport problems starting from a Boltzmann equation, a high-frequency electric field plays much the same role as a magnetic field. Remember that the cyclotron frequency is simply replaced by the circular frequency of the electric field. In this section, variational principles in the presence of magnetic, H, and high-frequency electric fields, E exp(iwt) are considered, starting from a linearized Boltzmann equation of the form (L+M+Q)Ø=-ir,

(6.40)

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153

where Q is the so-called frequency operator Q4,=iwçb.

(6.41)

In consequence, the function 4, is in general complex and is a function of H, E and w: 4,

=

4,(E, w; H).

(6.42)

In order to formulate a variational principle, it is convenient to introduce the inner product defined by <4,*,

41> =

Jd3k(~L~)4~4i~,

—

(6.43)

where 4, is the function in the magnetic and electric fields and the symbol * indicates complex conjugation. The function 4,1*) denotes the time-reversal of 4, with taking the complex conjugation; w; H)

(4,(E,

=

—

w; _H))*

(6.44)

.

Then, the expression (6.43) gives the identity <4,*, 41>

=

<41*, 4,>)*)

(6.45)

We can also show that the following equations hold:

<4)*, Li!’> <4,*, Mi/i> <4,*, Q’/’>

=

=

=

<41*, L4,>(*), —

—

(6.46a)

<~/,*p~,f4,>)*) <1/1*

(6.46b) (6.46c)

Q4,>(*)

Here, we have assumed that the collision operator L is invariant under time-reversal. It is remarkable that Eqs. (6.45) and (6.46) are formally analogous to the quantum-mechanical identities in a Hilbert space: that is, we have the formulae (cI’, ~t’)= (~P,P)* and (1iQ~P)= (~p Q+~Jj)* for a pair of wavefunctions ~Dand IF, and an operator Q with its Hermitian conjugation Q ~ Thus, for the sake of simplicity, we use the word “Hermitian” for the operator L and the word “antiHermitian” for M and Q. However, it is pointed out that we have in general the identity <4,*, 4,>

=

<4,, 4,*>(*)

(6.47)

and hence the inner product <4,*, 4,> is not in general real. The variational principle can now be formulated as follows: (I’) Find the functions 4,* and 4, which make the functional W[4,*; 4,]

=

<4,*, R4,>

—



—

<4,*,

it>

(6.48)

stationary. Here, R = L + M + Q. For the sake of brevity, where no misunderstanding can result, we use the same operator notation for it as before.

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It is seen that independent variations of 4,* and 4, yield, respectively, the two conjugate forms of the linearized Boltzmann equation (L + M + Q)4,

=

(L

it,

—

M

—

Q)4,(*)

—

afO/a6,,)~,,4,k=

=

(6.49a,b)

it.

The stationary value then is given by W5,[4,*, 4,]

=

—


~>

=

Jd3k(

X1J + X~Q,

(6.50)

which is equal to the Joule heat generated. Then, it is simple to verify that the transport coefficients are expressed as L,1(H, w)

=

<4,,(H, w), R4,J(H, w)> ,

(6.51)

where 4,, are the solutions to the Boltzmann equation of the form 4,1(H,w)=R~j and

4,2(H,w)=R’q.

(6.52)

The symmetry properties are obtained in the form L11(H, w)

=

L,~(—H, —w)

(6.53)

.

Here we have made use of Eqs. (4.46b) and (6.46c). Equation (6.53) is related to time-reversal invariance. We introduce the even and odd parts with respect to time reversal 4,(e)

=

(1/2)[4, + 4,(*)]

4,(O)

=

(l/2i)[4,

—

4,(*)]

.

(6.54a,b)

Then, from Eq. (6.49) we get L4,’°~ + (M +

Q)4~

L4,(e) + (M + Q)4,(o)

=

0,

(6.55)

=

it,

(6.56)

or, equivalently, [L

—

(M + Q)L’(M + Q)]4,(e)

=

it

.

(6.57)

As we have seen in the previous section, there is an extremal principle for Eq. (6.57). There is no difficulty in formally establishing variational principles in alternative forms. 6.4. Upper and lower bounth on transport coefficients The solution of transport equation always requires approximations. For instance, we have to rely on variational methods in order to calculate transport coefficients from a Boltzmann equation. In this section, we will show that by applying a variational principle we can determine upper and lower bounds on transport coefficients arising from a linearized Boltzmann equation.

M. Ichiyanagi/Phvsics Reports 243 (1994) 125—182

155

Let us first consider the linearized Boltzmann equation symbolically written in the form (6.57a)

it,

=

where L is a Hermitian operator. We are interested in a certain inner product of 4, and it denoted by T = (4,*,

it)

.

(6.57b)

The convention we use for an inner product of two functionsf(k) and g(k) is

(f*, g)

fd3kf*(k)g(k)w(k),

=

(6.58)

where w(k) is a real, positive weight factor to be specified in each case. Eq. (6.58) is equivalent to Eq. (6.14). Then, the variational principle [III] in Section 6.2 implies that T = (4,*,

it)

[Re(U*, it)]2/(U*, LU) ,

(6.59)

where U is an arbitrary trial function. This result is used to derive the lower bound on transport coefficients [50,51]. Let us assume that L can be separated into two positive and Hermitian operators H and G, i.e., L=H+G.

(6.60)

We assume that G LG’

1

exists. Then, from Eq. (6.60) we get at once

1 + HG~,

=

(6.61)

from which we have LG’H

=

H + HG’H

LG1H

=

(LG’

—

(6.62)

1)L.

Hence, the linearized Boltzmann equation can be rewritten as L’4,

=

m(LG’

it’

—

1)it,

(6.63)

which leads to a variational principle in the usual form. Then, we conclude that T’

=

(4,*,

it’)

[Re(U*, it’)]2/(U*, L’U)

from which we obtain T = (4,*,

it)

(it*,

G~it)

This is the upper bound on T.

—

{~‘

~)itI}2

(6.64)

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M. Ichiyanagi/Physics Reports 243 (1994) 125—182

We can obtain bounds on the inner product when L is not Hermitian. Remember that the linear operator L can always be written as the sum of a Hermitian part K and antiHermitian part A; L = K + A. Then Eq. (6.56) becomes (K + A)4,

(6.65)

it.

=

We now define 4,* by the equation (K

— A)4,*

=

(6.66)

it .

From these, under the assumption that K1 does exist, we obtain (K + iAK1iA)f= it.

(6.67)

Here,

f

(4,* + 4,)/2.

Similarly, if A

—

1

exists, we obtain

[iA + L(iA)~L]g = g

=

(6.68a)

it,

(1/2i)(4,* + 4,).

(6.69) (6.68b)

From Eqs. (6.67) and (6.69), we can get lower and upper bounds on the inner products (f*, it) and (g* it), respectively. This procedure applies to get bounds on frequency-dependent conductivity in the presence of a magnetic field. In this case, we simply put K = L and A = M + Q (see, Section 6.3). 6.5. Relation between the variational principles in the Boltzmann and the Onsager theories

of transport processes So far, we have discussed the variational principles in the context of obtaining the microscopic expressions for transport coefficients. In this section, we will discuss the relation of the solution of the Boltzmann equation to entropy production. In irreversible thermodynamics (extrinsic) entropy production enters, according to the following formula [52]: =

>JIX,

(6.70)

which defines the thermodynamic forces X. and the currents, f, (i = 1,2, ... ,f). In order to interpret the variational principle in terms of these quantities, we must find entities to which we may give these names. It should be noted that in the Onsager theory the currents are the independent variables and the forces are defined as the derivatives of the entropy with respect to the extensive variables. In what follows, we will show that in the theory based on the linearized Boltzmann equation the currents are given by response currents due to external forces. For our purpose it is useful to write the arbitrary extensive quantity ot.(t) as ot,(t)

=

j’d3k F,(k, t)f,,(t),

(6.71)

M. Ichivanagi/Physics Reports 243 (1994) 125—182

157

where F,(k, t) represents the density of ot,(t). Then, we have J~=

Jd3k E,(k, t)fk(t),

=

(6.72)

since we have set

af,,/at

=

(af,,/at)~+ (afk/at)d = 0

(6.73)

for nonequilibrium steady states. It is usual to demand that F,(k, t) is related to F,(k, t) as

Jd3k[E1(k. t)fk(t) + F,(k, t)(afk/at)d]

=

0,

(6.74)

because the quantity F,(k, t) should be determined by the proper dynamics of the considered system. Accordingly, we have J,

=

—

Jd3kF~(k~ t)(afk/at)d.

(6.75)

This definition coincides with Eqs. (6.17) and (6.18). The conjugate forces are defined in the following way. On the basis of the expression for the entropy S

=

—

kBj’d3kfklnfk~

(6.76)

we obtain =

_(1/T)Jd3~k(afk/at).

(6.77)

In the steady state we have considered, the total change of the distributionf,, is zero, so that ~ = 0. However, if we consider separately the two equal and opposite contributions to this rate of change, from collisions and drift, respectively, we may infer the corresponding contributions to the entropy production; i.e. for the collision process (aS/at)~=

—

~Jd3k

4,,,(af,,/at)~

(6.78)

and for the drift process —(aS/at)d

=

~Jd3k~k(afk/at)d.

(6.79)

In fact, since Eq. (6.79) must equal Eq. (6.78), the effect of the external fields is to reduce the entropy in the system, tending always to produce more order in the electron distribution. It is easy to verify that from Eqs. (6.7) and (6.79) —(aS/at)d

=

~(X~J + X2Q).

(6.80)

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M. Ichivanagi/Physics Reports 243 (1994) 125—182

This is what we should write for the entropy production in the form of Joule heat. In irreversible thermodynamics, this form is proposed by de Groot [29]. By substituting Eq. (6.5) into Eq. (6.73), we obtain 2 , 12 + R21)JQ + R22Q where the matrix R is the inverse of L. This is a quadratic function of 4,,,.

(aS/at)~

=

2 + (R

RI

(6.81)

1J

We call (aS/at)~the intrinsic entropy production, because it depends only upon the dissipative currents. On the other hand, we call (aS/at)d the extrinsic entropy production, because it can be calculated from measurements made outside the system, determining the thermodynamic forces and curents and does not depend upon the way in which electrons are scattered [22]. In terms of the inner product defined by Eq. (6.14), the two contributions can be written as (as/at)~ =

—

~(4,, L4,),

(aS/at)d

=

—

~,(it,

4,).

(6.82)

Then, if we note that T(aS/at)~= 2P[J, J] ,

(6.83)

we see that the variational principle [I] corresponds to the Onsager principle (3.6), which is used to obtain the phenomenological laws (2.6). The intrinsic entropy production is equal to twice the dissipation function (of the first kind). An alternative statement is that we have to maximize (aS/at)~,subject to (aS/at)~= —(aS/at)d. It should be emphasized that intrinsic entropy production cannot depend on any special way of counting collision processes and has the properties of a mathematical invariant of the problem. In this sense, the dissipation function ~b[f, f] is as fundamental as the entropy. The above discussion provides a solid basis for discussing some common aspects of the variational approach. Any considerations of variational principles must, however, emphasize a distinction of the hydrodynamic stage from the kinetic stage. Onsager’s variational principle acts in the hydrodynamic stage, and the Kohler—Sondheimer—Umeda principle in the kinetic stage. The former depends on the macroscopic currents which are much more contracted than the distribution function in the Kohler—Sondheimer—Umeda principle. The microscopic description should be reduced to some small set of variables, considered as relevant for the phenomena under study. Indeed, the contraction of information has given rise to innumerable studies and to an immense literature. Something of the general idea has long been known to many physicists. Nakano [7] has discussed the problem of irreversibility from the point of view of the variational principle. He has shown that if one contracts the information about the kinetic stage by restricting the variational function 4, to a relevant subspace of the function space, the Kohler— Sondheimer—Umeda variational principle leads to Onsager’s. The choice of such a subspace is usually guided by physics (see Section 7.2). As we have seen in Section 6.4, various approximation schemes help to deal explicitly with the Kohler—Sondheimer—Umeda variational principle. It thus appears possible to calculate transport coefficients in the kinetic stage. The relaxation-time approximation is a usual one to solve the (linearized) Boltzmann equation, which discards the irrelevant information at finite time steps.

M. Ichivanagi/Phs’sics Reports 243 (1994) 125—182

159

The resulting value of the entropy production will be proportional to the memory time. An alternative approach is to reduce the variational-function space to a relevant subspace. This approach, if employed in the kinetic stage, neglects a coupling between relevant and irrelevant variables. In this sense, this approach might be regarded as a kind of approximation. However, this contraction of information does lead the Kohler—Sondheimer—Umeda variational principle to Onsager’s principle. Accordingly, this reduction might not be regarded as an approximation in the usual sense of the word, since the two principles are entirely independent laws of statistical mechanics.

7. Variational principles in quantum theories of transport processes The conventional theory of electron transport in solids is developed in terms of the Boltzmann equation. The first use of a Boltzmann equation goes back to Lorentz [53] who formulated the Drude theory of metallic electrons as a kinetic theory. Sommerfeld [54] introduced Fermi statistics in the Lorentz theory. Bloch [55] opened the new area of the quantum theory of solids. It is to be noted that the Boltzmann equation is intrinsically defective at high frequencies and short wavelengths because its asymptotic treatment of atomic collisions ignores those incomplete collisions which occur on space and time scales comparable to interatomic distances and the duration of collisions, respectively. The rigorous quantum-mechanical analogy of the Boltzmann—Bloch equation in the kinetic stage of the molecular motion of constituents is the equation of motion for the single-electron density matrix. A theoretical study of transport phenomena in metals requires the solution of a complicated integral equation for the electron distribution function. The electrical conductivity of a pure metal can be obtained by a special method due to Bloch. The expression for the ideal electrical resistance obtained by such a method is identical with the Grüneisen—Bloch formula. We show that the quantum variational principle restricts the nonequilibrium density matrix to a class conforming to the requirement demanded by the second law of thermodynamics. A general formula for electrical conductivity in the dynamical stage has been given by Nakano [24], who used the von Neumann equation for a system exposed to external fields. Nakano and also Nakajima [56] showed that this formula leads to the Grüneisen—Bloch formula. Their work thus opened the new area of the quantum theory of transport phenomena, the linear response theory [25]. In relation to the linear response theory, the quantum variational principle has been developed by Nakano [7]. The primary purpose of this section is to review such principles, but we will also discuss their physical interpretation, in particular, their relation to entropy production and linearized Boltzmann equation approach. We show that the quantum variational principle restricts the nonequilibrium density matrix to a special class conforming to the requirements demanded by the second law of thermodynamics. 7.1. Nakano 1~variational principle The variational principle in the quantum theory of transport processes proposed by Nakano [7] is concerned with the von Neumann equation for the density matrix of a quantum system subject to

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M. Ichivanagi/Physics Reports 243 (1994) 125—182

external disturbances. The density matrix method has been applied by Nakano [24] and Kubo [25] to obtain general formulae for linear transport coefficients. Nakano’s variational principle is related to this type of investigation, the linear response theory, and has a fairly wide range of applicability in various areas of transport theory. Let p(t) be the density matrix giving the time evolution in the presence of an electric field E(t). If the Hamiltonian of the system in the absence of the external fields is H, the von Neumann equation reads

— PE(t),p(t)]

(a/at)p(t) + i[H

=

0,

(7.1)

where P is the electric polarization of the system. The operator P is related to the electric current by j =

i[H, P]

(7.2)

.

In the SchrOdinger picture, the time evolution is described by a unitary operator U(t, t0), which is a solution of the equation

(a/at)U(t, t0) =

i[H

—

—

PE(t)] U(t, t0) ,

(7.3)

with the initial condition at an initial time t0: U(t=t0,t0)= 1.

(7.4)

In particular, the formal solution of Eq. (7.1) is given by =

p(t)

U(t,

t0)

t0)p(t0)U~(t,

(7.5)

.

Then, since ln(1 + x) has a convergent power-series expansion for unitarity of U(t, t0), we obtain lnp(t)

=

U(t, t0)[lnp(t0)] U~(t,t0)

—

1
<

1, and invoking the

(7.6)

.

It should be noted that this is the formal solution of the equation of motion:

(a/at) ln p(t) + i [H —

PE(t), ln p(t)]

=

0,

(7.7)

which is equivalent to Eq. (7.1). We shall consider two situations: (1) The incoming disturbances: The external field has been growing exponentially from the infinite past E(t)

=

Eexp(st) ,

t <

0

(7.8a)

and the density matrix is required to satisfy the condition lim p(t) = p,, t—+

exp( —/JH)/Trexp( —f3H)

(7.9a)

—X

(2) The outgoing disturbance: The external field decreases exponentially towards the infinite future E(t)

=

Eexp( —st) ,

lirn p(t) =

.

t

>0 ,

(7.8b)

(7.9b)

M. Ichij’anagi/Physics Reports 243 (1994) 125—182

161

Here E denotes a constant field and s is a complex number with an infinitesimal positive real part. /3 denotes the inverse temperature. Let us write [8] lnp(t)

‘~PC+

=

flIF~~exp( ±st)

(7.lOa)

or equivalently p(t) = PC[l + flj’ dxp

IF~p~exp(±st)],

(7.lOb)

for the two cases (1) and (2) in the linear approximation with respect to the external field. As we will see in Section 7.2, this setting is quite analogous to definition (6.4) and is meant to be highly suggestive. The fact is suggested by the form which correctly treats the possible quantum effect. By substituting Eq. (7.10) into (7.7), we obtain L~IF~~=Ej, L_5W~~=Ej,

(7.lla,b)

to first order in E. Here the linear operator L~is defined by L2J~= s1 + i[H, cP]

(7.12)

.

It is convenient to define the inner product of two operators IF and ~Pas

<
~)

=

(7.13)

JdxTrp~IFP~.

This definition of the inner product is suggested by the form of Eq. (7.lOb). The linear operator L2 satisfies the relations as to this inner product

(7.14)

<>= —<<~,L~IF>>.

The variational principle can be formulated as follows. Find the operators ~ ± which make the functional W[P~ +); ~j~(

—)]

=

<<~J~(~); LSP~+)) + <
tp(

+))

—

~

~

El>>

(7.15)

stationary with respect to their independent variations. A pair of operators satisfying this condition are the solutions of Eqs. (7.lla) and (7.llb) and the stationary value equals W[IF~~IF~ ~]=EJ~=

(7.16)

—Ef~,

(7.17) are the phenomenological currents at t = 0 for the two cases. Ifwe take E to be a unit vector in the v direction (v = x, y, z) in Eq. (7.lla) and in the p direction in Eq. (7.11 b), we are led, in place of Eq. (7.15) to the variational principle expressed as a,[~~

~P~]

= <<~~

L5~~) +

<<~~jp)

—

(~~j~’)

.

(7.18)

It is quite simple to see that the operators which make this functional stationary are the solutions of the equations: LSIF~~~ =j~, and

LSIF~’

=j~, .

(7.19)

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M. lchivanagi/Phvsics Reports 243 (1994) 125—182

The stationary value is equal to a,AV[IFV,

IF~] = <> =

—

<>.

(7.20)

The solutions of Eq. (7.19) can be written as IF~A~ =

±t)

(7.21)

where j(t) is the Heisenbcrg operator: /(t) Eq. (7.20) yields

~~([IFY~ IF~] =

Jo

dtexp( —st)

=

exp(iHt)jexp(

— i Ht).

Substituting Eq. (7.21) into

(‘1~

Jo

(7.22)

TrpCjVjM(t + i)~)d).,

which is the time-correlation function expression for the conductivity tensor [25]. As we have seen above, the variational principle proposed by Nakano [7] possesses the remarkable structure, in which the incoming operator ~p(+) is necessarily combined with the outgoing operator i~— which is the time reversal of the former: ~ —) = 9 + ~,i9 being the time-reversal operator defined in Appendix A. Even in the case where the Hamiltonian is symmetric with respect to the reversal of time, two kinds of variational density operators have to be introduced in a pair. This feature is similar to the occurrence of incoming and outgoing waves in the Lippmann—Schwinger theory of scattering (see also Refs. [57, 58]). The equations for the unitary evolution operator U~(t) are replaced by the variational principle from which the collision operator S is obtained. These operators, are related by U+ (t) = U (t)S. Obtaining the two expressions, S = U+(co) and S~ = U( — ~), corresponds to deriving the two expressions, Ef, and — Ef - in Eq. (7.16), respectively. The former describes the up-hill process, the departure from equilibrium, whereas the other describes the down-hill process, the return to equilibrium. The problem of calculating dynamic susceptibilities is closely related to the fundamental questions about irreversibility. Sauermann [12, 58, 59] has used a similar principle to calculate susceptibilities. Balian and Veneroni [57] have established a similar variational principle in which a timedependent state and an observable are the conjugate variables. They have devoted a study to an algorithm obtaining a best estimate for expectation values of observables at time t1( > t0) from knowledge of the density matrix at time t0. The particulars of the problem then appear through the chosen boundary conditions; the variational density matrix should be equal to the initial density matrix at the time t0 and the variational observable should equal the final s tate value of the observable at the time t1. In their theory, the density matrix evolves from the initial density matrix onwards according to the von Neumann equation, while the observable evolves from the final value backwards according to the backwards Heisenberg equation of motion. As is well known, the duality between density matrices and observables in statistical mechanics is formally reminisent of the duality between bras and kets in the quantum mechanics of pure states [60]. Hence, the use of the backwards Heisenberg picture is quite natural in the Liouville formalism of statistical mechanics. However, there is an essential difference. The equations of motion result in from the Lippmann—Schwinger theory are the Schrödinger equation and its Hermitian conjugate, whereas the variational principle by Balian and Veneroni generates the von Neumann equation ~

—

—

M. Ichiyanagi/Physics Reports 243 (1994) 125—182

163

and the backwards Heisenberg equations of motion. Thus, variational functions of a time-dependent observable occur in addition to the variational density matrix. This is in contrast to Nakano’s variational principle. One consequence of Nakano’s theory has been the formulation of irreversibility in terms of entropy production. Unfortunately, the treatment by Balian and Veneroni is not helpful in elucidating the basis of irreversibility (see [61]). This point will be discussed in what follows. 7.2. Reduction from the quantum to the classical variational principles Friction and diffusion processes are manifestations of irreversibility of dynamical processes in the real world, and the problem of calculating transport coefficients is closely related to the fundamental questions about irreversibility. It is desirable to obtain a unified method for calculating transport coefficients that would be applicable to any system. The linear response theory has occasionally been criticized as to its validity for calculating transport coefficients [62, 63]. It is argued by van Vliet [63] that the linear response theory speaks of dissipation but nowhere is the dynamics commensurate with dissipation introduced. No one denies the fact that the linear response theory is essentially exact. However, this theory poses a dilemma in logic, since it applies to reversible as well as to irreversible processes. For applications Kubo’s expressions are too general, so that somewhere a statistical assumption, which is closely related to the fundamental questions about irreversibility, must be made. In this respect attempts to compare the Kubo—Nakano formulae with Boltzmann-type results have been made [64, One point that should be addressed here is that the Hamiltonian can be partitioned into two parts, the proper part of interest and interactions causing dissipation [63, 66]. In the problem of electron transport, Nakano’s variational principle reduces to that of Kohler—Sondheimer—Umeda in perturbation theory. We can use the one-electron representation if the scattering Bloch electrons is due to impurities. Let us consider the system whose Hamiltonian is expressed as 65].

H

=

H0 + H’,


1k>

=

==V(k;l),



=

(7.23a,b)

i:,,,

k~1,

(7.23c)

where 1k> is the eigenstate of the unperturbed Hamiltonian H0 with the Bloch energy c,,, and V(k; 1) the matrix element of the scattering potential. Then, we can evaluate the inner product (7.13). <<~IF>>

=

Jd~{~IFk,lexP[~l— ~k)]~1,kfki(i

—~)+ ~

(7.24)

wherefk =f(~,,)is the Fermi distribution, and IF,, and IF,,, stand for and , respectively. It is not difficult to show 2 W[;j=~~~[ 2 2 V(k,l)1 ~ICk[(~k~l) +~

—

—

)) +

~

—

~V~}J~E]~

(7.25)

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where in the limit s —

0

-~

E

1)~+ ~2]

itO(c,,

—

s.,) ,

(7.26)

and I V(k, 1)12 may be replaced by its average over a random distribution of impurities, N, N, being the number Then, can ~write 1 +); ‘~P1~] of= impurities. <<~ ~ LIi~ + ~>> we + <
— ); jE>> ,

I v,,1 12, (7.27) (7.28)

The operator L is positive definite. In consequence, the variational principle expressed in terms of Eq. (7.27) becomes the one of Kohler—Sondheimer—Umeda for the Boltzmann equation in which the variational functions are related to the distribution functionf,, by Eq. (6.4). We may therefore regard Eq. (7.10) as the quantum mechanical generalization of Eq. (6.4). The above reduction from the quantum to the classical variational principles involves, besides the reduction to the one-particle expressions, also the randomness assumption about the impurity distribution. This result naturally raises the question of why the use of randomness should cause irreversibility. It is a difficult and longstanding problem. The literature on this problem is extensive and varied (see especially Chester and Thellung [67]). With regard to this, it is interesting to note the fact that the reduced one from the quantum variational principle acquires extremum properties of which none existed before [7]. This fact gives us confidence that the use of the randomness assumption is sufficient for ensuing the Kubo—Nakano formulae to show irreversibility in the kinetic stage. As we will see soon, it is possible to formulate a maximum or minimum principle which shows the effects of dissipation in the dynamical stage, say those of quantum dissipation in a manner different from the above mentioned one. 7.3. Contraction of information Nakano’s variational principle possesses the structure in which the incoming operator +) in is 1 +) IP~ even necessarily combined with the outgoing operator p( ~, which is the time-reversal of IP the case of no magnetic field being applied, in contrast with the classical variational principle concerning the Boltzmann equation. We therefore rewrite the incoming operator IP~— simply as IP, and redefine the inner product as



— Tr

=

dxp~IFp~,

(7.29)

Jo

where the bar above the Hermitian operator denotes its time-reversal. Then, the variational functional (7.15) can be written in the form W(IP)

=

—

+ +

.

(7.30)

The condition that makes W(IP) stationary with respect to the variation of IP leads to Eq. (7.16). The stationary problem of the variational principle of this form can be reduced to an extremal problem if one applies a contraction of detailed microscopic information about the system without

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165

any approximation [26]. Let us decompose the incoming operator into odd and even components IP~°1and IP~regarding time reversal as = IP~°~ + IP(e) ; = IP~°~ and ~(e) = (7.31) —

Then, by applying Eq. (A.8) we obtain + ~(e) I

~°)

~>

=

,

(7.32)

where I m> denotes an eigenstate of H while I th> denotes an eigenstate of its time-reversal H (see Appendix C). In terms of these components, as shown in Appendix C, = ~pmnW~/n’[S(I
In> 12_ I 12)

IP~°~l n>]

.

(7.33)

Similarly, we have

=

>~PmnW~’ ,

(7.34)

Eq. (7.34) shows that the external field E only couples with the odd component IP~°~ and the even component IP~is not immediately affected by the external field. Hence, the even component ~ can be eliminated through the variational principle by expressing it in terms of the odd component as

=

~Wnms’.

(7.35)

This reminds us of the situation which we have seen in the Onsager—Machlup theory; the stationary variational principle becomes insensitive to the even variables as t 1 (an initial time) (see Section 5). By substituting Eqs. (7.33)—(7.35) into Eq. (7.24), we obtain the variational principle of the form: Make the functional —+

W(4,) = 2<4,;jE> maximum. Here, 4, =

=

—

<4,; 2~4,>

~

(7.36)

IP~°~ and

sq5 + [H, [H, 4,]s1].

(7.37)

The maximal principle is proved on the basis of the fact that the operator s is positive definite:

<4,; 2’~i/i>=


2~4,>,

<4,; 2’,4,>

0.

(7.38)

It should be noted that condition (7.38) formally corresponds to Eq. (6.16) [or Eq. (6.25)] and the operator £f. to the operator A defined by Eq. (6.39) for the classical cases. Accordingly, a clear analogy between the classical and quantum variational principles has been demonstrated. The deduction from the former to the latter principles will be executed systematically. Much more complicated variational principles have been developed by Blount [32], and Robinson and Bernstein [68]. Before concluding this section, we want to emphasize that the extremum problem comes about either from the usage of randomness or from the contraction of information. The use of randomness is evidently related to the “Stosszahl Ansatz” while the contraction of information, as we have

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M. Ichivanagi/Phvsics Reports 243 (1994) 125—182

seen in Sections 6.2 and 6.3, is related to the use of the retarded solutions to eliminate irrelevant degrees of freedom. The irreversibility created by the contraction of information is thus purely due to microscopic dynamics. This result is of cause in agreement with the known fact that we have to specify initial conditions but never require any final conditions for the microscopic equations of motion.

8. The variational principle for dynamical susceptibilities In Section 7.1, we have seen that Nakano’s variational principle can be used to derive the time-correlation function expression for the conductivity. It will be adequate to insert here the variational principle for dynamical susceptibilities to emphasize the general applicability of Nakano’s idea. Nakano [7] and Sauermann [12] have developed a variational principle for the response functions. Consider a Hamiltonian system subject to an external perturbation of the form Hext(t)

=

—

~ A,F,(t) ,

(8.1)

in which the time-dependence of F,(t) is assumed to be the same as E(t) in the previous section. The response of the observable A~is given by the dynamical susceptibility: X,1(s)

=

dtexp( —st)<>,

(8.2)

where A~(t)denotes the Heisenberg operator, exp(iHt)A3exp( —iHt). The dynamical susceptibility X,3(s) can be calculated by X,3(s)

=

(8.3)

/3<> + /3is<>.

Therefore, it suffices to establish a variational principle for the relaxation function, <>. Consequently, we can write down a variational functional of Nakano’s type 1 +); 4,1-)] = ~~— ~>,

(8.6)

which is the relaxation function. As a simple illustration, let us consider the case in which the Liouville operator L, has changed as =

L2 + L~’>

and the new variation functional is defined as 1 +); 41( —)] = (iL’i/i~~); A 1 F~[41 1>> + ~
(8.7) + I)

—

~ L~I/I1+))

.

(8.8)

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167

For example, let us consider a coupled electron—phonon system. In this case, L~stands for the electron and phonon systems and L~’~ describes the electron—phonon interaction: L,A

[He +

=

H~,A]+ sA,

L~’~A = [Hep,

A]

(8.9a,b)

.

He and H~denote the Hamiltonians of the electron and the phonon systems, respectively, and H~ is the electron—phonon interaction Hamiltonian. In the most general case, we must include interactions between electrons and scattering centers, interactions between phonons and scattering centers and phonon—phonon interactions. Then, it is easy to verify that F~1[~]is stationary for 1~1 = A 1~ = A, (8.10) L~IF 1, L’~IF and its stationary value is

F~

1~]= <
1>>

(8.11)

.

According to Eq. (8.5) it is suggestive to make the ansatz

(8.12) where C~±)are variational parameters. By making use of Eq. (8.12) in Eq. (8.8), together with Eq. (8.5) we obtain 1 — )* + C1 + )] F, 1 ~] — C1 — )*C( + )<> . (8.13) F~1[~]= [C 1[IP~ +); IP Its stationary value is found to be 1 + ); IP1 — )]) 2/<> . (8.14) 1[IP This corresponds to an approximation of Eq. (8.11), which may give the first-order expansion of a memory function. If we do not make the approximation involved in Eq. (8.12), it is interesting to rewrite Eq. (8.11) in the form =

F~

(F,

1 +); IF~~] = F, 1[IF

H,

=

1

+);

IF1~]

—

~
1[IF

1) ,

LL~S’~.

(8.15) (8.16)

In writing Eq. (8.15) we have used Eqs. (8.5) and (8.6). Equation (8.15) can be written in the form 1~ IF~1]—HF~ 1], (8.17) F~~[IF~’~ IF~] = F,1[IF 1[IF~’~ IF~ where TI is the operator defined by 11F

1 1[IF

~

IF1—)]

=

<>

.

(8.18)

Eq. (8.17) is analogous to the Dyson equation.

9. The variational principle for dynamical structure function For classical dynamical correlation functions of linear response, the variational principle in the Laplace-transformed Liouville equation has been formulated by Gross [69]. The same idea of such

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a variational principle has independently been developed by Davies [70] for coupled electronphonon system. These are not an extremal principle. 9.1. The variational principle in the Boltzmann equation Let us consider the linearized Boltzmann equation r, t) + v~F(v,r,

t)

=

I[F(v, r, t)],

(9.1)

where I[] denotes a linearized collision operator. The perturbed distribution functionf(v, r, t) is defined according to F(v, r, t)

IP(v)[l +f(v, r, t)]

=

(9.2)

in terms of the total distribution function F(v, r, t) and the Maxwellian distribution IP(v)

(/3/2it)3exp( _/3v2/2)

=

.

(9.3)

We are interested in the density—density correlation function C(k, t)

=

j’d3v IP(v) Jd3rexP(ikr)f(v~r, t)

(9.4)

or more directly its transformation C(k, s)

=

J

dtexp( —st)C(k, t),

(9.5)

to which the dynamical structure function S(k, w) is related via S(k, w)

2 Re C(k, s

=

=

iw).

(9.6)

Let us introduce the transformed distribution function which is defined as f(v, k, s)

=

j’d3rJ dtexp( —ikr — st)f(v, r, t).

(9.7)

Then, Eq. (9.1) can be written in the form ELf(v, k, s) I

=

=

(s + ikv)

1 —

(9.8)

,

I

.

(9.9)

The inner product in this case is defined as (a,b)C

=

Jd3vIP(v)a*(v)b(v).

(9.10)

Let us consider the functional [71] W[4,]

=

(1, 4,) + (4,, 1) — (4,, IL4,)~.

(9.11)

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169

The variational principle can then be formulated as follows: Make the functional W[4,] stationary, subject to arbitrary variation of the function 4,. It is easy to verify that a function satisfying this condition is a solution of Eq. (9.8) and the stationary value is given by W[f]

=

If we put 4, W[4,]

=

(l’f)C =

f+

C(k, s)

=

5f

C(k, z)

(9.12)

.

in Eq. (8.11), we have

—

(~5f,löf)C

.

(9.13)

In consequence, recalling the negative semidefinite character of the operator l, we see that Re { W[4,] } yields a lower bound to the dynamical structure function S(k, w). The above variational principle is used as follows. Let us choose the general form 4,(v, k, s) = >~A1(k,s)IP1(v) ,

(9.14)

in which A1 constitute the variational parameters, which will be determined by the Rayleigh—Ritz parameter variation scheme, and IP~(v)are known functions of v. This ansatz is analogous to Eq. (6.26). We often employ the Burnett polynomials for cP1. We often require that IP0(v) is an exact solution of Eq. (8.8) in the free-particle limit. Note that this solution represents the limiting behavior of the fluctuations in the regimes of extremely high frequencies and/or extremely short wavelengths in which the Boltzmann equation is intrinsically defective. 9.2. The variational principle for the Bethe—Salpeter equation An interesting variational approach for electron transport in an electron—phonon system has been developed by Davies [70]. This approach is different from those referred in Section 7 in several respects. The primary difference is that it is formulated from the Bethe—Salpeter equation, rather than the density-matrix approach. We consider a coupled electron—phonon system, whose Hamiltonian is given by H I_I

110

r~i

=

H0 + ~

(9.15)

_V + — L~,,a,,a,, T _V’~i \ — ~fL~q)ak

+

~‘i’-~ ~~~qUq

g+~.

IL.+

a,,qkuq

U,,

1\

+ Uq)

where a,,~and a,, denote the creation and annihilation operators for an electron with wave number k and b~and bq those for a phonon with wave number q. ).(q)denotes the coupling parameter of the electron—phonon interaction. When one develops the theory of transport processes in terms of a diagrammatic scheme, the transport equation usually emerges as an integral equation for certain vertex functions. In particular, we can obtain the equation for various correlation functions S’~(q)= _~vIL(k q/2)Av(q; k)S’(k)S’(k — q) , —

(9.18)

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itt. Ichivanagi/Phvsics Reports 243 (1994) 125—182

where we are using the four-dimensional notations: v~(k— q/2) S’(k)

=

~(k° — q~/2)/m, p = 1, 2, 3, PO~

S~(z,)= [z,

S’(k -— q) n

=

—

c,, — G,,(z,)] ,

[S~q(z~ — q0)j’,

=

(9.19a) (9.19b)

z~= p + iit(2n + 1)/fl,

(9.19c)

0, + 1, +2,

=

Because we are using the finite-temperature Green functions, we take q0 = 2it mo/fl, n0 being an integer. It is then to be understood that we will let q0 s = ~ + iw in the final stage of the calculation. Here in Eq. (9.19b), Gk(z~)is the proper self-energy part. It is simple to show that the vertex function AV (q; k) in Eq. (9.15) satisfies an integral equation of the form —*

(9.20)

A”(q;k)= vv(k_q/2)+~Iq(k,ki)Sf(ki)SI(ki_q)Av(q;ki), where Iq(k, k,) is the irreducible scattering function. For the sake of simplicity, let us take 2D(k— k,) , Iq(k, k,) = I(k — k,) = ,~.(q)I D(q) = 2Qq/(Q~— z(n)2) ,

(9.21) (9.22)

which yields the ladder approximation for the vertex functions. Correspondingly we evaluate the electron proper self-energy part in this approximation in the form G(k)

~I(k

=

—

k 1)S’(k,)

(9.23)

.

It is convenient to invoke the transformation to a new function Fv(q; k); A~(q;k)

Fv(q; k)[S’(k — q)’

=

—

S’(k)~].

(9.24)

By making use of Eq. (9.24), we can write Eq. (9.13) as 1”(q) = _~v~L(k q/2)Fv(q; k)[S’(k) — S’(k q)] S’ It is readily verified that F’~(q; k) obeys the equation —

—

— q/2)[S’(k) — S’(k — q)]

—

(qv,,q 12 —

(9.25)

.

qo)Fv(q;k)[Sf(k) — S’(k — q)]

— F’~(q;k)]Pq(k,ki)= 0,

Pq(k, k,)

=

[S’(k)

—

(9.26)

S’(k — q)]I(k — k1)[S’(k,) — S’(k1 — q)]

.

(9.27)

Equation (9.26) is reminiscent of the Boltzmann equation. The variational principle can now be formulated as follows: Make the functional S°’[F~F”] =

~ x F’

1(q; k)F”’(q; k)} + (l/2)~[F”(q; k,)

—

q/2)F~’(q;k)—(qv,,q12 —qo) F”(q; k)]Pq(k, k,)[Fv(q; k 1) — F”(q; k)] (9.28a)

M. Ichiyanagi/Phvsics Reports 243 (1994) 125—182

171

stationary, subject to arbitrary variations of the functions P and F”’. The functions satisfying this condition equal the solutions of Eq. (9.26) and the stationary value of the variational functional 5P~[.] then is given by Eq. (9.25). For the latter purpose it will be convenient to rewrite (9.28a) as S’~[A”;A”] =

~S’(k)S’(k — q) {v”(k — q/2)A’t(q; k) + v~(k q/2)A”(q; k) + A~(q;k)A”(q; k)} + ~~S’(k)S’(k q)A~(q;k)I(k — k’)A”(q; k’)S’(k’)S’(k’ — q) . (9.28b) —

—

—

The above formulation leads to alternative variational principle for the long-wavelength limit. To reformulate the variational principle, we will make use of the Ward—Takahashi identity of the form qF(q; k) q 0F°(q;k) = 1 . (9.29) —

Remember that the longitudinal conductivity is given by 2qo/q2)S°°(q). (9.30) a(q) = — (ie The above variational principle is very general. However, it is not easy to find a simple way to apply it for the case of arbitrary value of q. If we pass to the long-wavelength limit, some progress can be made. To do this, let us make an ansatz to order q2.

F°(q;k)= —(1/q

0)+f,(q;k)+f2(q;k)

(9.31)

.

Note that we have utilized the Ward—Takahashi identity for q = 0 in writing Eq. (9.31). The second order part f2(q; k) can be systematically eliminated in the calculation of S°°(q).By making use of Eq. (9.31) in Eq. (9.26), after some manipulations we obtain the equation forf,(q; k): [qv,,/q0 =

+ q0f1(q; k)] [S’(k)

S’(k

—

—

q0)] + ~[f,(q;

k1) —f,(q; k)]Pq(k, k,)

(9.32)

0.

For S°°(q), we have S°°(q)= —~A°(q0k)S’(k

q0)[S’(k) — S’(k — q)]q(a/ak)S’(k — q0)’f1(q; k)

—

.

(9.33)

If we make the transformation f1(q; k)

=

g,(q; k) + [S’(k)

—

S’(k

—

q0)]’-~-aS’(k— q0)/ak,

(9.34)

q0 from Eq. (9.18) we obtain S°°(q) = —~-~-v,,[S’(k)— S’(k q)]g,(q; k). q0 It is not difficult to verify from Eq. (9.20) that g1(q; k) must satisfy the equation —

A°(q0k)S’(k

—

q0)[S’(k) — S’(k

+ q0g,(q; k)[S’(k) — S’(k =

0.

—

—

q0)]qaS’(k

—

(9.35)

q0)’/ak

qo)] + ~[g1(q; k,)

—

g,(q; k)]Pq(k, k1) (9.36)

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172

If we discard the last term of Eq. (9.36), we would have f~(q;k)

=

—

qv,,/q~,

(9.37)

g?(q; k)

=

—

A°(q0k)S’(k

—

q)-~-ôS’(k qo)’/ök. —

(9.38)

We then reformulate the variational principle in the following form: Make the functional S°°[a,b]

=

(a + h

—

ab)4,(q) + abi/i(q)

(9.39)

stationary, subject to arbitrary variations a and b. Here, in Eq. (9.38), 0(q;k)g~(q;k)[S’(k) — S’(k — q)] , 4,(q) = >~qof1 ~

(9.40) (9.41)

and the variables a and b define the trial functions f 1(q; k)

=

af ,°(q;k),

g1(q; k)

=

(9.42)

bg~(q;k).

Accordingly, the condition that S°°[a,b] stationary gives 2(q)/[4,(q)— 41(q)] a = b = 4, and corresponding stationary value becomes S°°(q)= 4,2(q)/[4,(q)

— i/i(q)]

(9.43)

(9.44)

.

Then, with some manipulations, we obtain from Eqs. (9.30) and (9.44) the expression for the conductivity which is identical to the standard result obtained by the Kohler—Sondheimer—Umeda variational principle. For details, see the original article by Davies [70]. 9.3. Relation between the variational principles in the coupled electron—phonon system It is a simple matter to write the principle formulated by Davies [70] in the form proposed by Nakano [7, 72]. In relation to the form of the Hamiltonian (9.15) we can assume the variational operator P~±)in Nakano’s theory as ±~ =

~A1 ±1(k, k*; h)a~7’a,,~ , k*

=

k

—

Q

.

(9.45)

Here A1 ±~(k,k*; b) denote the operator valued wavefunctions, which may be written in the form ±>(k, k*;

b)

=

41 ±1(k,

k*) + ~{ii~ ±~(k,k*; q)bq + h.c.} ,

(9.46)

where h.c. means the Hermitian conjugate. In Eq. (9.44) we have assumed that the phonon system can be treated as if it were in thermal equilibrium (see [70]).

M. Ichiyanagi/Physics Reports 243 (1994) 125—182

173

By making use of Eq. (9.44) into Eq. (7.18), we obtain r~~[A1+); A1—~]= >~[A1 + 1(k, k’)(j~(Q);a,,~a,,>> A1 — 1(k, k’)<>] —

+ >~>[fl1 — 1(p, p’)A1

j,~(Q)

=

>(p

—

+ ~(k,k’)<>

+ ~A1

—

)(p, p’)A1

+ ~(k,k’)<
+ h.c.)

+ ~A1

—

1(p, p’; q)A1 + ~(k,k’)<>

+~

—

)(p, p’; q)A1 + )*(k, k’; q’)<
+ ~A1

—

)*(p, p’; q)A1

+ ~(k,k’; q)<>] ,

Q/2),,aa,,*,

(9.47) (9.48)

denotes the pth component of the electric current opci-ator, and L~°’ and L~’~ stand for the proper and interaction parts of the Liouville operator, respectively. By equating the derivatives of ~[~] with respect to 4( ±1(p, p’; q’) to zero owing to the stationary condition for the variational function, we get A1 + )*(k k’; q)

=

—

A1

+ ~<
A1

=

—

A1

—

—

)*(k, k’; q)

~~aap.btq;

11~(q)a,,~a,,>>,

(9.49a)

TI

5(q)a,,~a~>>, 1a,, a,,b~} L~ . TI~(q)= {~(aapbq; l~ By making use of Eqs. (9.49), we can write Eq. (9.47) as a,,~[IP1 ~

IP1

)] =

~[A1 —

+ 1(k,

(9.49b) (9.50)

k’)<> A1 ~(k,k’)<<,j~(Q);aa~>>] —

—

~~A1 ~(p, p’)A1 + ~(k,k’)<>

+ ~~A1

—

)(p, p’)A~+ )*(k, k’)(a~a,;H~(q)a~a,,.>>

J~+ x // \\a,,+ a,,,.Lw ak+ a,,’tiq

1

Eq. (9.51) is analogous to Eq. (9.28b) of the theory due to Davies [70]. As is well known, in the thermoelectric phenomenon in an electron—phonon system, phonons deviate more or less from the thermal equilibrium states. In fact, on the basis of the Boltzmann equation for a coupled electron—phonon system, Sondheimer [46] has proven the Kelvin relations even if the phonons are not in thermal equilibrium. Ziman [20] and Kasuya [73] have considered a more general case so as to take account of interactions other than the electron—phonon interaction. Nakano [72] has developed the quantum variational principle in the most general case of electron—phonon system which can be reduced to the classical theory of the type obtained by Ziman and Kasuya. In Nakano’s theory, we consider the variational operators of the following form

174

M. Ichivanagi/ Physics Reports 243 (1994)125-182

!P1 ~

=

cP1 ~ + ~A1 ±1(q, q’)b~bq,

(9.52)

where IP~±)are given by Eq. (9.45). The second term of the right-hand side of Eq. (9.52) describes the deviation in the phonon distribution function from its equilibrium value. A phonon—impurity interaction would be essential for such a system.

10. Concluding remarks In this report we have reviewed the variational principles associated with transport processes. We have concentrated in particular on linear processes and have discussed the three types of variational principles which are concerned with the three different stages of irreversible processes; the dynamical stage, the kinetic stage, and the hydrodynamic stage. A recent review article by Nakano [74] in which he has shown that contraction of microscopic information leads to irreversibility, may complement the present article. Onsager and Machlup [19] have demonstrated how the stationary principle of dynamics is transformed to the extremum principle of the irreversible thermodynamics. This can be achieved by solving the Euler Lagrange equation. We are interested in the solution which is a superposition of a regressive and antiregressive motion. The most probable path, which makes the action integral extremum, is the path that can be described by the solution. This is a point emphasized by Hasegawa [75]. The variational principle of quantum transport theory is a stationary principle. This leads to a disadvantage that nothing guarantees the variational solution for a given trial functions to be in any way better than some other member. It is to be noted here that the variational functional (7.15) has no directionarity in time while the stationary value of the solution has no time-reversal invariance. This is due to the usage of the retarded (or advanced) solution to the von Neumann equation. This solution describes only the time development of a class of solutions of the microscopic equations of motion. This is the usual way time-reversal invariance is broken in macroscopic description. Accordingly, if one wants to understand directionarity in time in the definition of the variational functional, one should eliminate the even component of the density matrix with respect to time reversal. Such an elimination reduces, as we have seen in Sections 5 and 7, the microscopic stationarity principle to an extremum principle. We next recall that the variational principle in linear irreversible thermodynamics can be formulated in the form of the classical Hamilton principle if the method due to Djukic and Vujanovic [13] is utilized. The outcome of Section 3.4 unifies the two approaches, both of which have been developed by Onsager and Machlup [19, 36]. Here we want to stress the fact that the hypothesis of a generalized entropy depending on the currents has opened the way to a thermodynamic theory of the third type, the extended irreversible thermodynamics (EIT) [76, 77]. The extended irreversible thermodynamics was born out of the double necessity to go beyond the hypothesis of local equilibrium and to avoid the paradox of propagation of thermal disturbances with an infinite velocity. We believe that the formalism set forth in Section 3.4 provides new perspectives of the extended irreversible thermodynamics. We have established the variational principle of the Fokker—Planck equation. We have used the variational functional ~[0; P], (5.19), in which 0(cç t) is the variational function while P(cz, t) is not.

M. Ichivanagi/Phvsics Reports 243 (1994)125—182

175

2’[O; P] plays the dual role: it determines the Fokker—Planck equation for fluctuations and gives the phenomenological laws. It has been demonstrated that the transport coefficients are the same L,~that determines the microscopic dynamics of fluctuations. Thus, it would seem that we have here the nucleus of an alternative interpretation for the usage of microscopic reversibility to proving the Onsager reciprocity relations. We have reviewed the variational principle which is formulated from the finite-temperature Bethe—Salpeter equation by making use of a generalized Ward—Takahashi identity. The hope in constructing such a theory is to find a thermodynamical counterpart of the so-called diffusion current which is closely related to the interference of the two electron propagators. An interesting question, which merits further consideration, is how to generalize the variational principle to take account of the collision—drag effect. For nonlinear irreversible processes [78, 79], we are unable to achieve in practice the programme reviewed in this report. It would be interesting to develop variational theories for the generalized (say, differential) transport coefficients [80, 81]. We remark additionally that Muschik and Trostell [2] give a short review of variational principles in thermodynamics.

Acknowledgments Anyone familiar with the original publications will understand just how much the author has been influenced by the work of H. Nakano. Indeed, the author has benefited from numerous conversations with Professor Nakano during the past six years. He wishes to thank H. Hasegawa, G. Sauermann, M. Sugiyama, M. Hattori and K. Kitahara. Much of the writing of this review was supported by the Grant-in-Aid for Scientific Research from Japanese Ministry of Education, Science and Culture.

Appendix A. A note on the Djukic—Vujanovic theory A simple version of the Djukic—Vujanovic theory applied to the phenomenological law is to chose the auxiliary function i/í(t; p) as 41(t;p)= PR/2J t,,

=

t/p,

dstanh(s+f(p)),

limf(p)

=

p >0,

0.

(A.1) (A.2,A.3)

0

0

Then, it is easy to see that (d/dt)41(t;p)

Rtanh(t/p +f(p)).

=

(A.4)

Accordingly, we obtain lim(d/dt)iJi(t;

~

=

(R ift>0 ~ —R , if ~<0

.

(A.5)

176

M. lchivanagi/Phvsics Reports 243 (1994) /25—182

By making use of the above results into Eq. (3.25), one is left with Rc~+ Scx

0,

=

0),

(t >

(A.6)

R~—S~=0, (t<0).

(A.7)

Equations (A.6) and (A.7) are describing the regression and the antiregression motion, respectively. It is to be noted that these equations are equivalent to 2~) (A.8) = (S/R)~( = y Verbally, in effect, the use of the auxiliary function 41(t; p) is to select the regression or the antiregression solution to Eq. (A.8). .

Appendix B. Analysis of the Onsager—Machiup theory We consider a simple, one-dimensional Gaussian Markoff process characterized by the stochastic equation Rc~+s~=r,

(B.l)

where ~ denotes the random force having a zero mean. We have for the conditional probability density p(~(2),t 2 I ~1), t,) [82]: 2),t p(~( 2 I ~(1),t1) (2itL/y~[ 1

=

x exp{

—

—

exp(

2y(t2

—

(y/2L) [~~exP(

—

(B.2)

—y(t2

Here y = s/R and L = 1/R. This probability density can be obtained by the path integral method. To see this, let us note that the Markoff process is described by the Fokker—Planck equation

~

(B.3)

for the probability density. In the limit of kB 2),t p(~ ~

~(12)

2 ~)

=

I~

t,)

~(cJ~)exp— (i/kB)

= ~

the solution to Eq. (B.3) is given by

—

=

(1/4L)(c~+

—~0,

.

dt .2i’~,c~),

(B.4) (B.5)

The value of the path integral (B.4) can be evaluated by making use of the classical path which is determined by the Euler—Lagrangian equation (d/dt)(a2’/aci) — ~/öx) = 0 .

(B.6)

M. Ichiyanagi/Phvsics

Reports 243 (1994) 125—182

That is to say, ~= +y2~,

177

(B.7)

the boundary conditions of which are specified by c~(t 1)=c~’~=Aexp(yt,)+Bexp(—yt1), ~x(t2)=

~(2)

=

(B.8)

Aexp(yt2) + Bexp( —yt2) ,

(B.9)

The classical path, which makes the “action integral” stationary, is evaluated by making use of the solution ~x(t)= Aexp(yt) + Bexp( yt) 1exp( y(t ~(2) — o~’ 2 — t,)) —

—

exp(y(t

=

1

—

exp(

(1)

+ ~

—

2y(t2

—

t2))

—

t1)) ~ —t “

(2)

exp~ Yk 2 “exp( —y(t + t,)). (B.iO) 1 exp( — 2y(t2 — t,)) This solution demonstrates that the stochastic process consists of the regression and the antiregression. Both are characterized by the stationary variational principle of dynamics! [75] Accordingly, by making use of Eq. (B.10) into the action integral, we obtain [ . 1 ~ [~(2) c~’~exp( y(t2 — I I dt~(o~,o~) (B.11) [J~ ~ 2L 1 exp( — 2y(t2 — t,)) Hence, by making use of Eq. (B.l1) into Eq. (B.4), one is left with —

~

—

—

(‘12

—

—

=—

.

—

2~, t P(o~ 2 I ~

r t,)

=

N exp~ L

1

/(‘t2 —

(I/kB) (

dt

\Jtj

~

~)

)

(B.12)

L

/m1nJ

N denoting the normalization constant, so that

I~

t1)d~= 1.

(B.13)

It is emphasized here that the minimum principle of the form (A. 12) emerges from the stationary principle of dynamics. The positiveness of ~ c~)is essential to obtain the minimum principle [75].

Appendix C. Time reversal problem It is clear that the canonical commutation relation between the position and momentum operators are not invariant with respect to the transformation t t since only the momentum operator changes sign. However, the invariance requirements demanded by the reversability of physical reality can be satisfied with an antiunitary operator i9 = TC, where T is a unitary operator representing the transformation t — t, and C is the antilinear operator of complex —‘

—~

—

M. Ichivanagi/ Physics Reports 243 (1994)125—182

178

conjugation, so that in terms of the wavefunctions characterizing the state I a> in the Schrodinger picture goes to OIa> = eJIq>dqi//b(q~t)

=

Jlq>d~O41b(~~t) =

Iq>dq41~*(q,

—

(C.l)

t).

Consider the Schrödinger equation expressed in the form

i(a/at)Ia>

=

HIa>

Then, by application of

(C.2)

,

e,

it is readily seen that

ei(a/at)Ia>= —ie(a/at)Ia>=i(a/at)eIa>. That is,

i(a/at)Ia>

=

HIa> ,

(C.3)

a> and H = e Hr1. For an eigenstate I n> with an eigenvalue E~satisfying =

I

HIn>=E~In>,

(C.4)

(CS)

application of 0 yields HIn>=E~In>.

(C.6)

That is, an eigenstate I n> immediately leads to an eigenstate I ii> of H with the same eigenvalue E~, where In> is equal to In>:

HIn>E,jn>,

EnEn.

(C.7)

It is to be noted that in the absence of magnetic field and rotation we have H = H, hence Eq. (C.6) states that both I n> and I n> are eigenstate of H with the same eigenvalue E~. It is quite simple to show that for a Hermitian operator Q

Jdqdq’41n(q’~t)41~(q, =

=

—

~dqdq’
—

t)

I Q I q’> 41~(q’,t)41~(q,t)

=.

(C.8)

The inner product (7.23) can be written as Prnn

=

=

~pmno~n’



—

,

Wmn

=

(C.9)

,

Em

—

E~.

M. Ichiyanagi/Phvsics Reports 243 (1994) 125—182

179

Therefore, application of Eqs. (C.7) and (C.9) yields

n’<~IIPI~><1~I IFIn>

~pmn0

=

=



(C.10)

,

where the bar on the upper right of the bracket shows that we should replace j~i,,and H in the inner product defined by Eq. (7.23) with their time-reversal operators Pc and H, respectively. Similarly, we can prove that = ~pmn ,n’ (s + lWnm) =

>~pmnwrnn’ (s + 10mm)

. In the absence of magnetic field, we write Eqs. (C.10) and (C.11) in the form =



=

,



=

.

(C.11) (C.12)

Appendix D. The relative entropy formula We want to solve Eq. (7.7) in the general form lnp(t)

=

lnp(to) + 4,(t; t0)

,

t

t0

(D.1)

.

By substituting this into Eq. (7.7), we obtain

(a/at)4,(t;

t0) + i[H

—

PE(t), 4,(t; t0)]

=

i[PE(t)

—

H, lnp(to)]

.

(D.2)

The formal solution of Eq. (D.2) is given by 4,(t; t0) = jds U(t, s)i[PE(s)

—

H, ln p(t0)] U~(t, s).

(D.3)

Here U(t, s) is the unitary evolution operator defined as a solution to the equation of motion

(a/at)U(t, s) = —i(H — PE(t))U(t, s) with the initial condition at t U(t

=

s, s) = I

.

=

(D.4)

s: (D.5)

We stress that in choosing the retarded solution of the inhomogeneous equation (D.2) we have broken the time-reversal invariance of Eq. (7.7). We have assumed that the system was in thermal equilibrium at an infinite past. As a consequence, the density matrix is expressed as an integral over the entire past. This is typical the way time-reversal invariance is broken in macroscopic equations, which describe only time development of a class of solutions to the original microscopic equations. However, it has been argued strenuously by van Kampen [62] that no irreversibility is entered the picture. It would be desirable to explain why the use of the density matrix should create a difference between past and future where none existed before. To reply van Kampen’s specific objection, we have to define an entropy production in the framework of the density matrix formalism. To do this, we utilize the concept of relative entropy [9, 83, 84]. A relative entropy of

180

M. Ichivanagi/Phvsics Reports 243 (1994) 125 182

two states described by p(t) and p(t0) is given by S[p(t) I p(to)]

=

kB Tr p(t) {ln p(t)

—

ln p(to)}

0

(D.6)

,

which is nonnegative by Klein’s inequality. The relative entropy is a measure which clarifies how far from the initial state p(to) the state p(t) evolves. By making use of Eq. (D.l) together with Eq. (D.3) into Eq. (D.6) we obtain S[p(t)Ip(to)]

=

Tr

dsp(s)i[PE(s)

—

H, lnp(t0)].

(D.7)

Jto

We now denote by P(t; t0) the entropy production defined as the time derivative of the relative entropy [8,9] P(t; t0)

=

(d/dt)S[p(t) I p(to)]

(D.8a)

or equivalently P(t; t0)

—

=

(d/dto)S[p(t) I p(t~,)].

We know that if lim P(t; t0) at t0

—*

(D.8b)

—cc exists then the initial-time average

(‘0

P(t)

lim (l/T0)

dt0 P(t; t0)

(D.9)

J—T0

also exists and is equal to the former. By making use of Eq. (D.8b) into Eq. (D.9), we have P(t)

=

lim (l/T0) {S[p(t) I p( To

—

T0)

—

S[p(t) I p(O)] }

—~X

Then, we obtain P(0)= lim S[p(0)Ip(—To)]/To0,

(D.I0)

-. ~

where the inequality is based upon Eq. (D.6). Substituting Eq. (D.6) into Eq. (D.lO) yields P(0)

=

lim (l/T0) T0-.cr~

(‘0

dsp(s)i[H, P]E(s)/T.

(D.ll)

J—T0

Here we have used the fact that p( —cc) = p~.Accordingly, P(0) is equal to the initial-time average of the entropy production (say the Joule heat) and is nonnegative in the steady state considered. The Joule heat is a manifestation of the irreversibility of dynamical processes in the real world. In short, a nonnegative entropy production is a consequence of the statistical assumption on the initial condition.

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