Hierarchical Structure of Irreversible Processes Deduced from the

Progress of Theoretical Physics, Vol. 96, No.1, July 1996

Hierarchical Structure of Irreversible Processes Deduced from the Dynamical Semi-Group Theory

Masakazu ICHIY AN AGI

37

Department of Mathematical Sciences, Gi/u University of Economics, Ohgaki 503

(Received January 22, 1996)

Some consequences of a hierarchical structure of irreversible phenomena are discussed in this paper. In particular, we examine the physical meaning of some results deduced from the semi·group theory of irreversible processes which uses two different limiting procedures, van Hove and singular coupling limits. It is shown that the information entropy deduced from the maximum entropy method is never the same as the statistical entropy defined in the dynamical semi·group theory. The difference between the two is measured in terms of the relative entropy. For an understanding of irreversibility as positive entropy production, two distinct rescalings of time turn out to be essential. These control the transition between macroscopic and microscopic levels. We show that the analysis of the motion of a system coupled to two reservoirs in the singular and van Hove limits can be performed. The two-reservoir system reflects the hierarchical structure in time of the dynamical process under study.

§ 1. Introduction

Macroscopic processes are, in many cases, irreversible and the equations describing macroscopic processes should not be repugnant to the second law of thermodynamics. On the other hand, dynamical processes in many-body systems are often approximately described by master equations which are asymmetrical under time reversal. Examples are the Pauli master equation for atoms in radiation fields, and the Fokker-Planck equation for a Brownian particle. Their analysis from the standpoint of thermodynamics, however, has not satisfactorily been made up to now. Pauli's original derivation of the master equation depends upon his hypothesis of repeated randomization. I) Since we believe in the microscopic description of a many-body system, it is not at all clear that Pauli's hypothesis of repeated randomization is consistent with the Liouville theorem. This weakness in Pauli's derivation was stressed by van Hove.2

) Some forty years ago, van Hove established that, for a large class of quantum many-body systems showing dissipative behavior, the macroscopic size manifest itself through a characteristic property of the Hamiltonian H; that is, H can be written as Ho+AV in such a manner that, when used a basis of the unperturbed Hamiltonian, the perturbation expansions with respect to A V display the so-called diagonal singularity if one uses the representation in which Ho is diagonalized. Then, van Hove demonstrated the importance of the time r which is scaled as A2 t for small A; that is, A--->O, while r=A2 t is fixed (the van Hove limit). We propose to call t and r the microscopic and macroscopic time, respectively. These names refer to the way in which microscopic and macroscopic quantities behave. A little thought shows that this is connected with the fact that an infinitesimal interval of the macroscopic time scale effectively corresponds to an infinite interval of the microscopic time. The present paper presents an attempt to give a possible ther-

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38 M. Ichiyanagi

modynamical interpretation of the macroscopic time r and the microscopic time t. Since Pauli's time, a number of derivations of the master equation have been

given. Rigorous derivation of the master equations have been given for a Brownian particle interacting with an infinite system of harmonic oscillators,3) for a system with a finite number of quantum states interacting with an infinite ideal gas etc.4

) These results are surveyed by Davies.5

) Most of the models for which the master equations have been derived are of the type of systems consisting of a simple system and an infinite reservoir. In general, the derivation depends upon the chosen initial state of the reservoir and the form of autocorrelation functions of fluctuations in the reservoir. One would like to study the time evolution of initially uncorrelated states for weak anharmonisities. For some of these models, therefore, the limiting procedure employed is not the van Hove limit but a related singular coupling limit in which the autocorrelation time of fluctuations in the reservoir tends to zero. The latter limit, which is for the limit when the relaxation time of the correlations of the reservoir goes to zero, operates with the Hamiltonian written as H = Ho+ )..-1 V. In the literature, only one of the two limits, the van Hove limit or the singular coupling limit, is used exclusively. It is, therefore, the main purpose of this article is to show that the combined use of these two limits is essential to develop a statistical mechanics of irreversible processes of many-body systems.

Quantum-dynamical semi-groups are the generalization of Markovian semigroups to quantum systems.G

) Since for the study of irreversible processes it is necessary to adopt a more general class of states than the trace class that admit a continuous energy spectrum, mathematical parts of the theory have been developed on the level of von Neumann and C* ·algebras. In this paper we shall restrict our attention to the naive level. We are more interested in physical problems than in problems of mathematical principle. The present author hopes that physics will not be hidden behind difficult mathematical techniques. If our consideration yields inter· esting results at the lowest level, it would then be imperative to extend to rigorous mathematical levels also. On the other hand, if our consideration does not yield interesting results at the lowest level, we should then abandon the approach.

The aim of nonequilibrium statistical mechanics is to relate the observable properties of macroscopic systems to dynamical properties of the molecules composing it. Then, one of conceptual problems is this; how can we reconcile the rever· sibility of microscopic mechanics with the irreversibility of macroscopic behaviors? Dynamical processes in many-body systems are often described by the Boltzmann equation for a single-particle distribution function, the derivation of which given by Boltzmann depends on the hypothesis of molecular chaos that is not invariant under time reversal. It is noted that in order to derive the Boltzmann equation, we must let the particle density of the system go to zero. However, the mean free path and mean free time, which are the typical length and time scale of the system, will tend to infinity. To obtain a well-defined limit, we therefore must adjust the length and time scale appropriately. That is to say, it is essential to keep in mind that one uses the two different time scales in the kinetic description.

Ojima7) clarified the physical meaning of the van Hove limit, through the scale·

changing transformation to control transitions between microscopic and macroscopic

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Hierarchical Structure of Irreversible Processes 39

levels and investigated the structure of nonequilibrium states in their relation to microscopic dynamics. He has pointed out the physical basis of his interpretation, that (in his words) "the notion of time emerges from the correlations among physical motions which are fibered into different levels with certain typical motions in each regime (i.e., different standard clocks at each size level}." He also has noted that, in the limit ,.1--->0, approximate descriptions are obtained for either microscopic dynamics or macroscopic processes, according to the choice between {t=finite with r-O} and {r=finite with t--->oo}. We may try to analyze the problem somewhat more deeply by asking how a relation between the second law of thermodynamics can be related to Boltzmann's H theorem. This is the real distinction between the two levels of motion, hydrodynamic and kinetic levels.

In § 2, we first review the elements of the maximum entropy method and general· ize it to extended irreversible thermodynamics (EIT).8) Since flux relaxation times in the EIT region are generally much shorter than the hydrodynamic relaxation time on which scale the conserved variables relax, it is true that steady states of the system in question are established in much shorter time than the hydrodynamic relaxation time. Following development of the dynamical semi-group theory, we focus our attention on transient processes changing with macroscopic time in § 3. In the present work we propose a new scheme of applications of the maximum entropy method which will determine the evolution equations of thermodynamic variables. All the proofs in this paper use physicists' language and have heuristic value only. Obviously this is a drawback, particularly in the field of dynamical semi-group, but it is not unreasonable to hope that similar results can be proved rigorously. In § 4, the singular coupling limit and the van Hove limit are used to model the two-reservoir system which consists of a mesoscopic and microscopic degrees of freedom. In § 5 we discuss the formal properties of the nonequilibrium entropy and entropy productions. It is shown that the assumed entropy formula in extended irreversible ther· modynamics (EIT) does not reproduce the results regarding the entropy production deduced from the dynamical semi-group theory. The conditions for the applicability of the assumed entropy formula in EIT are stated there. Section 6 is devoted to general discussion of the results obtained in this paper.

§ 2. Maximum entropy method

2.1. Definition of entropy change

We distinguish among systems which are: a) completely isolated, b) closed systems which can exchange energy with their surroundings under controlled conditions, and c) open systems which can exchange energy, particle, etc. with other systems. Classical thermodynamics has been concerned mainly with the study of closed systems. In this paper we consider an open system.

The properties of the information (say, statistical) entropy for quantum systems have been surveyed by Wehr1.9

) In order to have a basis for subsequent considerations, the definitions and some properties are recalled. As is well-known, there does not exist an operator with the property that its expectation value in some state is its

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entropy. The entropy is assumed to be a function of state. In quantum mechanics, states are characterized by a density matrix of the trace class. If the state is described by the density matrix p(t), its information entropy is defined by

SICt)= -kBTr[p(t)lnp(t)] . (2 -I)

This is the most immediate generalization of Shannon's information measure and generalizes the classical expression of Boltzmann's entropy to the quantum domain. The density matrix is normalized to unity:

Trp(t)=l. (2-2)

(A) Time-dependent dynamics

If the time evolution operator is unitary, then the time derivative of SI(t) vanishes (the so-called unitary invariance). Here, we are asking for the infinitesimal increase dSI(t) during an infinitesimal interval of time, dt. This implies at once that the information entropy (2 -I) is not a suitable candidate for an entropy which is intended to describe irreversible processes of (isolated) systems. By this we mean that we cannot describe the approach to equilibrium. Usually, therefore, one assumes that one should first take the thermodynamic (infinite-volume) limit.

The separation of dynamical behavior of a macroscopic system into a microscopic short time and a macroscopic long time behavior is a central and fundamental condition for considerations about nonequilibrium thermodynamics. Hence, we can ask the increase LlS(t) during the macroscopically small interval of time, Llt. A monotonic increase of entropy is usually achieved by the contraction of information in some way. As we have seen in a previous paper/OJ we can introduce a useful measure to clarify how far from an initial state the time-dependent state at t evolves and whether the entropic distance of p( t) from p( to) increases or decreases. This measure, the relative entropy, is defined by

S[p(t)lp(to)]=kB TrpCt)(Iogp(t) -logp(to» .

The time evolution is given by the von Neumann equation:

!t PCt)+ i[H(t; .-t), p(t)]=O,

HCt; .-t)=H - ~AjFj(.-t2t) ,

(2-3)

(2-4a)

(2-4b)

where F j (.-t2 t) denotes the time-dependent external fields, and the parameter .-t characterizes the slowness of their time dependence. H denotes the Hamiltonian of the system in the absence of the external fields. For obvious reasons that we control the driving forces from the outside they must vary on the time scale of the dissipation processes induced by its surroundings. The initial condition usually is

H(to; .-t)--->H and p(to)--->ppcx.exp/3H, as to--->-=. (2-5)

We take the limit, to--->-=, and define the (average entropy) production

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1 t U d a(p(t))=Tt)o ds S[p(t + s)lp( - CXJ )]ds ~O . (2·6)

Nonlinear response theory can be developed in this framework.9),[0)

(B) Reduced dynamics

This definition does not conform to the definition of Spohn and Lebowitz.l1) In the treatment due to Spohn and Lebowitz, it is set that to~CXJ, (return to equilibrium) and the entropy production is defined by

(2·7)

The evolution (2·4) is replaced by a Markovian master equation

:, p(t)=Dp(t)- i[H(t; ,-t), p(t)] , (2 ·S)

where D is a generator which describes the influence of the surroundings on the evolution of the state of the (open) system which is considered to be a subsystem of a large system (see Appendix A).

The purpose of the dynamical semi-group theory is to prove the approach to equilibrium. Hence, the final condition at t = to~CXJ has been specified as in (2·7), while, in terms of (2·6), we investigate the dynamics of the system initially in thermal equilibrium and the departure from equilibrium. On physical grounds, we expect that, in general, reservoirs will drive the system to several (quasi-)stationary states other than absolute equilibrium states. This is a crucial point of the present theory.

In practice, the detailed justification of such Markovian evolution for a reduced dynamics is extremely difficult. The dynamical semi-group theory3),6) involves limiting processes which do not have a clear thermodynamical interpretation. We are interested in the weak coupling limit (=van Hove limit) where the strength ,-t of the interaction between the system and its surrounding tends to zero; ,-t~O, t~CXJ, while r =,-t2t=constant.

Since we do not aim to treat response theory in the dynamical semi-group theory, the situation we consider in this paper is that of an (open) system coupled to inexhaustible reservoirs in the van Hove limit. We wish to describe nonequilibrium (quasi-)stationary states of the discontinuous system.

2.2. The generalized canonical density matrix

In an approach to irreversible thermodynamics it is thought that a useful theory can be constructed if the notion that the entropy is maximum at equilibrium is relaxed so that nonconserved variables are included among the constraints for the Lagrange variational method used for entropy maximization. In this section we wish to generalize the maximum entropy method for local equilibrium states to transient processes in open systems. Let us denote the set of conserved variables such as the Hamiltonian of the system, momentum, etc., by {Aj : j=1, 2, ''', fl. The fluxes B j of A j are defined as

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42 M. Ichiyanagi

(2°9)

The set of nonconserved variables other than those corresponding to the fluxes associated with the conserved variables is denoted by {bi: i = 1, 2, ... }. In the method of maximum entropy applied to a system in nonequilibrium12

) the information entropy SI(t; A) (= - kB Trp(t; A)lnp(t; A» is maximized subject to the constraints

aj(A2t)=Trp(t)Aj=Trp(t; A)Aj ,

JAA2t)=Trp(t)Bj=Trp(t; )")Bj ,

Gi()..2t)=Trp(t)bi=Trp(t; )..)bi.

(2 0 10a)

(2 0 10b)

(2 0 10c)

Here, in (2°10), ).. is a parameter which characterizes the interaction between the system and its surrounding and scales the (microscopic) time t in the form )..2t. In what follows, we call the scaled time r( =)..2t) a macroscopic time in the weak coupling limit, ),,---->0.

By definition, it is assumed that all of these variables, aj(r), Jj(r) and Gi(r), constitute the thermodynamic variables of interest. What we mean by this is that these can be measured by thermodynamic experiment and any macroscopic states of the system can be sufficiently described by these macroscopic variables. It is emphasized that this procedure tacitly assumes that the irreversible process in question is so slow that the system relaxes to the thermodynamic state at any instant of (microscopic) time t in a microscopically long but macroscopically short interval of time.

Maximization of SIC t;)..) subject to the constraints (2 °IOa) ~ (2 °10c) yields the density matrix in the form

(2 0 ll)

where F()..2 t) is the normalization factor defined by

F()..2t)=F[Xi)..2t), Yj()..2t), Yi()..2t)]

= -log{Tr exp[ - ~Xj(A2t)Aj- ~ Yj()..2t)Bj- ~Yi()..2t)bi]}, (2°12)

and Xj, Yj and Yi are the Lagrange multipliers which vary extremely slowly in time t. The factor F()..2t) is a functional of the Lagrange multipliers. We call Pm(t;)..) the generalized canonical density matrix. It is emphasized that the time dependence of Pm(t; )..) comes from the time dependence of the Lagrange multipliers. Hence, it is natural to assume that Pm(t;)..) is a slowly varying function of (microscopic) time t.

Information-theory-entropy concepts are related to observations or sequences of observations made on the system in question. As it stands, the Boltzmann entropy is thought to refer to the dynamics of the system for all time. We, on the other hand, are interested in the entropy of the system at a given instant and its time dependence. As can be seen from (2 oll), the generalized canonical density matrix Pm( t; )..) is defined in the space of observables, {Aj, Bj, bi}, which is the subspace of the space of observabIes of the system. This subspace defines an observation level in the sense of Fick and Sauermann}3) By this we mean the contracted description of the states of the system. Hence, in constructing a theory around Pm(t; )..), we are looking at the

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problem from the point of view of the contraction of information about the motion of the system. It is noted that these constraints are consistent, because we have chosen the conserved observables {A j } for which we have

(2·13a)

This is true only when we have chosen the variables A j which fulfill the conditions

(2·13b)

Before proceeding to the derivation of the evolution equation for Pm(t; A), it is necessary to give a self· consistent formulation of thermodynamics. If dF( r) refers to the infinitesimal change of F(r) which is caused by infinitesimal changes of the Lagrange multipliers, from (2 ,12) one gets at once

(2,14)

This shows that the Lagrange multipliers are obtained from the equations

air) aF(A2 t)

Trpm(t; A)Aj , (2'14a) aXiA2 t)

Jir) aF(A2 t) Trpm(t; A)Bj , (2·14b) aYj(A2 t)

Gi(r) aF(A2 t) Trpm(t; A)bi, (2'14c) d Yi(A2 t)

where the symbol a denotes a functional derivative. In performing the functional derivative, we consider each of the Lagrange multipliers to be an independent variable.

The maximized entropy Sm(t; A) is given by the formula

(2·15)

By use of (2 ·11), we may write

(2 ·16)

Since the Lagrange multipliers are functions of the thermodynamic variables considered, the derivative of the entropy Sm(t; A) is given by the form

dSm(t; A)=kB{~Xir)dair)+ ~ Yj(r)dJir) + ~y;(r)dGi(r)} . (2 ,17)

Here, we have used (2,14) for the infinitesimal change of F(r). In view of the integrability of dSm(t; A), the following relations can be readily obtained from (2 ,17):

Xir) aSm(t; A)

aaj( r) , YJ.(r) aSm(t; A) ( )_ aSm(t; A)

aJj(r) , Yi r - oGi(r) (2 ·18)

That is, to derive (2 ·18) one needs (2 ·14) which is a form of the integrability condition . for the differential form (2·17). In view of the similarities of these expressions with

the temperature formula, etc., in equilibrium thermodynamics, they are regarded as

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nonequilibrium generalizations of the temperature in equilibrium thermodynamics, of the pressure, etc. Such assignment of thermodynamic meanings to these expressions presumes that the equilibrium entropy is straightforwardly extendible to nonequilibrium situations, and therefore there exists a form of nonequilibrium entropy similar in mathematical structure to the equilibrium entropy. Because of this ther· modynamical analogy, the relation (2 '17) is called the extended Gibbs relation. 4).14)

Before concluding this subsection, we wish to pose a question; if the maximized entropy, as an acceptable nonequilibrium extension of the Clausius entropy, has something to do with thermodynamics, what is a definition of the entropy time derivative? Obviously we have the two options dSm/dt and dSm/dr. This will be discussed in what follows.

2.3. Irreversibility

Equation (2 '17) is taken to be the extended Gibbs relation and (2 '14) to be the Gibbs-Duhem equation whenever the Lagrange multipliers, Xj, Yj and Yi are made thermodynamically operational. As it stands, the connection of (2'17) with the second law of thermodynamics is obscure and has not been established. In extended irreversible thermodynamics (EIT) it is generally assumed that Selt=Sm is a nonequilibrium extension of the Clausius entropy, whereas in information theory and also in the kinetic theory of Boltzmann or along the line of Boltzmannian ideas one assigns the terminology "entropy" to SI(t) defined by (2·1) and considers it an extension of the Clausius entropy. There is obviously a confiict in the usage of the term entropy, since the same terminology is used for two different quantities of hierarchical order, one being subordinate to the other, as will be shown by the analysis made in the next section. There is a need for new distinctive terminology for them. Since there is already wide spread use of the term entropy for SI(t) in the literature on kinetic theory and information theory it is preferable to keep the term for SI(t). We then will have to use different terminology for the quantities descending from SI(t). However, we will not invent the names here since we can still carry out our discussion without doing so, as long as it is agreed that the latter are different from the statistical entropy SI(t).

In the previous subsection, we showed that the differential of the entropy Sm(t; A) obtained by the method of maximum entropy can be interpreted as the extended Gibbs relation in the space of the variables {aj, J j , Gi } if the Lagrange multipliers are endowed thermodynamic meanings and the integrability condition (2 '14) is fulfilled for the differentials of the Lagrange multipliers. The extended Gibbs relation (2 ·17) can be used to define the rate of the entropy change, which is obtained from it by replacing the symbol "d" by the derivative with respect to the scaled time r(=A2t).

ft Sm(t; A)=~X/r) ft !dA2t)+ ~ Yj(r) ft J/A2t)+ ~Yi(r) ft Gi (A2t).

(2'19)

Accordingly, the change of the maximized entropy in time t is negligible in the limit A->O, indicating that the states, to which the system relaxes in short interval of time, appear to be (quasi-)stationary. The relaxation times, depending upon the boundary

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conditions imposed on, are microscopically finite but are macroscopically infini tesimal.

As is seen from (2 '10) and (2 ·14), it is true that either p(t) or Pm(t; A) can be used to calculate the macroscopic values {a/r), Jj(r), Gi(r)}, (r=,12t). However, the two density matrices have important differences. On the one hand, p(t), the precise density matrix, involves all information about the motion of the system in question and is determined for all latter instants by an initial condition imposed on (2'4) or on an alternative equivalent to it, (2·8). Hence, in principle, memory is completely contained in p(t). On the other hand, it is emphasized, after Robertson/5

) that Pm(t; A) in (2·11) is nothing more than the initial density matrix describing only what is known about the initial state of the system. That is, the initial condition for (2·8) or for (2 '4) is that at t = to (= rO/,12) it equals a generalized canonical density matrix containing only macroscopic constraints of the motion. The point is that Pm(t;,1) does not contain memory of past values of the macroscopic values, {ait), Jit), Gi(t)}, and memory should be supplied by the equations of motion for them.

Before concluding this section, we mention the idea proposed by Mori, Oppenheim and Ross.16) It is easy to see that the quasi-stationary states described by the generalized canonical density matrix Pm(t;,1) correspond to the frozen states described by the local equilibrium density matrix Pt in their terminology. The density matrix Pt is used to define an initial state for the density matrix Pt(s) which equals

pt(s)=exp( - iHs)ptexp(iHs) . (2'20)

Since the evolution is unitary, the statistical entropy

(2·21)

is independent of s. This is in line with the fact that the frozen states are stationary. The crucial point is to assume that "the precise density matrix p(t) does not change appreciably in a time interval r very much shorter than the macroscopic (say, hydrodynamic) relaxation time rh." Hence, they write

(2·22)

Here, r is a characteristic time which is much shorter than the hydrodynamic relaxation time rho We can choose these such that rm~ r~ rho Obviously, (2'22) is an approximation which leads to a quasi-stationary state in a time on the order of the microscopic relaxation time rm. It has the defect that if one wants to consider an irreversible approach to the (quasi-)stationary state described by (2'22) from an initial state of the system, the approximate solution is useless.

The (macroscopic) entropy, as they define it, is -kBTrp(t)lnpt, the irreversible part of which is given by kBTrp(t)[lnpt, iH). It has been shown that this takes the form of the Joule heat. For some purposes, however, it is desirable to take a broader view of the subject (see § 5 below).

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§ 3. The semi-group theory of irreversibility

A system which is subject to uncontrolled random influences from its surroundings is often called an open system. There are two ways to model the interaction between the system and its surrounding quantitatively. The first of these two theories, the many-reservoir model, where the reservoirs are of a certain stochastic nature, was introduced by Bergmann and Lebowitz l7

) for the study of steady-state processes. The second is the dynamical semi·group theory, whose objective has been to treat the dynamics and thermodynamics of the total system S+ R using only the reduced description for the open system S. A physical interpretation is to consider the system S as a limited set of macroscopic degrees of freedom of a large system, S + R, and the reservoir as the uncontrolled degrees of freedom. The partial state of the system then suffers a time development which is not given by a unitary evolution in general.

The von Neumann equation (2·4) has a solution of the form

p(t)= U(t, s)p(s) (3 ·1)

for any "initial" density matrix pes), where U(t, s) is a unitary operator defined as the solution to the equation of motion with an initial value at t = s as follows:

1t U(t,s)=-iH(t;),)U(t,s) , U(t=s,s)=1. (3·2)

The evolution operator U(t, s) enjoys a semi-group under the law of composition

U(t,u)U(u,s)=U(t,s) for t>u>s. (3·3)

Since there exists the inverse U-l(t, s), and since U-l(t, s) equals U(s, t), the semigroup can be extended to a group.

The formula (3·1) can be generalized to more general dynamics, characterized by the following equations:

p(t)= T(t, s)p(s) ,

T(t,u)T(u,s)=T(t,s) for t>u>s.

(3·1a)

(3·3a)

If the semi-group (3·3a) cannot be extended to a group, the dynamics is said to be non· Hamiltonian.

When the evolution operator T(t, s) is a function of t-s, (3·3a) takes the form

(1) T(t+s)=T(t)T(s) for t,s>O. (3·4)

The main general results of the dynamical semi-group theory are that under certain conditions the time evolution of open systems can be described by a family of evolution operators T(t) for t >0, so chosen that (3·4) and the following properties are fulfilled:

(2) T(t---->O)=l. (3·5)

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(3) If p(O) is positive semi-definite and normalized, then so is p(t), defined by (3 ·1) (complete Positivity).

The property (3'4) defines the semi-group to be temporarily homogeneous. It is known that these conditions allow a complete description of the generator of the semi-group:

d T(t)=exp(tL), L=dfT(t)lt=o.

It can be proved that the generator L has the form

Lp(t =0)= Dp(t =0) - i[H, pet =0)] .

Here, D, for instance, is given by

Dp(t)= ~~{[Vj,p(t)Vj*]+[Vjp(t), Vj*]}, (t--->O)

(3·6)

(3-7)

(3'8)

where {Vj} is a sequence of bounded operators on the space of the operators of the system for which ~ Vj Vj* converges in the sense of means. A mathematically complete specification of these operators has been given by Lindblad.6

) The generator L given by (3'7) together with (3·8) describes the situation in which the typical variation time of pet) is much longer than the decay time of the correlation functions of the reservoir. We note that the generator obtained in the weak-coupling limit is of the form (3-8) (see, Appendix A). Then, the equation of motion to be considered is given by

(3-9)

It is mentioned that, for a given L, H is uniquely determined and that the operator L ensures that the positive definite character of a density operator is preserved in the course of time evolution. For the system which is not completely isolated from its surroundings, (3'9) describes the approach to thermal equilibrium.

The detailed justification of such a simple type of evolution operator is extremely difficult and involves a limiting procedure, which, while being mathematically well defined, does not have a very clear physical interpretation. The most used limiting procedure involves the van Hove limit. The basic content of the van Hove limit is that the limit t--->oo must be taken together with ,-1--->0 such that ,-12t = r (finite). In Appendix A, we will see the implication of the van Hove limit to define a Markovian evolution of the system. If we assume that the interaction between the system and a single reservoir at some temperature has the form A V = A~ Vj0 V/, we can derive the formula for D given above. In general, D depends upon the reservoir temperature. Some examples of dynamical semi-groups (quantum Poisson and Brownian processes) for the harmonic oscillator can be found in the paper by Kassakowski4

) to which the interested reader is referred. In this paper, we are simply taking the dynamical semi-group known in the literature and making it the object of study from the viewpoint of the maximum entropy procedure. For our purpose, it is sufficient to

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think of the Hamiltonian H as a dynamical system and D is such that for all conserved variables A j ,

TrAjDp(t)=O. (j=1, 2, ... , f) (3'10)

This condition gives rise to various conservation laws for the system when (3·9) is used for their derivation.

To proceed, it is important to note that the generalized canonical density matrix Pm(t; ,,) defined by (2·11) is nothing more than the initial density matrix describing only what is known about the initial state of the system in question. In order to qualify the irreversible process as a Markov process, we must introduce higher order conditional probabilities corresponding to repeated observations of the system. In general, the repeated observations are represented by a set of dynamical maps (which are often called operations). Let us consider the system to have been observed at fo < n < ... < fn. The corresponding (microscopic) times are given by to< tl < ... < tn (tk

=,,-2 fk ). We denote the macroscopically infinitesimal intervals as ek= fk - fk-I and the microscopically finite intervals as Llk=,,-2ek, respectively.

Let us define the dynamical map resulting from a sequence of observations on the system as a simple time-ordered composition of the maps of the semi-group evolution:

T(tn - to)p(to) = T(tn - tn-I)··· T(t2- t l ) T(tl- to)p(to)

= T(Lln)··· T(Ll2) T(LlI)p(to) .

The initial density condition for (3'9) is written as

(3'11)

(3 '12)

and the associated density matrix at t = tl is obtained from the following formula:

(3 ·13)

Here, we have introduced the notation ek=,,2Llk (k=l, 2, ... , n). If we take the van Hove limit in (3 ·13), we get at once

(3'14)

The result is physically interpreted as follows. The evolution of the system is followed over a microscopically infinite interval starting from the initial time fo from a microscopic point of view. The subscription tlof Tt,(oo) indicates that the limiting value of T(CI,,-2) as " goes to zero may be dependent on the final time tl. Hence, it is natural to expect that all microscopic (uncontrolled) information of the system has been contracted completely up to the next observation time n and a new density matrix, which is nothing but the initial value of p( t) at t = tl, can be constructed of the form

p(tI)=Pm(n)= V(el)Pm(fo),

V(CI)= Tt,(oo) .

(3 '15)

(3 ·16)

Repeating this procedure, we obtain the following scheme of quantum stochastic

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processes:

(3 ·17)

Next, let us determine the evolution operator Vee) defined by (3·16). By substituting (3·6) into (3·16), we take the van Hove limit

(3 ·18)

The second equality of (3 ·18) was proved by Palmer (Theorem 3.1).18) Hence, (3 ·17) can be rewritten in the form

(3·19)

As we will see in Appendix A, this limit defines the operator L which describes the case of the free motion of the system and dissipation are of the same order of magnitude. It is known that the dissipative part of L is independent of the system Hamiltonian H.

We end this section with a short summary. We have chosen the set of relevant variables {Aj , Bj, bJ "Slowness" of these relevant variables is built in by considering a line with well separated intervals of (microscopic) time t. The evolution is followed over microscopically infinite intervals of time from a dynamical semi-group theoretic point of view. Then, we construct the new density matrices P(tk) at every instant of time t = tk • Each construction of a p(tk ) requires that we do not deviate from the density matrix Pm( rk) at r= rk, in the van Hove limit. The evolution of the system in the microscopic time region is determined by L in the van Hove limit, while that in the macroscopic time region by L in the singular coupling limit.

§ 4. The singular coupling limit and temperature rescalings

The interaction between system and reservoir is assumed to be weak. In the singular coupling limit, the motion of the reservoir and the coupling between the system and reservoir are scaled in such a way as to produce white noise acting on the system. Let us consider, as an example, a system interacting with an (infinite) ideal Boson system:

(4 ·1)

where the Hamiltonian of the Boson system, as a reservoir, is

(4·2)

and the interaction is

(4 ·3)

where bk and M are the Boson operators, and Pk and P: are the properly defined system operators. This Hamiltonian is contrasted with the following Hamiltonians:

(4·4a)

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50 M. Ichiyanagi

and

(4·4b)

which are for the weak coupling limit. It is noted that, in terms of the Hamiltonian (4·4b), the motion of the system and dissipation are of the same order of magnitude. Then, it is easy to see thaeS

)

exp(iHr) =exp(iHw(2) tB) _ (r=A 2 tB) (4·5)

Here, in Eq. (4·5), tB is the time for the reservoir dynamics. We introduce the time (r )-dependent Hamiltonian defined by

H(r)=exp(iHBr)Hexp( - iHBr)

=Hc+A-l~(~: bk(r)+ h.c.)+ ,.1-2 HB(r) . (4 ·6)

This Hamiltonian and the one defined by (2·4b) obviously belong to the same category. More precisely, the time (r)-dependent external fields can be regarded as the condensation of the b-fields. Similarly, we can define the time (t)-dependent Hamiltonian

(4·7)

If the temperature of the reservoir of the singular coupling model is /3B, this Hamiltonian (or its equivalence, (4 ·4b)) suggests the definition of the rescaled temperature for the rescaled problem, IS)

(4·8)

This rescaling of temperature is based upon the requirement that /3BH = /3effHw(2). It is noted that when we rescale we must ensure that r//3eff= tB//3B in the singular coupling limit. Hence, in the case we connot define an imaginary time, t + i/3, for the system in the usual way, because the real and imaginary parts are scaled differently; t + i/3B--->A2t + iA-2/3B. It is noted that in the singular coupling limit the reservoir two-point functions tend to a a-function type and the reservoir state can be expected to be a KMS state in the limit /3B--->O. As we have noted at the beginning of this section, in this singular coupling limit we can expect the motion of the system to be Markovian. This is in line with the fact that when /3eff remains finite, /3B-4CO as ,.1-+0. However, it is believed that quantum random forces, in general, are never Markovian.

It is possible to consider a more realistic reservoir model, in which one assumes that HB consists of the three parts, a limited set of mesoscopic degrees of freedom (Hm), a set of the microscopic degrees of freedom (HR), and the interaction between the two sets:

(4 ·9)

Here, e( = e(A)-+O as ,.1-+0) denotes the coupling parameter. A simple model for it, for instance, is the independent-oscillator model, the Hamiltonian of which is given by

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(4 °10)

Here Pa and Xa stand for the mesoscopic coordinates, and pj and Qj are the reservoir coordinates, respectively.

When one adopts the usual method of the weak-coupling limit to the Hamiltonian H B, one defines the scalings7

)

(4 °11)

!3 being the temperature of the reservoir with the Hamiltonian HR. Note that t!3 = rB!3B in the weak coupling limit. This scaling is contrasted with the scaling (4 08) of the singular coupling limit.

The standard scales of time and temperature differ from the mesoscopic level to the reservoir level. Accordingly, the scale transformation (4 °11) suggests that a Markovian process in the mesoscopic time scale does not necessarily imply a corresponding Markovian process observed in the (microscopic) time t. This can be regarded as a simplification of the situation in which a mesoscopic subsystem, weakly interacting with a reservoir, relaxes to thermal equilibrium with the reservoir, while the (macroscopic) system is driven by the various dissipation mechanisms.

The new reservoir Hamiltonian HB describes the nontrivial structure of the reservoir in question. The interpretation of (4 °11) provides that the rescaled tempera ture !3eff, defined by (4 ° 9), is given by

(4°12)

This result clearly indicates that it is important to introduce the mesoscopic level between the macroscopic and microscopic levels_ Hence, the two limits, c---'O and ;1---'0, should not be independent to yield physically reasonable results_ A simple but nontrivial way is to set c(;1)=;1, and to consider the system with the Hamiltonian

(4°13)

This is similar in form to a two-reservoir model with different temperatures !3 (for HR) and !3B(=;12!3 for Hm).

Alternatively, one can formally construct the model Hamiltonian:

H=Hc+cV +c2 HB (4 °14)

and

(4 °15)

The temperatures in this case are rescaled according to the formula (4 °12)_ As before, if one sets c=;1, one gets

(4°16)

This is a form of the many-reservoir model with two temperatures !3B (for Hm) and ;1-2!3B (for HR)'

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52 M. lchiyanagi

The latter model Hamiltonian describes the situation in which the mesoscopic system is driven by the various dissipation mechanisms due to the interaction between it and the reservoir, and the macroscopic system is not completely isolated from its surroundings. Accordingly, one may use this Hamiltonian for the statistical mechanical study of phenomena in the EIT region. We do not, however, have anything in this regard, except for a few hints on the many reservoir models discussed by Lebowitz et al. ll ) Thus, the idea expressed above should be regarded largely as motivation for further research along the lines indicated here.

§ 5. Formal properties of nonequilibrium entropy

In kinetic theory, Boltzmannl9) introduced for the first time a nonequilibrium

generalization of the Clausius entropy in the form of the H function. The existence of the H function prompts the thought that nonequilibrium entropies might be obtained by generalizing the single-particle distribution functions to nonequilibrium density matrices. Indeed, the notion of information entropy was introduced by Shanon.20

) In the information theory approach to statistical mechanics, the entropy of a system is defined in terms of the density matrix p(t) by (2·1). We have seen in § 2 that the maximum entropy method yields the maximized entropy Sm(t; A) given by (2 '15) subject to the constraints (2 ·10). In this section we wish to discuss the possible relationship between the two entropies, SIU) and Sm(t; A).

First of all, one must define the entropy production which is the source term in the entropy balance equation for an open system. To do this, one must note that there is an entropy flow through the boundaries of the system in question, which can be defined as

(5'1)

Hence, the entropy production is given by

d a(t; A)=dFS1(t)-J/,(t)

d = -dFS[p(t)IPm(t; A)] , (5'2)

where S[p(t)IPm(t; A)] is the relative entropy defined by (2·3). Equation (5·2) is the value of entropy production at an instant of time t. We are interested in its long time average, defined by

a(r)=lim TIlT dsa(s; A)=log ktB S[p(O)IPm(t; A)] 20.

T ...... oo 0 t-DO (5·3)

Since we have assumed the constraints (2 '10), it is easy to prove the following equality

(5·4)

It is easy to verify that A) if one assumes a unitary evolution for p(t), then one gets

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entropy production of the form

or B) if one assumes a dynamical semi-group law for pet), then one arrives at

It Slt)= - It S[p(t)IPm(t; A)]. (in the limit A-->O)

This remark is relevant to the discussion of the entropy production given in § 2. The generalized canonical density matrix Pm(t; A) is expressed in terms of the slowly varying variables, {Aj , Bj, bi}. Hence, in the van Hove limit, we can put

(5'5)

as long as dSm(t; A)/dr is finite. This result can be interpreted in the following way. The maximized entropy Sm(t; A) changes extremely slowly in (microscopic) time t. That is, the states described by the generalized canonical density matrix Pm(t; A) are effectively stationary in the scale of microscopic time t. On the other hand, by utilizing the Gibbs relation (2 ·17), it is easy to verify that

This is compared with the previous result, (2'19). The right-hand side of (5'6) is nothing but the macroscopic entropy production of the louIe heat type, which seriously differs from the entropy-production concept defined by (5'2) (or equivalently by (5·3».

Consequently, the formula for the total entropy production in the macroscopic time scale is given by

(5' 7)

The total entropy production consists of two different terms. The first term is due to the microscopic dynamics, which are outside external control and the other the louie heat which is related to the dissipative flows in the system. This clearly demonstrates that the extended Gibbs relation (2 '17), in general, does not give us the correct expression for the entropy production.

§ 6. Discussion of the results

In this paper we have taken for granted that there is a mathematical similarity of the van Hove and singular coupling limits. The singular coupling limit has been considered by Hepp and Lieb,22) who has presented a class of mean-field model Hamiltonians consisting of a macroscopic (finite) system of many components interacting with an infinite reservoir for the study of phase transitions in driven open systems.

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54 M. Ichiyanagi

In developing our discussion of transient processes, we have relied heavily on techniques and arguments borrowed from the dynamical semi-group theory. A Markovian reduced dynamics gives a formally exact description of the irreversible processes of an open system coupled to its surroundings under the limiting conditions (the van Hove limit and the singular coupling limit). In this paper, we have demonstrated that the singular coupling limit is useful to develop a theory for fast transient processes in the region of lET.

One may wonder whether the efforts to build a thermodynamic description for transient processes is worthwhile, or whether one should be satisfied with a kinetic description. Since kinetic approaches, in general, give much more information than that needed for the interpretation of macroscopic measurements, contraction of information is always implemented in practice. Such contractions are based upon the hierarchical structure in time. Hence, our effort to build a thermodynamics of transient processes has been directed at generalizing the second law of thermodynamics for them which is consistent with a kinetic theory such as the Boltzmann theory.

It is known that transport phenomena in nonuniform fluids consist of two coupled relaxation processes; one is the hydrodynamic process of attaining spatially uniform states and the other is the microscopic process of attaining local equilibrium states of the system. It is expected that the microscopic process forms a quasi-stationary state in very short intervals of time. If one includes the dissipative fluxes corresponding to the conserved variables to the basic thermodynamic variables, as was done in extended irreversible thermodynamics (EIT), one is forced to take into account the flux-relaxation processes also. In this paper, we have developed a statistical mechanical theory for transient processes which are not so slow that the microscopic processes cannot form a stationary state in a time interval much shorter than a typical flux-relaxation time r¢. In general, the characteristic times satisfy the relation rmS: r¢

s: rho At the end of § 2, we saw that in the usual treatment of the local equilibrium theory, the precise density matrix is assumed not to change appreciably in the interval of time, r. This has been a crucial point in the present consideration.

However, in the region of EIT, it is not certain that one can find such a characteristic time r which should satisfy the relation rmS: rS: r¢. If it is true that one can define such a characteristic time r, one can follow an approach which may be an eminently reasonable extension of the local equilibrium theory, since microscopic processes on times shorter than r¢ could be used to produce the entropy of the system. In turn, this fact indicates that the total entropy production is expressed as the sum of the two effects, shown in (5 -7). Consequently, it is concluded that the extended Gibbs relation (2 -17) for the maximized entropy Sm(t; A) in extended irreversible thermodynamics (EIT) is not justified for the fast transient processes discussed above. Regarding this point, the present author21) has pointed out that the theories of EIT based upon the extended Gibbs relation are possible only for rather slow transient processes for which one can make a hypothesis that the irreversible process carries with it an infinity of possible (quasi-}stationary states to which the system can relax in a very short interval of time r.

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Appendix A -- Reduced Dynamics--

For the sake of completeness of the paper, we present here a simplified derivation of the semi-group dynamics in the van Hove limit given by Davies.23

) The model is of the S+ R type. Hs denotes the Hilbert space associated with the system Sand Hr the Hilbert space of the reservoir R. Then, the formal Hamiltonian of a model is given by

H=Hs®l +;1V +l®Hr,

=Ho+;1V,

on Hs®Hr, where V is a sum of weak interaction terms.

(A·la)

Use of this Hamiltonian presumes that the interaction between the system and its surroundings is weak, the motion of the system is on the order of one, and the dissipation on the order ;12_ An alternative point of view is to scale the motion in such a way that the uncoupled motion of the system and dissipation are the same order of magnitude. The latter case is described by a Hamiltonian of the form

(A'lb)

As the evolution operator of the total system S + R is Hamiltonian, we start with the von Neumann equation for the Hamiltonian (A 'la)

~ M(t)=LM(t)=LoMCt)+;1AM(t)

L o+;1A=-i[Ho+;1V,] (A'2)

where M(t) is the density matrix of the total system. The formal solution of (A·2) IS

MCt)= V/M(O) , V/=exp( - i[Lo+;1A]t).

The evolution operator V/ is expressed in the form

V/= Vt+;1 ltdSUt-sA VsA,

where

V t =exp( - iLot) .

(A·3)

(A'4)

(A'5)

To proceed, it is convenient to introduce the projection operator P, which projects onto the Hilbert space of the system Hs. If we define

P V/P= W/, PVtP=Xt ,

then, from (A '4) we obtain

(A·6)

(A'7)

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56 M. Ichiyanagi

Here,

AOl=PA(I-P), A lO=(I-P)AP. (A-8)

Changing the variables t and s into the scaled times r( =A2 t) and 6( =A2S) we can write (A-7) as

(A-g)

where

(A-lO)

If the second term of the right-hand side of (A-g) converges in the limit A-O, we have a limit operator Wr as A-D. To prove this, a more sophisticated approach is necessary. The definition (A -10) suggests to introduce the quantity

(A-ll)

From (A -10) it follows that

=1 + [A-2r d6X-A-2 lYK(A, r-6)XA-2 lYd>(A, 6). (A-12)

An analogy to (A -12) is

~(A, r)=l + [A-2r d6X-A-2lYKXA-2lY~(A, 6). (A -13)

From (A -13) we obtain

X-A-2r~(A, r)=exp{ - i[A-2Lo+ K]r}. (A-14)

Davies23) has proved that

(A-15)

in the van Hove limit where the limit t_oo is taken together with A-O such that A2 t = r(finite). In physical terms, (A -14) is nothing but the evolution operator of a Markovian process. It is noted that, as can be seen from (A -11), the operator K may depend on a particular initial state of the reservoir R, since there appears the projection operator 1- P on the right-hand side.

Let us consider the second Hamiltonian, (A -lb). One should go over to the scaled time r( =A2 t). Let V/ be the evolution operator of the system defined as in (A -3) but with the Hamiltonian scaled as

(A-16)

Then, it was proved by Palmer that the van Hove limit defines the operator t,

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lim V/p(to)=exp( - iit)p(to) . .1-0

(A-17)

For instance, if we assume V = Q@q" we have for L

Lp=-i[Hs, p]+iS[Q2, p]+h[Q, [Q, p]], (A-IS)

where h+ is characterizes the reservoir correlation function. In general, the reduced dynamics for the system does not provide an interesting

description, because it often depends on the chosen initial state of its surroundings at a particular instant in past, as just described. Rather than try to consider specific interactions between the system and its surroundings explicitly, we can also consider general modifications of the von Neumann equation, which are thought of as modeling the contraction of information about the initial state as the motion proceeds. A dynamical semi-group is defined as a one-parameter semi-group of linear endomorphisms of the set of all density matrices of the system in question. A physical system, the dynamical semi-group of which cannot be extended to a group is called a nonHamiltonian system.

References

1) W. Pauli, Probleme der Moderenen Physik, Festschrift zum 60 Geburtstage A. Sommer/elds (Leipzig, 1928), p. 30.

2) L. van Hove, Physica 21 (1955), 517. 3) J. Ford, M. Kac and P. Mazur, J. Math. Phys. 4 (1965), 1293. 4) A. Kossakowski, Rep. Math. Phys. 3 (1972), 247.

V. Gorini, A. Frigerio, M. Verri, A. Kossakowski and E. C. G. Sudarshan, Rep. Math. Phys. 13 (1978), 149. R. Alicki, Phys. Rev. A40 (1989), 4077.

5) E. B. Davies, Quantum Theory of Open Systems (Academic, London, 1976). 6) G. Lindblad, Nonequilibrium Entropy and Irreversibility (Reidel, Dordrecht, 1983). 7) I. Ojima, J. Stat. Phys. 56 (1989), 203. 8) D. Jou, J. Casas-Vazquez and G. Lebon, Extended Irreversible Thermodynamics (Springer, Berlin,

1993). B. C. Eu, Kinetic Theory and Irreversible Thermodynamics (Wiley, New York, 1992). I. Muller and T. Ruggeri, Extended Thermodynamics (Springer, Berlin, 1993).

9) A. Wehrl, Rev. Mod. Phys. 50 (1978), 221. 10) I. Ojima, H. Hasegawa and M. Ichiyanagi, J. Stat. Phys. 50 (1988), 633. 11) H. Spohn and J. L. Lebowitz, Adv. Chern. Phys. 38 (1978), 109. 12) W. T. Grandy Jr., PhYs. Rep. 62 (1980), 175. 13) E. Fick and G. Sauermann, The Quantum Statistics of Dynamic Processes (Springer, Berlin, 1990).

Chapter 5. 14) D. Jou, C. Perez·Garcia and J. Casas· Vazquez, J. of Phys. A17 (1984), 2799. 15) B. Robertson, Phys. Rev. 144 (1966), 151. 16) H. Mori, I. Oppenheim and J. Ross. in Studies in Statistical Mechanics, vol. 1, ed. J. De Boer and

G. E. Uhlenbeck, (North Holland, Amsterdam, 1962), p. 217. 17) P. G. Bergmann and J. L. Lebowitz, Phys. Rev. 99 (1955), 578.

J. L. Lebowitz, Phys. Rev. 114 (1959), 1192. 18) P. F. Palmer, J. Math. Phys. 18 (1976), 527. 19) L. Boltzmann, Lectures on Gas Theory (Univ. California, Berkely, 1964). 20) E. T. Jaynes, Information Theory and Statistical Mechanics, in Statistical Physics (1962 Brabdeis

Lectures), ed. W. K. Ford (Benjamin, 1963). See also, E. T. Jaynes, in The Maximum Entropy Formalism, ed. R. D. Levine and M. Tribus (MIT Press, 1978), p. 15.

21) M. Ichiyanagi, J. Phys. Soc. Jpn. 64 (1995), 4628. 22) K. Hepp and E. H. Lieb, Helv. Phys. Acta 46 (1973), 573. 23) E. B. Davies, Commun. Math. Phys. 39 (1974), 91.

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