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Journal of the Physical Society of Japan

On the Nonequilibrium Ensemble of Nakajima and Zubarev

Masakazu Ichiyanagi

Published Apr 15, 1987• 0 citations
Publisher

Abstract

It is argued that in the formalism of the revised Zubarev, an irreversible production of entropy is manifest. An exact solution of the Liouville-von Neumann equation is given, in terms of which we can define the two equal, but opposite, contributions to the entropy production. We propose the procedure in which we look upon the irreversible process as the production of entropy by stochastic (microscopic) processes occurring in a many-body system.

DOI
10.1143/jpsj.56.1264

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Journal of the Physical Society of Japan•Apr 1, 1984•1 citation
It is argued that in the formalism of the revised Zubarev, an irreversible production of entropy is manifest. An exact solution of the Liouville-von Neumann equation is given, in terms of which we can define the two equal, but opposite, contributions to the entropy production. We propose the procedure in which we look upon the irreversible process as the production of entropy by stochastic (microscopic) processes occurring in a many-body system.
In this contribution we prove that the entropy of an N-body isolated system can not decrease and the entropy production should be non-negative provided the system possesses an equilibrium state. We define the entropy as a functional of the set of n-particle reduced density operators (n=0,....,N) generalizing the von Neumann fine-grained entropy formula. Additionally, as a consequence of our analysis we find the expression of the equilibrium n-particle reduced density operators which enter the definition of the entropy as well as the dissipated energy in an irreversible process.
The method of Zubarev allows one to construct a statistical operator for the nonequilibrium. The von Neumann equation is modified introducing a source term that is considered as an infinitesimal small correction. This approach provides us with a very general and unified treatment of nonequilibrium processes. Considering as an example the electrical conductivity, we discuss the modification of the von Neumann equation to describe a stationary nonequilibrium process. The Zubarev approach has to be generalized to open quantum systems. The interaction of the system with the irrelevant degrees of freedom of the bath is globally described by the von Neumann equation with a finite source term. This is interpreted as a relaxation process to an appropriate relevant statistical operator. As an alternative, a quantum master equation can be worked out where the coupling to the bath is described by a dissipator. The production of entropy is analyzed.
The microscopic theory of irreversible processes that we developed is summarized and illustrated, using as a simple example the Friedrichs model. Our approach combines the Poincaré's point of view (dynamical interpretation of irreversibility) with the Gibbs-Einstein ensemble point of view. It essentially consists in a nonunitary transformation theory based on the symmetry properties of the Liouville equation and dealing with continuous spectrum. The second law acquires a microscopic content in terms of a Liapounov function which is a quadratic functional of the density operator. In our new representation of dynamics, which is defined for a restricted set of observables and states, this functional takes a universal form. We obtain, in this way, a semi-group description, the generator of which contains a part directly related to the microscopic entropy production. The Friedrichs model gives us a simple field theoretical example for which the entropy production can be evaluated. The thermodynamical meaning of life-times is explicitly displayed. The transition from pure states to mixtures, as well as the occurrence of long tails in thermodynamic systems, are also briefly discussed.
Abstract We consider the temporal evolution of an isolated system of N particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy, S = S max, S′ ≤ S max. The application of usual density matrix theory to the temporal development of S leads us to dS/dt = 0: The entropy S does not change in time t. Thereby it is irrelevant, whether we consider a non-equilibrium state or an equilibrium state. Consequently, the system cannot irreversibly change by entropy production dS/dt > 0 from S′ to S max. This is a paradoxial result, which contradicts the experience. It can be traced back to the von Neumann equation, which in principle describes reversible processes and hence is unsuitable for calculating the irreversible evolution of the entropy S in time t. Each irreversible process is accompanied by a positive entropy production P = dS/dt ≥ 0 inside the system, which only vanishes in case of the equilibrium state. In order to overcome the above mentioned difficulties, we assign an operator P to the entropy production P, which is defined by the eigenvalue equation P |u〉 = P|u〉 with the state vector |u〉 of the N-particle system. There was an extensive discussion about the relation between the production of entropy P on one hand and the progression of time t on the other. Making use of this concept, we combine the operator of entropy production P with the time-development operator U (t, t 0) of the system and finally deduce the infinitesimal unitary operator U (t+τ,t) = 1 + (i/k)τ P by means of very general assumptions. Here P means the generator of the infinitesimal progression of time τ = t − t 0 and k the Boltzmann constant, representing the atomic entropy unit. Similarly to P we also treat the time t as an observable, defined by t |u〉 = t|u〉. We apply the infinitesimal time-evolution operator U (t+τ,t) to the operator of time t and finally obtain the P – t commutation relation i[ P , t ] = k, which is independent of τ. It shows that the operators P and t do not commute, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties ΔP and Δt on measuring P and t, which are given by the P–t uncertainty relation ΔPΔt ≥ k/2. It readily allows a discussion of the evolution of the entropy S of the isolated system in time t from S′ to S max. Now, the irreversible steps are correctly described by the entropy production P = dS/dt > 0, while the thermal equilibrium is given by P = 0, ΔP = 0, and thus the lifetime of the equilibrium state Δt = ∞. According to the P–t uncertainty relation, the Boltzmann constant k is similarly important to the quantum thermodynamics of irreversible processes like Planck′s constant h to usual quantum mechanics.
We address the emergence of entropy production in the nonequilibrium process of an open quantum system from the viewpoint of the environment. By making use of a dilation-based approach akin to the Stinespring theorem, we derive an expression for the entropy production that comprises two fundamental contributions. The first is linked to the rate of the creation of correlations between the system and environment whereas the second highlights the possibility for the environment to modify its state in light of its coupling to the system. Both terms are shown to be associated with irreversible currents within the system and the environment, which pinpoint the emergence of irreversibility in the Markovian limit. Finally, we discuss how such a change of perspective in the study of entropy production has fecund implications for the study of non-Markovian open-system dynamics.
In this article we propose a new theory for collective modes of a many-body system which contains linear response theory and supplies us with systematic methods for the analysis of Brownian motion. The essential idea is to develop a method to find the possible justification for Zubarev's nonequilibrium ensemble from the new standpoint which is to draw attention to the usage of the dynamical invariance. The present theory has many interesting features which are relevant to irreversible processes. In this article we are more interested in problems of principle than in problems of applicability.