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The Source Term of the Non-Equilibrium Statistical Operator

GERD RÖPKE

Published Jun 6, 2019• 1 citation
Publisher

Abstract

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.

DOI
10.3390/particles2020020

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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 method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.
The method of the nonequilibrium statistical operator developed by D. N. Zubarev is employed to analyze and derive generalized transport and kinetic equations. The degrees of freedom in solids can often be represented as a few interacting subsystems (electrons, spins, phonons, nuclear spins, etc.). Perturbation of one subsystem may produce a nonequilibrium state which is then relaxed to an equilibrium state due to the interaction between particles or with a thermal bath. The generalized kinetic equations were derived for a system weakly coupled to a thermal bath to elucidate the nature of transport and relaxation processes. It was shown that the "collision term" had the same functional form as for the generalized kinetic equations for the system with small interactions among particles. The applicability of the general formalism to physically relevant situations is investigated. It is shown that some known generalized kinetic equations (e.g. kinetic equation for magnons, Peierls equation for phonons) naturally emerges within the NSO formalism. The relaxation of a small dynamic subsystem in contact with a thermal bath is considered on the basis of the derived equations. The Schrödinger-type equation for the average amplitude describing the energy shift and damping of a particle in a thermal bath and the coupled kinetic equation describing the dynamic and statistical aspects of the motion are derived and analyzed. The equations derived can help in the understanding of the origin of irreversible behavior in quantum phenomena.
In this paper, we formally demonstrate that the nonequilibrium density matrix developed by Hershfield for the steady state has the form of a McLennan-Zubarev nonequilibrium ensemble. The correction term in this pseudoequilibrium Gibbs-like ensemble is directly related to the entropy production in the quantum open system. The fact that both methods state that a nonequilibrium steady state can be mapped onto a pseudoequilibrium, permits us to develop nonequilibrium quantities from formal expressions equivalent to the equilibrium case. We provide an example: the derivation of a nonequilibrium distribution function for the electron population in a scattering region in the context of quantum transport.
We propose a systematic approach to the dynamics of open quantum systems in the framework of Zubarev’s nonequilibrium statistical operator method. The approach is based on the relation between ensemble means of the Hubbard operators and the matrix elements of the reduced statistical operator of an open quantum system. This key relation allows deriving master equations for open systems following a scheme conceptually identical to the scheme used to derive kinetic equations for distribution functions. The advantage of the proposed formalism is that some relevant dynamical correlations between an open system and its environment can be taken into account. To illustrate the method, we derive a non-Markovian master equation containing the contribution of nonequilibrium correlations associated with energy conservation.
A central open question in quantum thermodynamics is the identification of thermodynamic entropy away from equilibrium. Although thermodynamic entropy coincides with the von Neumann entropy for a Gibbs state at equilibrium, the thermodynamic role of quantum coherence generated during nonequilibrium evolution remains unclear. Here, we show that von Neumann entropy does not, in general, represent the thermodynamic entropy governing heat exchange in coherent nonequilibrium quantum systems. By integrating the resource theory of quantum coherence with thermodynamic principles, we develop a general framework for nonequilibrium quantum thermodynamics. We identify thermodynamic entropy with the energy entropy derived from energy-measurement statistics and introduce a dynamical temperature extending the equilibrium concept to nonequilibrium states. Consequently, we obtain an exact entropy-balance relation separating the entropy flux associated with heat exchange from the coherence-loss contribution to entropy production, establishing an operational connection between quantum coherence, heat flow, and total entropy change. Unlike approaches relying on equilibrium reference states or stationary Gibbs ensembles, our framework uses instantaneous dynamical quantities and remains valid throughout transient nonequilibrium evolution. It provides consistent dynamical definitions of temperature, work, heat, and free energy. Applying the formalism to an exactly solvable open quantum system, we show that the first and second laws hold throughout transient evolution and that equilibrium thermodynamics emerges dynamically in the weak-coupling long-time limit. Our results establish a unified framework for coherent nonequilibrium quantum thermodynamics and suggest opportunities for controlling heat dissipation through coherence engineering in quantum thermal technologies.