Information entropy in problems of classical and quantum statistical mechanics
type: Item
instance of: scholarly article
title: Information entropy in problems of classical and quantum statistical mechanics
zbMATH Open document ID: 1206.82054
DOI: 10.1134/S1064562406060317
author: O. G. Smolyanov
author: Valery V. Kozlov
published in: Doklady Mathematics
publication date: 2010-12-08T00:00:00Z
review text: In this short paper we have the considerations about a generalization of information entropy with respect to a set of \(F\) of observables (the \(F\)-entropy). Its special cases are called the structurally stable entropy [cf. e.g. \textit{H. Poincaré}, Selected works in three volumes. Volume III: Mathematics. Theoretical physics. Analysis of the works of Henri Poincaré on mathematics and the natural sciences. Moskva: Izdat. Nauka. 771 p. R. 4.00 (1974; Zbl 0526.01033)] or the entropy of a macroscopic state (used by von Neumann). Taking into account that it is generally accepted that the information entropy of a Gibbs state of both quantum and classical systems coincides with the thermodynamic entropy of an equilibrium state, the \(F\)-entropy of a depending on time state of the system is defined as the information entropy of some ensemble similar to the microcanonical state. But the \(F\)-entropy is defined for any state of the system, which may not be quasi-equilibrium one. The main purpose in this paper is the study of conditions for the \(F\)-entropy of a quantum system to stabilize with increasing time. The paper consists of four sections. The first one describes the definitions and shows the preliminaries in the case of both the quantum and classical cases. It starts with the asumption that each state of the system is determined by a Borel probability measure on the separable complex Hilbert space of a system. If it is assumed that \(\nu\) is a state of the quantum system and \(A\) is a bounded quantum observable, then \(\int(Ax,x)\nu(dx)=\)tr\(AT\), where \(T\) is the correlation operator of the measure \(\nu\) defined as \(T=\int(x\otimes x)\nu(dx)\). Moreover, if \(\nu\) and \(f\) are a state and an observable of the Hamiltonian system, then \(\mu=\int f(x)\nu(dx)\) and \(\int(Ax,x)\nu(dx)=\mu\). If \(\nu\) is a probability and together with \(\mu\) they are measures on measurable space with \(\nu(\Omega)=1\) then the information entropy is defined as \(S(\n…
reviewed by: Dominik Strzałka
Mathematics Subject Classification ID: 82C03
Mathematics Subject Classification ID: 82B03
Mathematics Subject Classification ID: 94A17
Mathematics Subject Classification ID: 82C10
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 94A15
zbMATH DE Number: 5824772
zbMATH Keywords: B-G-S information entropy
zbMATH Keywords: \(F\)-entropy
zbMATH Keywords: quantum systems
zbMATH Keywords: statistical mechanics
zbMATH Keywords: non-equilibrium systems
MaRDI profile type: Publication
cites work: Quantum dynamical systems
cites work: Q4377723
cites work: Q3696277
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description: scientific article; zbMATH DE number 5824772