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Information entropy in problems of classical and quantum statistical mechanics. (English. Russian original) Zbl 1206.82054
Dokl. Math. 74, No. 3, 910-913 (2006); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 411, No. 5, 587-590 (2006).
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. 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 \(S(\nu,\mu)=-\int \frac{d\nu}{d\mu}\ln\frac{d\nu}{d\mu}d\mu\). When \(\mu\) is a counting measure, then the information entropy \(S(\nu)=S(\nu,\mu)\). The second Section gives the definition of \(F\)-entropy (denoted by \(S_F\), the third one a possible use of the notations and assumptions introduced in previous sections. The last section gives the scheme of two proofs given in the third Section.
Reviewer’s remark: Because in this paper the numbers of equations were not used sometimes it is hard to find an appropriate reference to exact equation given in the text.

82C03 Foundations of time-dependent statistical mechanics
82B03 Foundations of equilibrium statistical mechanics
94A17 Measures of information, entropy
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
82B10 Quantum equilibrium statistical mechanics (general)
94A15 Information theory (general)
Full Text: DOI
[1] R. Alicki and M. Fannes, Quantum Dynamical Systems (Oxford Univ. Press, Oxford, 2001). · Zbl 1140.81308
[2] M. Ohya and D. Petz, Quantum Entropy and Its Use (Springer-Verlag, Berlin, 1993). · Zbl 0891.94008
[3] H. Poincaré, in Selected Works (Nauka, Moscow, 1974), Vol. 3, pp. 385–412.
[4] J. Von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Univ. Press, Princeton, N.J.; Nauka, Moscow, 1964).
[5] V. V. Kozlov, Heat Equilibrium According to Gibbs and Poincaré (Izhevsk, Moscow, 2002) [in Russian].
[6] D. V. Zubarev, V. G. Morozov, and G. Röpke, Statistical Mechanics of Nonequilibrium Processes (Moscow, 2001) [in Russian].
[7] N. N. Bogolyubov and N. N. Bogolyubov, Jr., Introduction to Quantum Statistical Mechanics (Nauka, Moscow, 1984; World Scientific, Singapore, 1982). · Zbl 0576.60095
[8] J. W. Gibbs, Elementary Principles in Statistical Mechanics (Yale Univ. Press, New Haven, Conn., 1902; Gostekhizdat, Moscow, 1946). · JFM 33.0708.01
[9] J. von Neumann, Z. Phys. 57, 30–70 (1929).
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