On N. N. Bogolyubov's works in classical and quantum statistical mechanics
type: Item
instance of: scholarly article
title: On N. N. Bogolyubov's works in classical and quantum statistical mechanics
zbMATH Open document ID: 0806.01013
DOI: 10.1007/BF01060975
author: D. Ya. Petrina
author: Yu. A. Mitropol'skij
published in: Ukrainian Mathematical Journal
publication date: 1994-09-20T00:00:00Z
review text: Nicolai Nicolajewich Bogolyubov (born 21. 8. 1909 in Nishni-Novgorod (Gorki)) founded new directions in nonlinear mechanics, statistical physics, and quantum field theory. The authors list these results, the ideas of his proofs and the prehistories. This begins with the theorem on the existence of invariant measure for compact dynamical systems and its history (Boltzmann's ergodic hypothesis, and von Neumann and Birkhoff). For the Hamiltonian systems, the required measure is the volume of phase space. Together with his teacher N. M. Krylov he showed that the invariant measure exists for any dynamical system. They introduced the concept of ergodic sets and showed that the invariant measure can be decomposed into invariant measures that are concentrated on ergodic sets and cannot be decomposed. Fine properties of complete continuous positive operators were used. A model for the transition to statistical equilibrium in a system connected with a heat bath is being considered. In his monograph `Problems of Dynamic Theory in Statistical Physics' the classical statistical mechanics is formulated in terms of sequences of distribution functions and equations for these (=Bogolyubov equations). The states of infinite systems were determined in terms of the states of finite systems by the thermodynamic limit procedure. These papers gave modern form to classical statistical mechanics and opened a new stage. The method of justification of the thermodynamic limit procedure suggested by Bogolyubov in 1949 was reanimated in the sixties and applied to the case of grand canonical ensemble. In his `Lectures in Quantum Statistics' (1949) he introduced a new concept of states in infinite systems in terms of an infinite sequence of statistical operators. A new approach to the approximate quantization was elaborated and applied to the polar theory of metals and the theory of magnetism. It is shown that the methods developed by Bogolyubov for classical equilibrium systems can be applied to …
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 82-03
Mathematics Subject Classification ID: 82D50
Mathematics Subject Classification ID: 82B30
zbMATH DE Number: 635710
zbMATH Keywords: Bogolyubov
zbMATH Keywords: statistical mechanics
zbMATH Keywords: superfluidity
zbMATH Keywords: superconductivity
reviewed by: Hilmar Grimm
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 635710