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Thermal equilibrium in the sense of Gibbs and Poincaré. (Тепловое равновесие по Гиббсу и Пуанкаре.) (Russian) Zbl 1099.82001
Sovremennaya Matematika. Izhevsk: Institut Komp’yuternykh Issledovanij (ISBN 5-93972-187-7). 320 p. (2002).
Author’s summary: This monograph extends the concepts of thermal equilibrium in mechanical systems suggested by Gibbs and Poincaré. Although Gibbs’ ideas are generally known, many problems he proposed remain unsolved. On the other hand, the profound results on kinetics given by Poincaré have not found any application and are even unknown to the specialists in the statistical mechanics. The problems discussed in the book are connected with the three interrelated topics: weak convergence of probability measures (the densities are solutions of the Liouville equation), the hierarchy of chaotic behavior of Hamiltonian dynamic systems, the theory of perturbation for an ensemble of weakly interacting subsystems. The results presented in the book provide better insights into the nature of the irreversible behavior of thermodynamic systems and allow a new interpretation of the second law of thermodynamics (which states an increase in the value of entropy) and to derive rigorously the Gibbs canonical distribution without resort to the ergodic hypothesis. The main text is organized essay-style into four appreciably autonomous chapters. Each chapter is supplied with comments and an individual list of references.
The ten appendices deal with the properties of invariant measures with smooth density, conditions for the existence of additional conservation laws (first integrals of Hamiltonian equations) and diffusion in nonlinear dynamical systems.

MSC:
82B03 Foundations of equilibrium statistical mechanics
80Axx Thermodynamics and heat transfer
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics
82C03 Foundations of time-dependent statistical mechanics
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