Mathematical Foundations of Modern Statistical Mechanics

Abstract

The book is devoted to the systematic description of the mathematical foundations of modern statistical mechanics.The approach is based on methods of the infinite dimensional analysis, which most adequately meet the mathematical needs of describing physical systems with a large number of elements. A characteristic feature of the description is the application of infinite-dimensional integrals, which makes it possibleavoid cumbersome combinatorial formulas and make the proof of many theorems and statements more transparent. The issue of interaction between point particles is covered in detail, new sufficiency criteria for potentials, mathematical problems of the thermodynamic limit for correlation functions (ordinary, connected, partially connected) by the methods of integral equations and methods of cluster expansions. The quasi-lattice approximation for thermodynamic functions and correlation functions of continuous systems are described within the framework of the so-called {\it cell gas} model. Systems of ions and dipoles are described. A rigorous justification of the theory of Debye–Hückel of screening inteructions is given. Quantum continuous systems are briefly considered from the point of view of the technique of cluster expansions for the reduced density matrix. For senior year students, graduate students and scientists who seek to deepen the understanding of mathematical problems of statistical mechanics.