Mathematical problems of statistical mechanics of a system of elastic balls

Abstract

CONTENTS § 1. Introduction § 2. Hamiltonian dynamics of a system of elastic balls 2.1. Definitions 2.2. Theorem on the existence of phase trajectories 2.3. Evolution operator 2.4. Cauchy problem for the Liouville equation 2.5. Non-equilibrium ensembles § 3. Cauchy problem for Bogolyubov equations 3.1. Evolution operator for Bogolyubov equations 3.2. Solutions of Bogolyubov equations 3.3. Derivation of Bogolyubov equations 3.4. One-dimensional system of elastic rods 3.5. A series of iterations § 4. Thermodynamic limit 4.1. Thermodynamic limit of equilibrium states 4.2. Cauchy problem for Bogolyubov equations of an infinite system 4.3. Existence of the thermodynamic limit 4.4. Global solutions 4.5. Methods of solution of Bogolyubov equations § 5. The Boltzmann-Grad limit 5.1. Formulation of the problem and the basic result 5.2. The Boltzmann equation 5.3. The Boltzmann-Grad limit for equilibrium states 5.4. Technical lemmas 5.5. The Boltzmann-Grad limit of solutions of Bogolyubov equations References