Many-Particle Dynamics and Kinetic Equations: (420 Mathematics and Its Applications)

This work is devoted to the evolution of infinite systems interacting via a short range potential. The Hamilton dynamics is defined through its evolution semigroup and the corresponding Bogolubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is constructed. The existence of global in time solutions of the BBGKY hierarchy for hard spheres interacting via a short range potential is proved in the Boltzmann-Grad limit and by Bogolubov's and Cohen's methods. This volume should be of interest to graduate students and researchers whose work involves mathematical and theoretical physics, functional analysis and probability theory.

Table of Contents:
I: The Bbgky Hierarchy.- 1.1. Introduction.- 1.2. The Hamilton dynamics of a system of particles with hard core.- 1.3. The evolution operator for a system with finite number of particles.- 1.4. The derivation of the BBGKY hierarchy.- 1.5. The steady BBGKY hierarchy.- Appendix I. The infinitesimal operator of the groupSN (t).- Appendix II. The formal derivation of the infinitesimal operator [BB] of the group SN(t).- II: The Initial Value Problem For The Bbgky Hierarchy of a System of a Finite Number of Particles.- 2.1. Introduction.- 2.2. The evolution operator of the BBGKY hierarchy.- 2.4. Existence of solutions for a one-dimensional BBGKY hierarchy.- 2.5. The iteration series.- III: The Initial Value Problem For L? Data: Thermodynamic Limit.- 3.1. Introduction.- 3.2. A local existence theorem for the BBGKY hierarchy of hard spheres.- 3.3. Global existence theorems.- 3.4. Method of the interaction region.- IV: The Derivation of The Boltzmann Equation.- 4.1. Introduction: On the Boltzman-Grad limit.- 4.2. The Boltzmann-Grad limit for equilibrium states.- 4.3. The Boltzmann hierarchy and the Boltzmann equation.- 4.4. The Boltzmann-Grad limit for solutions of initial value problem for the BBGKY hierarchy.- 4.5. The Boltzmann-Grad limit for equilibrium states of systems of hard spheres in the framework of the canonical ensemble.- V: On the Derivation of Kinetic Equations From the Bbgky Hierarchy.- 5.1. Introduction: kinetic equations.- 5.2. Bogolubov’s method of constructing kinetic equations.- 5.3. The non-equilibrium cluster expansions method.- 5.4. Justification of the generalized kinetic equation.- References.