Stochastic dynamics and Boltzmann hierarchy. Translated from the Ukrainian by Dmitry V. Malyshev and Peter V. Malyshev (Q2272289)

The problems of the BBGKY for a system of hard spheres are discussed in this eight chapters' book. General results are based on the Boltzmann equation. In the introduction, fundamental mathematical problems of the nonlinear Boltzmann equation solutions are reviewed. The BBGKY hierarchy for a system of hard spheres, the stochastic Boltzmann hierarchy and justification of the Boltzmann-Grad limit are discussed. The stochastic dynamics is derived from the Hamiltonian dynamics of hard spheres in the Boltzmann-Grad limit. The stochastic Boltzmann hierarchy and boundary conditions is derived from the stochastic dynamics of point particles. In the fourth chapter, the existence and uniqueness of the Boltzmann hierarchy is proved and the stochastic property is established. The fifth chapter is devoted to obtain the stochastic Kac dynamics in the momentum space from the stochastic dynamics in the phase space. In the sixth chapter, a connection between the scattering of hard spheres and systems of particles with arbitrary cross sections is discussed. In the seventh chapter, the system of spheres with inelastic scattering cross sections is considered. Readers can find some comparisons of the microscopic results that are obtained from the regular and the stochastic kinetic of particles. This book may be useful for advanced graduate students and for scientists who are interested in mathematical problems of the statistical mechanics and rarefied gases flow.