| 概要 |
Petrina (formerly of the Institute of Mathematics of the Ukrainian Academy of Sciences) begins this monograph on stochastic dynamics and the Boltzmann hierarchy by surveying the results concerning the...existence of solutions of the BBGKY (Bogolyubov-Born-Green-Kirkwood-Yvon) hierarchy of a system of hard spheres and presents a justification of the Boltzman-Grad limit, giving attention to the boundary conditions for both the BBGKY hierarchy and the stochastic Boltzmann hierarchy. He then derives the stochastic dynamics from the Hamiltonian dynamics of hard spheres in the Boltzmann-Grad limit and derives the stochastic Boltzmann hierarchy with boundary conditions from the stochastic dynamics of point particles. The existence of a solution of the stochastic Boltzmann hierarchy is then proved and the property of chaos is established in order to deduce the Boltzmann equation. After obtaining the stochastic Kac dynamics in the momentum space from the stochastic dynamics in the phase space, he shows that the spatially homogenous Boltzmann equation can be derived from the stochastic Boltzmann hierarchy in the phase space without using mean-field approximation. Next, results obtained for a system of hard spheres are generalized to systems of particles with arbitrary scattering cross section; a system of spheres with inelastic scattering is used as a model of granular flows; and, finally, a solution of the Cauchy problem for the hierarchy in the space of sequences of summable functions is constructed and the stochastic dynamics for granular flows corresponding to the Boltzmann equation is introduced. Annotation ©2009 Book News, Inc., Portland, OR (booknews.com) 続きを見る
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