Stochastic Dynamics and Hierarchy for the Boltzmann Equation with Arbitrary Differential Scattering Cross Section
Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1629-1653
The stochastic dynamics for point particles that corresponds to the Boltzmann equation with arbitrary differential scattering cross section is constructed. We derive the stochastic Boltzmann hierarchy the solutions of which outside the hyperplanes of lower dimension where the point particles interact are equal to the product of one-particle correlation functions, provided that the initial correlation functions are products of one-particle correlation functions. A one-particle correlation function satisfies the Boltzmann equation. The Kac dynamics in the momentum space is obtained.
Spatially-Homogeneous Boltzmann Hierarchy as Averaged Spatially-Inhomogeneous Stochastic Boltzmann Hierarchy
Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 78-93
We introduce the stochastic dynamics in the phase space that corresponds to the Boltzmann equation and hierarchy and is the Boltzmann–Grad limit of the Hamiltonian dynamics of systems of hard spheres. By the method of averaging over the space of positions, we derive from it the stochastic dynamics in the momentum space that corresponds to the space-homogeneous Boltzmann equation and hierarchy. Analogous dynamics in the mean-field approximation was postulated by Kac for the explanation of the phenomenon of propagation of chaos and derivation of the Boltzmann equation.
Ukr. Mat. Zh. - 1999. - 51, № 5. - pp. 614-635
We consider the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of a system of hard spheres. A new concept of averages over states of stochastic systems is introduced, in which the contribution of the hypersurfaces on which stochastic point particles interact is taken into account. We give a rigorous derivation of the infinitesimal operators of the semigroups of evolution operators.
Boltzmann-Enskog limit for equilibrium states of systems of hard spheres in the framework of a canonical ensemble
Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1195–1205
We prove the existence of the Boltzmann-Enskog limit for an equilibrium system of hard spheres. On the basis of analysis of the Kirkwood-Salsburg equations, we show that the limit distribution functions are constants that can be represented as series in density.