Abstract
We consider different methods for the derivation of the stochastic Boltzmann hierarchy corresponding to the stochastic dynamics that is the Boltzmann-Grad limit of the Hamiltonian dynamics of hard spheres. Solutions of the stochastic Boltzmann hierarchy are the Boltzmann-Grad limit of solutions of the BBGKY hierarchy of hard spheres in the entire phase space. A new concept of reduced distribution functions corresponding to the stochastic dynamics are introduced. They take into account the contribution of the hyperplanes of lower dimension where stochastic point particles interact with one another. The solutions of the Boltzmann equation coincide with one-particle distribution functions of the stochastic Boltzmann hierarchy and are represented by integrals over the hyperplanes where the stochastic point particles interact with one another.
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D. Ya. Petrina and K. D. Petrina, “Stochastic dynamics and Boltzmann hierarchy. I, II, III,” Ukr. Mat. Zh., 50, 2–4, (1998).
M. Lampis, D. Ya. Petrina. and K. D. Petrina, “Stochastic dynamics as a limit of Hamiltonian dynamics of hard spheres,” Ukr. Mat. Zh., 51, No. 5, 614–635 (1999).
D. Ya. Petrina and K. D. Petrina, Stochastic Dynamics and Boltzmann Hierarchy [in Russian], Preprint No. 96.7, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996).
D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev. Mathematical Foundations of Classical Statistical Mechanics. Continuous Systems, Gordon and Breach, New York 1989.
C. Cercignani, V. I. Gerasimenko, and D. Ya. Petrina, Many-Particle Dynamics and Kinetic Equations, Kluwer, Dordrecht 1997.
I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. 5, Integral Geometry and Representation Theory [in Russian], Nauka, Moscow (1967).
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Petrina, D.Y. Methods for derivation of the stochastic Boltzmann hierarchy. Ukr Math J 52, 543–563 (2000). https://doi.org/10.1007/BF02515396
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DOI: https://doi.org/10.1007/BF02515396