The generalized Vlasov kinetic equation

V. V. Kozlov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 63 (2008) no. 4, pp. 691-726

Résumé

This paper is concerned with the investigation of a generalized kinetic equation describing the evolution of the density of a probability measure. In the general case this is a non-linear integro-differential equation. On the one hand, this equation includes as a special case the simpler linear Liouville equation (which underlies classical statistical mechanics) and the equation of a self-consistent field (the Vlasov kinetic equation). On the other hand, some other well-known equations also reduce to this equation, for instance, the vorticity equation for plane flows of an ideal incompressible fluid. The main aim of the paper is to study the problem of the weak limits, as the time tends to infinity, of solutions of the generalized kinetic equation. This problem plays a significant role in the transition from a micro- to a macrodescription, when the behaviour of the averages (most probable values) of dynamical quantities is considered. The theory of weak limits of solutions of the Liouville equation is closely connected with ideas and methods of ergodic theory. The case under consideration presents greater difficulties, which stem from the non-trivial problem of the existence of invariant countably-additive measures for dynamical systems in infinite-dimensional spaces. General results are applied to the analysis of continua of interacting particles and to the investigation of statistical properties of plane flows of an ideal fluid.

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