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Classical markovian kinetic equations: Explicit form and H-theorem

par Tzanakis, Constantinos;Grecos, Alkis
Référence Transport theory and statistical physics, 28, 4, page (325-348)
Publication Publié, 1999

Article révisé par les pairs

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Résumé :

The probabilistic description of finite classical systems often leads to linear kinetic equations. A set of physically motivated mathematical requirements is accordingly formulated. We show that it necessarily implies that solutions of such a kinetic equation in the Heisenberg representation, define Markov semigroups on the space of observables. Moreover, a general H-theorem for the adjoint of such semigroups is formulated and proved provided that at least locally, an invariant measure exists. Under a certain continuity assumption, the Markov semigroup property is sufficient for a linear kinetic equation to be a second order differential equation with nonegative-definite leading coefficient. Conversely it is shown that such equations define Markov semigroups satisfying an H-theorem, provided there exists a nonnegative equilibrium solution for their formal adjoint.