Classical Markovian kinetic equations: Explicit form andH-theorem

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Abstract
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 semi groups on the space of observable. Moreover, a general H-theorem for the ad joint of such semi groups is formulated and proved provided that at least locally, an invariant measure exists. Under a certain continuity assumption, the Markov semi group property is sufficient for a linear kinetic equation to be a second order differential equation with no negative-definite leading coefficient. Conversely it is shown that such equations define Markov semi groups satisfying an H-theorem, provided there exists a nonnegative equilibrium solution for their formal ad joint.

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