Classical Markovian Kinetic Equations: Explicit Form and HTheorem
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 semigroups on the space of observables. Moreover, a general Htheorem 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 nonegativedefinite leading coefficient. Conversely it is shown that such equations define Markov semigroups satisfying an Htheorem, provided there exists a nonnegative equilibrium solution for their formal adjoint, vanishing at infinity.
 Publication:

arXiv eprints
 Pub Date:
 August 1997
 DOI:
 10.48550/arXiv.physics/9708031
 arXiv:
 arXiv:physics/9708031
 Bibcode:
 1997physics...8031T
 Keywords:

 Mathematical Physics
 EPrint:
 25pp, LATEX