Classical Markovian Kinetic Equations: Explicit Form and H-Theorem
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 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, vanishing at infinity.
- Publication:
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arXiv e-prints
- Pub Date:
- August 1997
- DOI:
- 10.48550/arXiv.physics/9708031
- arXiv:
- arXiv:physics/9708031
- Bibcode:
- 1997physics...8031T
- Keywords:
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- Mathematical Physics
- E-Print:
- 25pp, LATEX