dorsal/arxiv
View SchemaClassical Markovian Kinetic Equations: Explicit Form and H-Theorem
| Authors | Constantinos Tzanakis, Alkis P. Grecos |
|---|---|
| Categories | |
| ArXiv ID | physics/9708031 |
| URL | https://arxiv.org/abs/physics/9708031 |
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.
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"abstract": "The probabilistic description of finite classical systems often leads to\nlinear kinetic equations. A set of physically motivated mathematical\nrequirements is accordingly formulated. We show that it necessarily implies\nthat solutions of such a kinetic equation in the Heisenberg representation,\ndefine Markov semigroups on the space of observables. Moreover, a general\nH-theorem for the adjoint of such semigroups is formulated and proved provided\nthat at least locally, an invariant measure exists. Under a certain continuity\nassumption, the Markov semigroup property is sufficient for a linear kinetic\nequation to be a second order differential equation with nonegative-definite\nleading coefficient. Conversely it is shown that such equations define Markov\nsemigroups satisfying an H-theorem, provided there exists a nonnegative\nequilibrium solution for their formal adjoint, vanishing at infinity.",
"arxiv_id": "physics/9708031",
"authors": [
"Constantinos Tzanakis",
"Alkis P. Grecos"
],
"categories": [
"math-ph",
"math.MP"
],
"title": "Classical Markovian Kinetic Equations: Explicit Form and H-Theorem",
"url": "https://arxiv.org/abs/physics/9708031"
},
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