Classical Markovian Kinetic Equations: Explicit Form and H-Theorem

TL;DR

This paper characterizes classical Markovian kinetic equations, proving they generate Markov semigroups and establishing an H-theorem, linking the equations' structure to probabilistic and thermodynamic properties.

Contribution

It provides a rigorous mathematical framework connecting kinetic equations with Markov semigroups and proves an H-theorem under specific conditions.

Findings

Solutions define Markov semigroups on observables.

An H-theorem is valid if an invariant measure exists.

Second order differential equations correspond to Markov semigroups with equilibrium solutions.

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.