Markovian processes in classical and quantum statistical mechanics

Abstract

On the basis of the ninth axiom of Mackey a unified description of a (reversible and irreversible) time evolution of classical and quantum systems is formulated in terms of one-parameter semigroups (called dynamical semigroups) of contracting positive linear operators acting in the partially ordered real Banach space generated by states of the system. In the case of classical statistical mechanics our description generalizes the well-known approach via Markovian processes, while in the quantum case it coincides with the Kossakowski's recent axiomatic definition of time evolution of quantum systems. In this paper we find a necessary and sufficient condition for a classical dynamical semigroup to be a Markovian one, and show that an analogous condition for a quantum dynamical semigroup is automatically satisfied, i.e. it always describes a quantum Markovian process.