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Journal of Mathematical Physics

Diffusion processes, quantum dynamical semigroups, and the classical KMS condition

Alberto Frigerio, Vittorio Gorini

Published Apr 1, 1984• 15 citations
Publisher

Abstract

We study some properties of the generator of a diffusion process describing the reduced dynamics of a classical system weakly coupled to an external reservoir. We show that such processes may be equivalently obtained as classical limits of quantum dynamical semigroups derived in the weak coupling limit. We prove that the stationary state is independent of the coupling if and only if the reservoir is in a state of thermal equilibrium.

DOI
10.1063/1.526274

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We study some properties of the generator of a diffusion process describing the reduced dynamics of a classical system weakly coupled to an external reservoir. We show that such processes may be equivalently obtained as classical limits of quantum dynamical semigroups derived in the weak coupling limit. We prove that the stationary state is independent of the coupling if and only if the reservoir is in a state of thermal equilibrium.
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\bad Quantum dynamical semigroups represent a noncommutative analogue of (sub)Markov semigroups in classical probability: while the latter are semigroups of maps in functional spaces, the former are semigroups of maps in operator algebras having certain properties of positivity and normalization. In this paper we describe quantum dynamical semigroups, which are the noncommutative analogues of classical diffusions on RRR and R+R_{+}R+​, and demonstrate various properties of the semigroup and its generator depending on the boundary condition. We also give a proof of a result describing the domain of the generator of “noncommutative diffusion on R+R_{+}R+​ with extinction at 0” and give an explicit example of the trace-class operator in this domain, which does not belong to the domain of closure of the initial operator.
In the weak-coupling limit approach to open quantum systems, the presence of the bath is eliminated and accounted for by a master equation that introduces dissipative contributions to the system reduced dynamics: within this framework, there are no bath entropy contributions to the entropy balance. We show that, as a consequence, the entropy production fails to be positive for a class of physically legitimate, that is completely positive and trace preserving, non-Markovian dynamical maps. Moreover, in absence of the semigroup property, if the reduced dynamics has a thermal asymptotic state, this need not be stationary. Then even the integrated entropy production becomes negative. These observations imply that, when the conditions leading to reduced dynamics of semigroup type are relaxed, a consistent formulation of the second law of thermodynamics requires that the environment contribution to the entropy balance be explicitly taken into account.
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The classical limit of the infinitesimal generators of quantum dynamical semigroups obtained from quantum diffusions are shown to be second order semi-elliptic differential operators and thus (Markov) generators of classical diffusions in phase space. Necessary and sufficient conditions are established, in terms of the quantum generators, under which (a) the Markov generator is strictly elliptic, b) the classical diffusion is deterministic, and c) the classical flow is symplectic On montre que la limite classique du generateur infinitisimal d'un semigroupe dynamique quantique, obtenu a partir d'une diffusion quantique, et un operateur differentiel semi-elliptique d'ordre 2 et donc, le generateur (Markov) d'une diffusion classique dans l'espace de phase. On etablit des conditions necessaires et suffisantes pour que: a) le generateur soit (positivement) elliptique; b) la diffusion classique soit deterministe; c) le flot classique soit symplectique