Stochastic Liouville Equations

TLDR

The stochastic Liouville equation describes a dynamical system perturbed by a stochastic process, and its ensemble‑averaged formal integral defines a relaxation operator Φ(t)=exp K(t) where K(t) is the cumulant function. The authors first analyze a simple oscillator with random frequency modulation to illustrate general properties, then introduce the concepts of slow and fast modulation and extend these ideas to more general stochastic Liouville equations. They demonstrate that by defining generalized exponential functions, this framework can unify and clarify essential features of stochastic Liouville problems.

Abstract

When a dynamical system has a perturbation which is considered as a stochastic process, the Liouville equation for the system in the phase space or the space of quantum-mechanical density operators is a sort of stochastic equation. The ensemble average of its formal integral defines the relaxation operator Φ(t) of the system. By the definition Φ(t) = exp K(t), the cumulant function K(t) may be introduced. Some general properties are first discussed for a simple example of an oscillator with random frequency modulation and then, concepts of slow and fast modulation are considered. These concepts can be generalized to more general types of stochastic Liouville equations. It is shown that by various possibilities of defining generalized exponential functions, this approach may be useful to understand some essential features of the problem from an unified point of view.