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Physical Review A

Stochastic theory of relaxation and approach to thermal equilibrium for phonon reservoirs

Karim Faid, Ronald F. Fox

Published Mar 1, 1987• 22 citations
Publisher

Abstract

A molecular system is stochastically coupled to a phonon thermal reservoir. The equation for its averaged density matrix describes the approach to thermal equilibrium.Received 15 September 1986DOI:https://doi.org/10.1103/PhysRevA.35.2684©1987 American Physical Society

DOI
10.1103/physreva.35.2684

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A molecular system is stochastically coupled to a phonon thermal reservoir. The equation for its averaged density matrix describes the approach to thermal equilibrium.Received 15 September 1986DOI:https://doi.org/10.1103/PhysRevA.35.2684©1987 American Physical Society
Abstract We have previously developed this framework in the context of static response, here we will generalize this perspective to dynamic response, and the evolution between distinct thermal equilibrium states. Of particular importance will be the derivation of the fluctuation-dissipation theorem, which relates the response of a system to its time dependent fluctuations, clarifying an equivalence between how a system responds to a small perturbation and how it regresses back to equilibrium in the absence of a perturbation. With this framework we can develop linear models of relaxation, classically and quantum mechanically, and derive equa- tions of motion suitable for describing the dynamics of thermal ensembles held at fixed temperature rather than fixed energy. These models are Gaussian, and approximate the dynamical correlations of interacting systems. These effective equations of motion are rendered stochastic due to the incomplete description of the microscopic degrees of freedom that evolve the system.
Stochastic methods are ubiquitous to a variety of fields, ranging from physics to economics and mathematics. In many cases, in the investigation of natural processes, stochasticity arises every time one considers the dynamics of a system in contact with a somewhat bigger system, an environment with which it is considered in thermal equilibrium. Any small fluctuation of the environment has some random effect on the system. In physics, stochastic methods have been applied to the investigation of phase transitions, thermal and electrical noise, thermal relaxation, quantum information, Brownian motion and so on. In this review, we will focus on the so-called stochastic Schrödinger equation. This is useful as a starting point to investigate the dynamics of open quantum systems capable of exchanging energy and momentum with an external environment. We discuss in some detail the general derivation of a stochastic Schrödinger equation and some of its recent applications to spin thermal transport, thermal relaxation, and Bose-Einstein condensation. We thoroughly discuss the advantages of this formalism with respect to the more common approach in terms of the reduced density matrix. The applications discussed here constitute only a few examples of a much wider range of applicability.