The second fluctuation-dissipation theorem for the generalized Langevin equation

TL;DR

This paper establishes conditions for stationary solutions of the generalized Langevin equation and characterizes when these solutions satisfy a fluctuation-dissipation relation, linking stochastic forces with system response.

Contribution

It provides necessary and sufficient conditions for stationarity and introduces a second fluctuation-dissipation theorem for the generalized Langevin equation.

Findings

Stationary solutions exist under specific conditions on the memory kernel.

Solutions approach a stationary process when driven by correlated Gaussian noise.

Stationarity of the solution requires a particular fluctuation-dissipation relation.

Abstract

Necessary and sufficient conditions are presented for the existence of (second order) stationary solutions of the generalized Langevin equation under appropriate assumptions on the associated memory kernel. When this stochastic equation is formulated as an initial value problem, then it is shown that the solution approaches a stationary process as time goes to infinity, whenever the fluctuating force term is taken to be a combination of a white noise process and a mean-square continuous centered stationary Gaussian process (which may be correlated with each other). The limiting process can be any centered stationary Gaussian process with sufficiently smooth spectral density. On the other hand, the solution itself is only stationary when the fluctuating force satisfies a certain fluctuation-dissipation relation, and this stationary solution is uniquely specified by its covariance.