Markovian perturbation, response and fluctuation dissipation theorem

TL;DR

This paper rigorously analyzes the Fluctuation Dissipation Theorem (FDT) within Markov processes, characterizing response functions out of equilibrium and near equilibrium, and comparing different perturbations across various stochastic systems.

Contribution

It formalizes the response function concept for Markov processes, characterizes all possible responses out of equilibrium, and compares perturbations in different stochastic models.

Findings

At equilibrium, response functions satisfy FDT.

Response functions depend on the perturbation for out-of-equilibrium processes.

Comparison of perturbations in jump processes, diffusions, and spin systems.

Abstract

We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of "linear response function" in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is, if the initial measure is invariant for the given Markov semi-group, then for any pair of times s<t and nice functions f,g, the dissipation, that is, the derivative in s of the covariance of g(X(t)) and f(X(s)) equals the infinitesimal response at time t and direction g to any Markovian perturbation that alters the invariant measure of X(.) in the direction of f at time s. The same applies in the so called FDT regime near equilibrium, i.e. in the limit s going to infinity with t-s fixed, provided X(s) converges in law to an invariant measure for its dynamics. We provide the response function of two generic Markovian perturbations which we then compare and contrast for pure jump processes on a discrete space, for finite dimensional diffusion processes, and for stochastic spin systems.