Nonlinear susceptibilities and the measurement of a cooperative length

TL;DR

This paper derives a comprehensive fluctuation dissipation relation for Markov systems and proposes a second order susceptibility as a tool to measure cooperative effects in glasses, validated through numerical simulations.

Contribution

It provides an exact, general fluctuation dissipation relation applicable to equilibrium and nonequilibrium systems, and introduces a new measurable quantity for cooperative effects.

Findings

Derived exact fluctuation dissipation relation for Markov systems.

Proposed second order susceptibility as a measure of cooperativity.

Validated the approach with numerical simulations of the Edwards-Anderson model.

Abstract

We derive the exact beyond-linear fluctuation dissipation relation, connecting the response of a generic observable to the appropriate correlation functions, for Markov systems. The relation, which takes a similar form for systems governed by a master equation or by a Langevin equation, can be derived to every order, in large generality with respect to the considered model, in equilibrium and out of equilibrium as well. On the basis of the fluctuation dissipation relation we propose a particular response function, namely the second order susceptibility of the two-particle correlation function, as an effective quantity to detect and quantify cooperative effects in glasses and disordered systems. We test this idea by numerical simulations of the Edwards-Anderson model in one and two dimensions.