The fluctuation-dissipation relation: how does one compare correlation functions and responses?

TL;DR

This paper explores the fluctuation-dissipation relations in complex systems, highlighting how their verification depends on the choice of correlation functions and degrees of freedom, especially in systems with memory or spatial inhomogeneities.

Contribution

It extends the understanding of fluctuation-dissipation relations by analyzing a generalized framework and illustrating ambiguities in their verification in systems with memory and in granular media.

Findings

Usual FDRs are recovered only in specific cases.

Correlation functions must be carefully chosen to verify FDRs.

Inhomogeneities can cause violations of standard FDRs.

Abstract

We discuss the well known Einstein and the Kubo Fluctuation Dissipation Relations (FDRs) in the wider framework of a generalized FDR for systems with a stationary probability distribution. A multi-variate linear Langevin model, which includes dynamics with memory, is used as a treatable example to show how the usual relations are recovered only in particular cases. This study brings to the fore the ambiguities of a check of the FDR done without knowing the significant degrees of freedom and their coupling. An analogous scenario emerges in the dynamics of diluted shaken granular media. There, the correlation between position and velocity of particles, due to spatial inhomogeneities, induces violation of usual FDRs. The search for the appropriate correlation function which could restore the FDR, can be more insightful than a definition of ``non-equilibrium'' or ``effective temperatures''.