Fluctuation-Dissipation relation in sub-diffusive systems: the case of granular single-file

TL;DR

This paper investigates the fluctuation-dissipation relation in a granular gas of hard rods on a ring, demonstrating its validity in elastic/dilute cases and its breakdown in dense/inelastic regimes, proposing a generalized relation.

Contribution

It provides the first analysis of fluctuation-dissipation relations in sub-diffusive granular single-file systems, highlighting conditions for validity and failure.

Findings

FD relation holds in elastic/dilute regimes

Failure of Einstein formula in dense/inelastic regimes

Proposes a generalized FD relation for complex cases

Abstract

We study a gas of hard rods on a ring, driven by an external thermostat, with either elastic or inelastic collisions, which exhibits sub-diffusive behavior $<x^2 > \sim t^{1/2}$. We show the validity of the usual Fluctuation-Dissipation (FD) relation, i.e. the proportionality between the response function and the correlation function, when the gas is elastic or diluted. On the contrary, in strongly inelastic or dense cases, when the tracer velocity is no more independent of the other degrees of freedom, the Einstein formula fails and must be replaced by a more general FD relation.