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

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 $ \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.