Fluctuation-Dissipation relations in Driven Granular Gases

TL;DR

This study investigates fluctuation-dissipation relations in a driven 2D granular gas, demonstrating that a modified Kubo relation with an effective temperature equal to the granular temperature holds in non-equilibrium steady states.

Contribution

It shows that the classical Kubo relation applies to driven granular gases when the temperature is replaced by the granular temperature, extending fluctuation-dissipation concepts to non-equilibrium systems.

Findings

Modified Kubo relation holds with granular temperature

Response functions match autocorrelations for low inelasticities

Effective temperature equals granular temperature in steady state

Abstract

We study the dynamics of a 2d driven inelastic gas, by means of Direct Simulation Monte Carlo (DSMC) techniques, i.e. under the assumption of Molecular Chaos. Under the effect of a uniform stochastic driving in the form of a white noise plus a friction term, the gas is kept in a non-equilibrium Steady State characterized by fractal density correlations and non-Gaussian distributions of velocities; the mean squared velocity, that is the so-called {\em granular temperature}, is lower than the bath temperature. We observe that a modified form of the Kubo relation, which relates the autocorrelation and the linear response for the dynamics of a system {\em at equilibrium}, still holds for the off-equilibrium, though stationary, dynamics of the systems under investigation. Interestingly, the only needed modification to the equilibrium Kubo relation is the replacement of the equilibrium temperature with an effective temperature, which results equal to the global granular temperature. We present two independent numerical experiment, i.e. two different observables are studied: (a) the staggered density current, whose response to an impulsive shear is proportional to its autocorrelation in the unperturbed system and (b) the response of a tracer to a small constant force, switched on at time $t_w$, which is proportional to the mean-square displacement in the unperturbed system. Both measures confirm the validity of Kubo's formula, provided that the granular temperature is used as the proportionality factor between response and autocorrelation, at least for not too large inelasticities.