Frequency-dependent fluctuation-dissipation relations in granular gases

TL;DR

This paper investigates the Green-Kubo relation in granular gases, showing that the effective temperature derived from it is frequency-independent and equal to the granular temperature in specific models, but overall the relation is violated.

Contribution

It demonstrates the frequency dependence of the Green-Kubo relation in granular gases and clarifies conditions under which the effective temperature matches the granular temperature.

Findings

Effective temperature is frequency independent in the Maxwell model.

In the mean-field model, the high-frequency effective temperature equals the granular temperature.

The Green-Kubo relation is generally violated in granular gases.

Abstract

The Green-Kubo relation for two models of granular gases is discussed. In the Maxwell model in any dimension, the effective temperature obtained from the Green-Kubo relation is shown to be frequency independent, and equal to the average kinetic energy, known as the granular temperature. In the second model analyzed, a mean-field granular gas, the collision rate of a particle is taken to be proportional to its velocity. The Green-Kubo relation in the high frequency limit is calculated for this model, and the effective temperature in this limit is shown to be equal to the granular temperature. This result, taken together with previous results, showing a difference between the effective temperature at zero frequency (the Einstein relation) and the granular temperature, shows that the Green-Kubo relation for granular gases is violated.