Frequency-dependent fluctuation-dissipation relations in granular gases

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.

Figure 1

Collision of two particles in the rest frame of particle $i$ before the collision.

Figure 2

(Color online) FD temperature in the Maxwell model and in the mean-field granular gas model. In the former, the temperature obtained from the Green-Kubo relations is frequency independent and equal to the granular temperature. In the latter, the temperature obtained depends on the measurement frequency. The schematic curve interpolates between the high- and low-$\omega$ limits. The difference between the two curves is not to scale.