Fluctuations of Internal Energy Flow in a Vibrated Granular Gas

The nonequilibrium fluctuations of power flux in a fluidized granular media have been recently measured in an experiment [Phys. Rev. Lett. 92, 164301 (2004)], which was announced to be a verification of the fluctuation relation (FR) by Gallavotti and Cohen. An effective temperature was also identified and proposed to be a useful probe for such nonequilibrium systems. We explain these results in terms of a two-temperature Poisson process. Within this model, supported by independent molecular dynamics simulations, power flux fluctuations do not satisfy the FR and the nature of the effective temperature is clarified. In the pursuit of a hypothetical global quantity fulfilling the FR, this points to the need of considering candidates other than the power flux.

Figure 1

Left: Snapshot of the system considered for MD simulations, with the inner region marked by the solid rectangle. Right: Corresponding vertical profiles of density [$\Phi (y)$, dashed line] and temperature [$T(y)$, solid line]. The dotted lines mark the bottom and top boundaries of the inner region. Here $N=270$ and $\alpha =0.9$. The mean free path is $\sim 5.7d$.

Figure 2

(a) PDFs of injected power $f_q(q_{\tau })$ from MD simulations for different values of $\tau =(1,2,4,8,16,32)\times {\tau }_{\min }$ with ${\tau }_{\min }=0.015{\tau }_{\mathrm{box}}$. Here $N=270$ and $\alpha =0.9$. The distributions are shifted vertically for clarity. The dashed lines put in evidence the exponential tails of the PDF at $\tau ={\tau }_{\min }$. (b) plot of $(1/\tau )\log [f_q(q_{\tau })/f_q(-q_{\tau })]$ vs $q_{\tau }$ from MD simulations (same parameters as above) at large values of $\tau$. The solid curve is a linear fit (with slope ${\beta }_{\mathrm{eff}}$) of the data at $\tau =128{\tau }_{\min }$. The dashed line has a slope ${\beta }_{\mathrm{gran}}=1/T_{\mathrm{gran}}$. In the inset, the same graph is shown for small values of $\tau =(1,2,4,8)\times {\tau }_{\min }$ (from bottom to top).

Figure 3

PDF of transverse ($xy$) injected power $f_q(q_{\tau })$ for low values of $\tau =(1,2,4,8)\times {\tau }_{\min }$ with ${\tau }_{\min }=0.00015{\tau }_{\mathrm{box}}$ (resp., circle, squares, diamond, and triangles) from MD simulations with $N=270$ and $\alpha =0.9$. The distributions are shifted vertically for clarity. The (red) solid lines show the solution of the “two-temperatures” model. The dashed lines indicate exponential tails $\sim \exp (\mp {\beta }_{\pm }\tau q_{\tau })$. Left inset: same plot for a large value $\tau =6400{\tau }_{\min }$. Right inset: plot of $(1/\tau )\log [f_q(Q_{\tau })/f_q(-Q_{\tau })]$ vs $q=Q/\tau$ from MD simulations (same parameters as above) at large values of $\tau =(1,2,4,8)\times {\tau }_{\min }^{\prime }$ with ${\tau }_{\min }^{\prime }=3200{\tau }_{\min }$, together with a dashed line of slope ${\beta }_{\mathrm{eff}}$ predicted by Eq. (4) and the dotted line of slope ${\beta }_{\mathrm{gran}}=1/T_{\mathrm{gran}}$.