Fluctuation-dissipation relations for motions of center of mass in driven granular fluids under gravity

We investigated the validity of fluctuation-dissipation relations in the nonequilibrium stationary state of fluidized granular media under gravity by two independent approaches, based on theory and numerical simulations. A phenomenological Langevin-type theory describing the fluctuation of center of mass height, which was originally constructed for a one-dimensional granular gas on a vibrating bottom plate, was generalized to any dimensionality, even for the case in which the vibrating bottom plate is replaced by a thermal wall. The theory predicts a fluctuation-dissipation relation known to be satisfied at equilibrium, with a modification that replaces the equilibrium temperature by an effective temperature defined by the center of mass kinetic energy. To test the validity of the fluctuation-dissipation relation, we performed extensive and accurate event-driven molecular dynamics simulations for the model system with a thermal wall at the bottom. The power spectrum and response function of the center of mass height were measured and closely compared with theoretical predictions. It is shown that the fluctuation-dissipation relation for the granular system is satisfied, especially in the high-frequency (short time) region, for a wide range of system parameters. Finally, we describe the relationship between systematic deviations in the low-frequency (long time) region and the time scales of the driven granular system.

Figure 1

Top: Snapshots of the two-dimensional simulation with $N=5000$, $L=100$, and $g={10}^{-3}$ for different values of the restitution coefficient (a) $r=0.999$ and (b) $r=0.992$. Bottom: Area-fraction profiles averaged over a long time period for $r=0.999$ and $r=0.992$.

Figure 2

Kinetic energy per particle $\overline{K}$ (circles) and kinetic energy of the COM ${\overline{K}}_{\text{COM}z}$ (squares), plotted vs $\Lambda =\frac{\sqrt{\pi }}{2}50{(1-r^2)}^{1/2}$ for $r=0.9999,0.9996,0.999,0.998,0.996,0.994,0.992$, and $0.99$ from left to right. The solid line gives a numerical fit of the form $1.04\times {\Lambda }^{-1.48}$.

Figure 3

The average height of the center of mass $\overline{Z}$ vs $\overline{K}$ for $r=0.999,0.998,0.996,0.994,0.992$, and $0.99$ from right to left. The error bars are smaller than the size of the marks. The solid line gives a linear fit with the slope ${\left(mg\right)}^{-1}$, where $m=1$ and $g={10}^{-3}$.

Figure 4

Kinetic energy per particle $\overline{K}$ as a function of $g$. The error bars are smaller than the sizes of the marks.

Figure 5

Probability distribution of the scaled COM velocity $C\equiv V_z/{(2{\overline{K}}_{\text{COM}z}/M)}^{1/2}$. The solid line is Gaussian with unity dispersion.

Figure 6

Top: Power spectrum for the COM height vs angular frequency $\omega$. Averages were taken over 400 realizations. Bottom: Scaled power spectrum for the COM height. The solid line depicts the theoretical prediction given in Eq. (15) with $\widehat{\mu }=0.50$ and $\widehat{\Omega }=1.7$.

Figure 7

Power spectrum (PS) of $\delta K\left(t\right)/mg$ and of $\delta Z\left(t\right)$, $S\left(\omega \right)$ for (a) $r=0.999$ and (b) $r=0.992$.

Figure 8

The real part ${\chi }^{\prime }$ (top) and the imaginary part ${\chi }^{\prime \prime }$ (bottom) of the frequency response function vs angular frequency $\omega$ for $r=0.992$. Circles show the data for $\Delta g/g={10}^{-1}$ using Eq. (21) with averages obtained over 800 realizations. Squares are for $\Delta g/g={10}^{-2}$ without using Eq. (21); averages were obtained over $41000$ realizations.

Figure 9

Real (top) and imaginary (bottom) parts of the scaled frequency response function vs the scaled angular frequency $\widehat{\omega }=\omega c/g$. Averages were taken over 800 realizations. The thick lines are the theoretical prediction Eqs. (16) and (17) with fitting parameters $\widehat{\mu }=0.50$, $\widehat{\Omega }=1.7$.

Figure 10

Left-hand side $\omega S\left(\omega \right)/2k_B{T}_{\text{eff}}$ and right-hand side ${\chi }^{\prime \prime }\left(\omega \right)$ of Eq. (13) for $r=1$. $N=5000$, $L=100$, and $g={10}^{-2}$, with averages over 400 realizations for $S\left(\omega \right)$ and over 800 realizations for ${\chi }^{\prime \prime }\left(\omega \right)$.

Figure 11

Left-hand side $\omega S\left(\omega \right)/2k_B{T}_{\text{eff}}$ and right-hand side ${\chi }^{\prime \prime }\left(\omega \right)$ of Eq. (13) for (a) $r=0.999$, (b) $r=0.998$, (c) $r=0.996$, (d) $r=0.994$, (e) $r=0.992$, and (f) $r=0.99$. $N=5000$, $L=100$, and $g={10}^{-3}$, with averages over 400 realizations for $S\left(\omega \right)$ and over 800 realizations for ${\chi }^{\prime \prime }\left(\omega \right)$.

Figure 12

(a) Schematic of the system observed in the laboratory frame of reference, (b) the same system observed in the center of mass frame, and (c) the Rayleigh piston: a piston that undergoes random collisions with a one-dimensional heat bath of particles.