Linear response in the uniformly heated granular gas

We analyze the linear response properties of the uniformly heated granular gas. The intensity of the stochastic driving fixes the value of the granular temperature in the nonequilibrium steady state reached by the system. Here, we investigate two specific situations. First, we look into the “direct” relaxation of the system after a single (small) jump of the driving intensity. This study is carried out by two different methods. Not only do we linearize the evolution equations around the steady state, but we also derive generalized out-of-equilibrium fluctuation-dissipation relations for the relevant response functions. Second, we investigate the behavior of the system in a more complex experiment, specifically a Kovacs-like protocol with two jumps in the driving. The emergence of an anomalous Kovacs response is explained in terms of the properties of the direct relaxation function: it is the second mode changing sign at the critical value of the inelasticity that demarcates anomalous from normal behavior. The analytical results are compared with numerical simulations of the kinetic equation, and a good agreement is found.

Figure 1

Coefficients $a_{\pm }$ as a function of the restitution coefficient $\alpha$. Both the three-dimensional (solid line) and the two-dimensional (dashed line) cases are plotted. The coefficient $a_{+}$ is always positive, whereas $a_{-}$ changes sign at the critical value $\alpha ={\alpha }_c$—as given by Eq. (29). Note that $|a_{-}|$ is quite small throughout, with its least small value at $\alpha =0$, and thus $a_{+}=1-a_{-}$ is always very close to unity.

Figure 2

Relaxation function of the granular temperature as a function of the scaled time $\tau$. Panels (a)–(c) display ${\varphi }_{\theta }\left(\tau \right)$ for three different values of the restitution coefficient, namely $\alpha =0.3$, 0.7, and 0.9, respectively—symbols correspond to simulation results in a system of ${10}^6$ particles averaged over 10 runs, and lines to the analytical prediction (27). The three simulation curves are plotted together in panel (d), where it is clearly seen that they superimpose. Therefore, the linear relaxation of the granular temperature to the corresponding NESS is “universal” in the $\tau$ scale.

Figure 3

Relaxation rates ${\lambda }_{\pm }$ as a function of the restitution coefficient. Again, both $d=3$ (solid line) and $d=2$ (dashed) are plotted. The largest rate ${\lambda }_{+}$ is always close to $-3$ throughout. The smaller rate ${\lambda }_{-}$ has a stronger dependence on $\alpha$, diverging in the elastic limit $\alpha \to 1^{-}$.

Figure 4

Check of the FDR for the relaxation function of the granular temperature. The left panel shows the correlation function $R_2\left(\tau \right)$ for three different values of the restitution coefficient $\alpha$: from top to bottom, $\alpha =0.3$ (squares), 0.7 (circles), and 0.9 (triangles). Also shown is the theoretical prediction for the direct relaxation function of the granular temperature ${\varphi }_{\theta }\left(\tau \right)$ (line). As discussed in the text, the decay of ${\varphi }_{\theta }$ is well described by $R_2$, although the initial values do not exactly match. This is illustrated in the right panel, where we plot $R_2\left(\tau \right)/R_2\left(0\right)$ for the specific case $\alpha =0.3$. Therein, $R_2\left(\tau \right)/R_2\left(0\right)$ is compared with both ${\varphi }_{\theta }\left(\tau \right)$ (solid line) and the diagonal part of the correlation $R_2^{\text{dia}}\left(\tau \right)$ (dashed line). It is clearly observed that the diagonal part equals unity for $\tau =0$, but it decays faster than the whole correlation $R_2$: roughly, its relaxation time is divided by 2.

Figure 5

Relaxation function of the scaled excess kurtosis $A_2=a_2/a_2^{\text{s}}$ as a function of $\tau$ for $\alpha =0.3$. In the simulations, the driving is instantaneously changed from ${(\xi +\delta \xi )}^2=0.25$ (squares) and ${(\xi +\delta \xi )}^2=0.35$ (solid circles) to ${\xi }^2=0.2$ at $\tau =0$. Despite the jumps being quite large, the agreement with the theoretical prediction, as given by Eq. (35) (line), is remarkably good.

Figure 6

Relaxation function of the granular temperature for the same situation considered in Fig. 5. Note the logarithmic scale on the vertical axis. Although the jumps in the driving are quite large—relative size of 10–$30\%$—the linear response prediction (solid line) fits perfectly well the simulation results (symbols). The coding for the different lines is the same as in Fig. 5.

Figure 7

Check of the FDR relation for the relaxation function of the fourth-moment. The squares represent the correlation function $R_4\left(\tau \right)$ for $\alpha =0.3$, measured in DSMC simulations, while the solid line is the theoretical prediction given by Eq. (38). The decay of ${\varphi }_{\langle c^4\rangle }$ is well described by $R_4$, although their initial values do not exactly match. On the other hand, the diagonal part of the correlation $R_4^{\text{dia}}\left(\tau \right)$ (dashed line) correctly gives the initial value ${\varphi }_{\langle c^4\rangle }\left(0\right)$, but it decays faster than the whole correlation $R_4$.

Figure 8

Qualitative plot of the Kovacs protocol. The system starts from a NESS with granular temperature $T_0=T_{\text{s}}+\delta T$, corresponding to a driving ${\xi }_0=\xi +\delta \xi$. In the waiting time window, the driving is suddenly decreased to ${\xi }_1=\xi -\delta {\xi }^{\prime }<\xi$ and the granular temperature approaches the corresponding steady-state value $T_1=T_{\text{s}}-\delta T^{\prime }<T_{\text{s}}$. At the waiting time, the granular temperature equals its steady value for the driving $\xi$. However, for longer times the granular temperature does not remain flat: it displays a nonmonotonic behavior, the Kovacs hump. The normal case is depicted, in which the temperature displays a maximum, but anomalous behavior—a minimum instead of a maximum—can also be observed. Note that $\delta T$ and $\delta T^{\prime }$ are defined in such a way that they are both positive.

Figure 9

Kovacs hump for large inelasticity. Specifically, the restitution coefficient is $\alpha =0.3$. The circles are simulation results obtained with $N={10}^6$ particles averaged over ${10}^4$ runs, while the solid line is our theoretical prediction (67). The Kovacs response is anomalous for this large inelasticity, $\alpha <{\alpha }_c$.

Figure 10

Kovacs hump for small inelasticity. Specifically, the restitution coefficient is $\alpha =0.8$. The response is normal in this case, $\alpha >{\alpha }_c$. The circles are simulation results obtained with $N={10}^6$ particles averaged over $5\times {10}^4$ runs, whereas the solid line is again Eq. (67).