Memory effect in uniformly heated granular gases

We evidence a Kovacs-like memory effect in a uniformly driven granular gas. A system of inelastic hard particles, in the low density limit, can reach a nonequilibrium steady state when properly forced. By following a certain protocol for the drive time dependence, we prepare the gas in a state where the granular temperature coincides with its long time value. The temperature subsequently does not remain constant but exhibits a nonmonotonic evolution with either a maximum or a minimum, depending on the dissipation and on the protocol. We present a theoretical analysis of this memory effect at Boltzmann-Fokker-Planck equation level and show that when dissipation exceeds a threshold, the response can be called anomalous. We find excellent agreement between the analytical predictions and direct Monte Carlo simulations.

Figure 1

Sketch of the drive time dependence for the cooling and heating protocols. The resulting normal temperature evolution is depicted. The system is first in a nonequilibrium steady state at temperature $T_s({\xi }_0$) under a drive ${\xi }_0$. $T\left(t_w\right)$ coincides with $T_s\left(\xi \right)$. (a) Cooling protocol: The driving ${\xi }_1$ in the waiting time window $0<t<t_w$ is smaller than its initial value ${\xi }_0$, and the granular temperature will display a maximum before returning to its steady value for $t>t_w$. (b) Heating protocol: We have ${\xi }_1>{\xi }_0$, and the granular temperature will display a minimum for $t>t_w$.

Figure 2

Plot of $a_2^{\mathrm{HCS}}/a_2^{\mathrm{s}}$ as a function of the restitution coefficient $\alpha$ for a system of inelastic hard disks ($d=2$), following from the accurate expressions obtained in [12]. The top and bottom insets show the excess kurtosis for the steady state $a_2^{\mathrm{s}}$ and the parameter $B$ as functions of $\alpha$, as given by Eqs. (12) and (17), respectively.

Figure 3

Plot of the Kovacs hump for $\alpha =0$ (top) and $\alpha =0.3$ (bottom). The simulation curves (points) have been averaged over ${10}^5$ trajectories, and they are compared to (i) the raw theoretical curve (34), evaluated with the theoretical expressions for the parameters $a_2^{\mathrm{s}}$, $B$, and $a_2^{\mathrm{HCS}}$ (dashed line) and (ii) the improved theory obtained by inserting into (34) the value of the $B$ parameter given by the Monte Carlo simulation (solid line). The second route improves the agreement between theory and simulation. The specific values of the parameters for each of the plotted curves are given in Table 1. Note the smallness of $\beta -1$, which is of the order of ${10}^{-3}$ in both cases.

Figure 4

Decay of the excess kurtosis from its initial value to its steady state value. Plotted is the simulation curve obtained by direct simulation Monte Carlo (DSMC) (points) for $\alpha =0.3$. The long time limit is very close to its predicted value $a_2^{\mathrm{s}}=0.00638$, following from Eq. (12) and shown by the dashed line. In the inset, the same decay is shown, but on a logarithmic scale (points). From the linear slope, we directly measure the parameter $B$ to be inserted into the theoretical expression for the Kovacs hump, Eq. (34). The obtained values are given in Table 1.

Figure 5

Plot of the Kovacs hump for $\alpha =0.8$. The meaning of the different symbols and lines is the same as in Fig. 3. Note that the sign of $\beta -1$ is reversed, $\beta -1<0$, as the restitution coefficient $\alpha >{\alpha }_c\simeq 0.707$.

Figure 6

Evolution of excess kurtosis ratio, $A_2\left({\tau }_w\right)\equiv a_2\left({\tau }_w\right)/a_2^{\mathrm{s}}$, as a function of waiting time within the cooling protocol at $\alpha =0.3$. From bottom to top, the curves correspond to $T_s\left({\xi }_0\right)/T_s\left({\xi }_1\right)=2,4,9,25$, and 200. The upper dashed curve is for the limit $T_s\left({\xi }_1\right)/T_s\left({\xi }_0\right)\to 0$. Note that $A_2\left({\tau }_w\right)$ defines the quantity $A_2^{\mathrm{ini}}$ used throughout. For a given value of $\alpha$, the maximum possible $A_2$ is $a_2^{\mathrm{HCS}}/a_2^s$. For $\alpha =0.3$, Fig. 2 indicates that this ratio is close to $2.33$, which is consistent with the maximum of the dashed curve.

Figure 7

Same as Fig. 6, but for the heating protocol. Here, from top to bottom, $T_s\left({\xi }_1\right)/T_s\left({\xi }_0\right)=2,4,9,25$. The lower dashed curve is for $T_s\left({\xi }_1\right)/T_s\left({\xi }_0\right)\to \infty$.

Figure 8

Excess kurtosis ratio as a function of waiting time (cooling protocol) for different dissipations and $T_s\left({\xi }_1\right)/T_s\left({\xi }_0\right)=1/25$.

Figure 9

Phase diagram of the Kovacs hump. The line ${\xi }_1/{\xi }_0=1$ (solid) separates the cooling (${\xi }_1<{\xi }_0$) and the heating (${\xi }_1>{\xi }_0$) protocols. The dashed line $\alpha ={\alpha }_c=1/\sqrt{2}$ separates systems with high inelasticity ($\alpha <{\alpha }_c$) from those with low inelasticity ($\alpha >{\alpha }_c$). Note that the plots are for the granular temperature $T$; a maximum in $T$ corresponds to a minimum in the $\beta$ variable defined in Eq. (14).

Figure 10

Tendency to the universal reference state for very long times. We show a zoom of the long time behavior ($\tau -{\tau }_w\ge 1$) of the decay of the excess kurtosis to its steady value, $|a_2-a_2^{\mathrm{s}}|\le 2\times {10}^{-5}$. The overall picture is that of Fig. 4, which also corresponds to $\alpha =0.3$, for which $a_2^{\mathrm{ini}}-a_2^{\mathrm{s}}\simeq 0.086$. Plotted here is the excess kurtosis decay obtained from (i) the numerical integration of Eq. (23) with initial conditions (25) (solid line) and (ii) the asymptotic behavior given by Eqs. (41) and (39) (dashed line).