Memory effects in the relaxation of a confined granular gas

The accuracy of a model to describe the horizontal dynamics of a confined quasi-two-dimensional system of inelastic hard spheres is discussed by comparing its predictions for the relaxation of the temperature in a homogenous system with molecular dynamics simulation results for the original system. A reasonably good agreement is found. Next the model is used to investigate the peculiarities of the nonlinear evolution of the temperature when the parameter controlling the energy injection is instantaneously changed while the system was relaxing. This can be considered as a nonequilibrium generalization of the Kovacs effect. It is shown that, in the low-density limit, the effect can be accurately described by using a simple kinetic theory based on the first Sonine approximation for the one-particle distribution function. Some possible experimental implications are indicated.

Figure 1

Noise strength ${\xi }_0$ of the stochastic thermostat model for the two-dimensional dynamics as a function of the velocity $v_b$ of the vibrating wall in the three-dimensional system, obtained in the way discussed in the main text. Dimensionless quantities have been defined as indicated in the respective labels, with $T\left(0\right)$ being the initial temperature of the (three-dimensional) gas. The coefficient of normal restitution is $\alpha =0.8$ and the density of the gas $n=0.02{\sigma }^{-3}$.

Figure 2

Characteristic speed $\Delta$ of the collisional model for the two-dimensional dynamics as a function of the velocity $v_b$ of the vibrating wall in the three-dimensional system, obtained in the way discussed in the main text. Dimensionless quantities have been defined as indicated in the respective labels, with $T\left(0\right)$ being the initial temperature of the (three-dimensional) gas. The coefficient of normal restitution is $\alpha =0.8$ and the density of the gas $n=0.02{\sigma }^{-3}$. The straight dashed line is just a guide for the eye.

Figure 3

Relaxation of the temperature in a system with a coefficient of normal restitution $\alpha =0.8$, height $h=1.5\sigma$, and three-dimensional density $n=0.02{\sigma }^{-3}$. The velocity of the vibrating wall is $v_b={10}^{-3}{\left(T\left(0\right)/m\right)}^{1/2}$. Both temperature and time are measured in the dimensionless units indicated in the labels. The solid line is the result from MD simulations, while the dot-dashed (blue) line and the (red) dashed line are the theoretical predictions from the stochastic and collisional models, respectively. The used values of ${\xi }_0$ and $\Delta$ are from Figs. 1 and 2.

Figure 4

The same as described in the legend of Fig. 3 but now for a system with a vibrating wall with a velocity $v_b={10}^{-2}{\left(T\left(0\right)/m\right)}^{1/2}$.

Figure 5

MD simulation results for the Sonine coefficient $a_{2,\mathrm{st}}$ of the steady velocity distribution function in the horizontal plane of the quasi-two-dimensional granular gas as a function of the coefficient or normal restitution $\alpha$. The velocity of the vibrating wall is $v_b=5\times {10}^{-3}{(T\left(0\right)/m)}^{1/2}$, the density of the three-dimensional gas is $n=0.02{\sigma }^{-3}$, and the distance between the two confining walls is $h=1.5\sigma$.

Figure 6

MD simulation results for the Sonine coefficient $a_{2,\mathrm{st}}$ of the steady velocity distribution function of the quasi-two-dimensional granular gas as a function of the dimensionless velocity of the vibrating wall. The coefficient of normal restitution is $\alpha =0.8$, the density of the three-dimensional gas is $n=0.02{\sigma }^{-3}$, and the distance between the two confining walls is $h=1.5\sigma$.

Figure 7

Logarithmic plot of the MD simulation values (black circles) of the marginal one-particle velocity distribution ${\phi }_x\left(c_x\right)$ defined by Eq. (18). The (black) dashed line is the Gaussian and the (red) solid line is the Sonine approximation as given by Eq. (20), with the value of $a_{2,\mathrm{st}}$ measured also in the simulations. The parameters of the system are the same as described in the legend to Fig. 6, and $v_b={10}^{-3}{(T\left(0\right)/m)}^{1/2}$.

Figure 8

Sketch of the protocol followed in the Kovacs-like experiment described in the main text. Initially, the system is in the steady state corresponding to a value ${\Delta }_1$ of the characteristic speed parameter, with $T_1$ as its temperature. Then, at $t=0$, the speed parameter is suddently changed to ${\Delta }_2<{\Delta }_1$, and the system relaxes towards a new steady temperature $T_2<T_1$. At $t=t_0$, when the temperature of the system is $T\left(t_0\right)$, the characteristic speed is again suddently changed, now to a value ${\Delta }_0$ such that $T\left(t_0\right)$ is the steady temperature for it. The solid line describes the time evolution of the temperature of the system, assuming that it exhibits a negative hump before tending again towards its steady value.

Figure 9

Hydrodynamic Sonine coefficient ${\overline{a}}_2$, as a function of the dimensionless characteristic speed ${\Delta }^{*}$ for a system with a coefficient of normal restitution $\alpha =0.8$. Shown in the figure are the steady values of $a_2$ and ${\Delta }^{*},a_{2,\mathrm{st}}$ and ${\Delta }_{\mathrm{st}}^{*}$, respectively. Also plotted is the value ${\Delta }_b^{*}$, defined by Eq. (38). The arrows on the curve indicate the way in which it is described as the system approaches its steady state. For constant $\Delta$, interval I corresponds to cooling and intervals II and III to heating. If the value of $a_2$ at the moment $t=t_0$ in which $\Delta$ is modified so ${\Delta }^{*}\left(t_0\right)={\Delta }_{\mathrm{st}}^{*}$, lies in regions I or II, the system keeps on cooling and heating, respectively. On the other hand, if it lies in region III the system instantaneously changes from heating to cooling (rebound effect).

Figure 10

The Kovacs effect in a system under cooling. Temperature $T^{*}$ and time $s$ are measured in the dimensionless units indicated in the main text. The characteristic speeds verify ${\Delta }_2=0.5{\Delta }_0$, where ${\Delta }_2$ is the speed in the former (no plotted) evolution, and ${\Delta }_0$ is the speed in the final relaxation starting at the steady temperature. The dashed line is the theoretical prediction given by the solution of Eqs. (A1) and (A2), while the solid line has been obtained by the DSMC method.

Figure 11

The Kovacs effect without rebound in a system under heating. Temperature $T^{*}$ and time $s$ are measured in the dimensionless units indicated in the main text. The characteristic speeds verify ${\Delta }_2=1.2{\Delta }_0$, where ${\Delta }_2$ is the speed in the former evolution, and ${\Delta }_0$ is the speed in the final relaxation starting at its steady temperature. The dashed curve is the solution of the evolution equations given in the Appendix.

Figure 12

The Kovacs effect with rebound in a system under heating. Temperature $T^{*}$ and time $s$ are measured in the dimensionless units indicated in the main text. The characteristic speeds verify ${\Delta }_2=2{\Delta }_0$, where ${\Delta }_2$ is the speed in the former evolution, and ${\Delta }_0$ is the speed in the final relaxation starting at its steady temperature. The dashed line is the theoretical prediction given by the solution of Eqs. (A1) and (A2), while the solid line has been obtained by the DSMC method. Note that the temperature of the system was increasing for $s<0$.

Figure 13

Kurtosis in the hydrodynamic regime, ${\tilde{a}}_2$, as a function of the dimensionless parameter, $\beta$. The left figure is for $\alpha =0.95$ and the right one for $\alpha =0.5$. The stationary value of the kurtosis, $a_{2,st}$, and of the dimensionless parameter, ${\beta }_s$, is also shown.