Kinetic glass transition in granular gases and nonlinear molecular fluids

In this paper, we investigate, both analytically and numerically, the emergence of a kinetic glass transition in two different model systems: a uniformly heated granular gas and a molecular fluid with nonlinear drag. Despite the profound differences between these two physical systems, their behavior in thermal cycles share strong similarities, which stem from the relaxation time diverging algebraically at low temperatures for both systems. When the driving intensity—-for the granular gas—or the bath temperature—for the molecular fluid—is decreased to sufficiently low values, the kinetic temperature of both systems becomes “frozen” at a value that depends on the cooling rate through a power law with the same exponent. Interestingly, this frozen glassy state is universal in the following sense: for a suitable rescaling of the relevant variables, its velocity distribution function becomes independent of the cooling rate. Upon reheating, i.e., when either the driving intensity or the bath temperature is increased from this frozen state, hysteresis cycles arise and the apparent heat capacity displays a maximum. The numerical results obtained from the simulations are well described by a perturbative approach.

Figure 1

Left: Dynamical evolution of the granular temperature $\theta$ as a function of the bath temperature ${\theta }_{\text{s}}$. Symbols correspond to DSMC data for the linear cooling protocol (15) with different cooling rates $r_c$, namely, $r_c=0.05$ (red squares), 0.025 (green up triangles), 0.01 (orange circles), 0.005 (blue down triangles), 0.0025 (black rectangles), and 0.001 (purple diamonds) for two values of the restitution coefficient: $\alpha =0.9$ (empty symbols) and $\alpha =0.3$ (filled symbols). The dashed line corresponds to the instantaneous NESS curve $\theta ={\theta }_{\text{s}}$. Right: Limit values of the kinetic temperature at the frozen state ${\theta }^{\text{Frz}}$ as a function of $r_c$. The plotted points have been extracted from the DSMC data on the left panel. The lines correspond to the best fits to the function ${\theta }^{\text{Frz}}=a{r}_c^b$, with $a=0.741$ and $b=0.666$ for $\alpha =0.9$ (dashed), and $a=0.781$ and $b=0.666$ for $\alpha =0.3$ (solid), both in excellent agreement with the theoretical prediction (25a). We have considered a granular gas in the three-dimensional case $d=3$. The same parameter values are employed in the remainder of the numerical simulations for the granular gas.

Figure 2

Universality of the frozen state. Left: VDF at the frozen state for different values of the cooling rate $r_c$. The color code and symbols are the same as in Fig. 1. For each value of $\alpha$, its corresponding VDFs are superimposed over a unique, universal, curve independent of $r_c$, in agreement with our theoretical prediction. In the inset, we show the VDF at the frozen state divided by the equilibrium Maxwellian, with the lines corresponding to the polynomials in Eq. (6) within the first Sonine approximation for $\alpha =0.9$ (blue dashed) and $\alpha =0.3$ (red solid), respectively. Right: Excess kurtosis at the frozen state $a_2^{\text{Frz}}$ as a function of the restitution coefficient $\alpha$. Here, for the sake of clarity, we show DSMC data corresponding to only two values of the cooling rate, $r_c=0.01$ (squares) and $r_c=0.001$ (circles). The numerical values of $a_2^{\text{Frz}}$ are compared with both the NESS value $a_2^{\text{s}}$ (blue dashed line) and the HCS value $a_2^{\text{HCS}}$ (red solid line), being very close to the latter.

Figure 3

Scaled granular temperature $Y$ as a function of the scaled bath temperature $X$. We have employed the linear cooling protocol (15) with different cooling rates and two values of the restitution coefficient $\alpha$. The color codes and symbols for the DSMC data are the same as those employed in the left panel of Fig. 2. The dashed, purple vertical line marks the fictive temperature $X_f={\theta }_f/r_c^{2/3}=1$ from Eq. (21) for $c=3/2$.

Figure 4

Hysteresis cycles in the granular gas. The system is first cooled with rate $r_c$ and later reheated from the frozen state with rate $r_h=r_c=r$. Top blue (bottom red) symbols and lines correspond to the cooling (heating) protocol, as explicitly stated in both panels. Left: Scaled kinetic temperature $Y=r^{-2/3}\theta$ as a function of the scaled bath temperature $X=r^{-2/3}{\theta }_{\text{s}}$. Specifically, we present results for $r=0.01$ (squares) and $r=0.001$ (diamonds), and for two values of $\alpha$: 0.9 (open symbols) and 0.3 (filled symbols). Symbols are simulation results of the Boltzmann equation (3), while the solid curves correspond to the numerical integration of Eqs. (27) and (32) for cooling and heating, respectively. The dashed straight line corresponds to the instantaneous NESS curve $Y=X$. The purple vertical line marks the bath temperature $X_g$ at which the heat capacity reaches its maximum in the reheating program—see right panel. Both the solid curves and the purple vertical line were obtained for $\alpha =0.9$, as they superimpose with the ones corresponding to the $\alpha =0.3$ case. Right: Associated apparent heat capacity $C=d\theta /d{\theta }_{\text{s}}=dY/dX$. Again, the symbols have been obtained from the simulation data, and the lines correspond to the numerical integration of Eqs. (27) and (32). Note the logarithmic scale used for the horizontal axis on both panels.

Figure 5

Hysteresis cycles for reheating with rate $r_h$ from the frozen states corresponding to different cooling rates $r_c$. All reheating curves correspond to $r_h=0.01$, and the different cooling rates employed are symbols $r_c$ = red squares, 0.05; orange circles, 0.01; blue diamonds, 0.005; and purple triangles, 0.001. Symbols correspond to DSMC simulation data, whereas the solid line corresponds to the perturbative expression for the normal curve in Eq. (37).

Figure 6

Hysteresis cycles for reheating with different rates $r_h$ from the common frozen state corresponding to a given value of $r_c$. Specifically, the plotted DSMC simulation data correspond to $r_c=0.01$ with different reheating rates: symbols, $r_h$ = red squares, 0.05; orange circles, 0.1; blue diamonds, 0.005; and purple triangles, 0.001. The solid curves correspond to Eq. (37) for each value of $r_h$, from left to right $r_h$ increases from 0.001 to 0.05.

Figure 7

Scaled glass transition temperature $X_g=r_h^{-2/3}{\theta }_g$ for the granular gas as a function of the ratio $r_c/r_h$. Only plotted is the theoretical prediction for $X_g$, obtained via the numerical integration of the evolution equations for the scaled variables, in a hysteresis cycle characterized by a cooling with rate $r_c$ followed by reheating with rate $r_h$. The dashed horizontal line marks the value for $r_c=r_h$, which corresponds to the vertical line in both panels of Fig. 4.

Figure 8

Plot of the dimensionless VDF at the frozen state for the nonlinear fluid. Symbols correspond to the numerical integration of the Langevin equation with $N={10}^5$ stochastic trajectories for different cooling rates: $r_c=0.005$ (purple diamonds), 0.01 (blue triangles), 0.05 (orange circles), and 0.1 (red squares). The dashed curve corresponds to the equilibrium Maxwellian, whereas the dotted vertical line marks the position of the LLNES Dirac-delta peak, as given by Eq. (49).

Figure 9

Hysteresis cycle in the nonlinear molecular fluid. Both panels are the transposition to the case of the nonlinear molecular fluid of those in Fig. 4 for the granular gas, with the same values of the cooling and heating rate $r$ in dimensionless variables. For the molecular fluid, the numerical data (symbols) corresponds to the simulation of the Langevin equation (44), while the theoretical curves (solid lines) correspond to the numerical integration of Eqs. (52) and (55). On the left panel, the vertical line marks the bath temperature $X_g$ at which the heat capacity reaches its maximum in the reheating program—see right panel. Therein, an additional inset shows the anomalous behavior of the apparent heat capacity in the cooling process, which is discussed in the text.

Figure 10

Hysteresis cycles for the nonlinear fluid upon reheating with rate $r_h$ from the frozen states corresponding to different cooling rates $r_c$. All simulation data corresponds to $r_h=0.01$, whereas the cooling rates are symbols, $r_c$ = red squares, 0.05; orange circles, 0.01; blue triangles, 0.005; and purple diamonds, 0.001. The solid curve corresponds to the perturbative expression (57a) for the normal curve.

Figure 11

Hysteresis cycles for the nonlinear molecular fluid upon reheating with different rates $r_h$ after a common cooling protocol with rate $r_c$. Specifically, the plotted simulation data correspond to $r_c=0.01$ with different reheating rates: symbols, $r_h$ = red squares, 0.05;, orange circles, 0.01; blue triangles, 0.005; and purple diamonds, 0.001. The solid curves correspond to Eq. (57a) for each value of $r_h$, from left to right $r_h$ increases from 0.001 to 0.05.

Figure 12

Scaled glass transition temperature $X_g=r_h^{-2/3}{\theta }_g$ for the molecular fluid as a function of the ratio $r_c/r_h$. Similarly to Fig. 7 for the granular gas, we only plot our theoretical prediction for $X_g$, stemming from the numerical integration of the evolution equations in scaled variables, and the dashed horizontal line marks the value for $r_c=r_h$.

Figure 13

Excess kurtosis $a_2$ as a function of the scaled bath temperature $X$ for the family of cooling programs (A1) with different values of $k$. The solid lines correspond to the numerical integration of the evolution equations in the first Sonine approximation for the values of $k$ depicted in the legend—i.e., from top to bottom $k$ increases from $-5$ to 1. Left: behavior found in a granular gas. Right: Behavior found in a molecular fluid. In both cases, a dimensionless cooling rate $r_c=0.1$ has been employed. On the left (right) panel, (i) the dotted line represents the value of the excess kurtosis at the NESS $a_2^{\text{s}}$ (equilibrium value of the excess kurtosis $a_2^{\text{eq}}=0$), from which all the curves depart for large $X$, and (ii) the dashed line accounts for the value at the HCS (LLNES) within the first Sonine approximation.