Large Mpemba-like effect in a gas of inelastic rough hard spheres

We report the emergence of a giant Mpemba effect in the uniformly heated gas of inelastic rough hard spheres: The initially hotter sample may cool sooner than the colder one, even when the initial temperatures differ by more than one order of magnitude. In order to understand this behavior, it suffices to consider the simplest Maxwellian approximation for the velocity distribution in a kinetic approach. The largeness of the effect stems from the fact that the rotational and translational temperatures, which obey two coupled evolution equations, are comparable. Our theoretical predictions agree very well with molecular dynamics and direct simulation Monte Carlo data.

Figure 1

Density plots of $\mathrm{\Phi }(T^{*},\theta )$ as defined by Eq. (4a). Two representative examples of the coefficient of restitution, (a) $\alpha =1$ and (b) $\alpha =0.9$, are considered, for $\beta =0$. The contour lines (solid for negative $\mathrm{\Phi }$, dashed for positive $\mathrm{\Phi }$) are separated by an amount $\mathrm{\Delta }\mathrm{\Phi }=0.05$. The thick solid line is the locus $\mathrm{\Phi }(T^{*},\theta )=0$ and the circle marks the steady-state point ($T^{*},\theta )=(1,{\theta }^{\text{st}})$.

Figure 2

Locus $\tilde{\theta }(\alpha ,\beta )=0$ in the $(\beta ,\alpha )$ plane. We have a nonmonotonic behavior of $\mathrm{\Phi }(T^{*},\theta )$ vs $\theta$ with a minimum at $\tilde{\theta }>0$ above the curve, where $\tilde{\theta }$ is given by Eq. (7), whereas $\mathrm{\Phi }(T^{*},\theta )$ has a monotonically increasing behavior below the curve. Here, $\kappa =\frac{2}{5}$ (uniform solid spheres).

Figure 3

Phase diagrams of the Mpemba effect for $\alpha =0.9$ and $\beta =0$. In panel (a), we consider the $({\theta }_{0B},{\theta }_{0A})$ plane: The Mpemba effect is present (absent) for points below (above) the plotted lines, which correspond to several choices of the initial conditions; specifically, (i) $(T_{0A}^{*},T_{0B}^{*})=(5,4)$, (ii) $(T_{0A}^{*},T_{0B}^{*})=(4,3)$, and (iii) $(T_{0A}^{*},T_{0B}^{*})=(5,3)$. Note the different scales in both axes. In panel (b), we plot the line in the $(T_{0B}^{*},T_{0A}^{*}/T_{0B}^{*})$ plane below which the Mpemba effect may appear, provided that the kinetic energy of the initially hotter (colder) sample is concentrated in the translational (rotational) degrees of freedom to a sufficient extent.

Figure 4

Mpemba effect in the granular gas of rough hard spheres. The relaxation of the scaled temperature $T^{*}$ to its steady-state value is shown for different values of the coefficients of restitution $\alpha$ and $\beta$. The MD data (open circles) and, especially, the DSMC data (triangles) are in very good agreement with the theoretical values (lines). Dashed blue, solid green (light gray), and solid red lines refer to initial states given by $(T_{0A}^{*},{\theta }_{0A})=(5,0.01)$, $(T_{0B}^{*},{\theta }_{0B})=(4,10)$, and $(T_{0C}^{*},{\theta }_{0C})=(3,100)$, respectively. The Mpemba effect is neatly observed, with the initially hottest system being the first one to reach the steady state.