When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids

Under certain conditions, two samples of fluid at different initial temperatures present a counterintuitive behavior known as the Mpemba effect: it is the hotter system that cools sooner. Here, we show that the Mpemba effect is present in granular fluids, both in uniformly heated and in freely cooling systems. In both cases, the system remains homogeneous, and no phase transition is present. Analytical quantitative predictions are given for how differently the system must be initially prepared to observe the Mpemba effect, the theoretical predictions being confirmed by both molecular dynamics and Monte Carlo simulations. Possible implications of our analysis for other systems are also discussed.

Figure 1

(a) Crossover time ${\tau }_c$ as a function of the ratio $\mathrm{\Delta }{\theta }_0/\mathrm{\Delta }{a}_{2,0}$ for $\alpha =0.9$. (b) Phase diagram in the plane $\mathrm{\Delta }{\theta }_0/\mathrm{\Delta }{a}_{2,0}$ vs $\alpha$. The regions of the plane inside which there appears or does not appear the ME are separated by the curve $(\mathrm{\Delta }{\theta }_0/\mathrm{\Delta }{a}_{2,0}{)}_{\max }$.

Figure 2

Relaxation of the scaled temperature to the steady state for $\alpha =0.9$. The upper and the lower curves correspond to the ME for the cooling and heating processes, respectively (see text). The direct simulation Monte Carlo (DSMC) (open symbols) and molecular dynamics (MD) (filled symbols) data show an excellent agreement with the theoretical prediction (lines).

Figure 3

Relaxation time ${\tau }_{\epsilon }$ (with $\epsilon ={10}^{-4}$) as a function of the initial scaled temperature ${\theta }_0$ for $\alpha =0.9$. Three values of the initial excess kurtosis are considered, $a_{2,0}=0.5$ (solid line), $a_{2,0}=0$ (dotted line), and $a_{2,0}=-0.35$ (dashed line). The horizontal (grey) segments join values of initial temperatures that share the same value of the relaxation time and, thus, mark the onset of either the ME (${\theta }_0>1$) or the inverse ME (${\theta }_0<1$).

Figure 4

Evolution of the temperature in the free cooling case. Again, the agreement between the theory (lines) and both DSMC (open symbols) and MD (filled symbols) simulation data is excellent. The inset shows $t_c^{*}$ as a function of $\mathrm{\Delta }{T}_0^{*}/\mathrm{\Delta }{a}_{2,0}$.