Velocity Distribution of a Homogeneously Cooling Granular Gas

In contrast to molecular gases, granular gases are characterized by inelastic collisions and require therefore permanent driving to maintain a constant kinetic energy. The kinetic theory of granular gases describes how the average velocity of the particles decreases after the driving is shut off. Moreover, it predicts that the rescaled particle velocity distribution will approach a stationary state with overpopulated high-velocity tails as compared to the Maxwell-Boltzmann distribution. While this fundamental theoretical result was reproduced by numerical simulations, an experimental confirmation is still missing. Using a microgravity experiment that allows the spatially homogeneous excitation of spheres via magnetic fields, we confirm the theoretically predicted exponential decay of the tails of the velocity distribution.

Figure 1

Experimental setup. (a) The diagonals of a PMMA cube containing 2796 ferromagnetic spheres are aligned with four pairs of electromagnets that are used to inject energy into the granular gas. The time evolution of the granular gas is then observed with a camera at $165\mathrm{frames}/\mathrm{s}$. Panel (b) shows a typical example image.

Figure 2

All four experiments are well described by Haff’s cooling law. Each panel displays first the last second of the magnetic driving with fluctuations in the average velocity. At $t=0$ the magnetic driving is switched off and the subsequent evolution of the average velocity is fit with Haff’s law [Eq. (1)] assuming a constant coefficient of restitution (red line). Each fit provides two parameters: $v_0$, the average particle velocity at $t=0$, and $\tau$, the time by which the velocity has decreased to $v_0/2$.

Figure 3

The distribution of particle velocities in the homogeneous cooling state displays an exponential tail for $c>2.2$. Data are averaged over all four experiments shown in Fig. 2 and the time interval 1.8–3 s after onset of cooling.