Shock-Like Dynamics of Inelastic Gases

Gases of inelastically colliding particles model the dynamics of granular materials [1,2], geophysical flows [3], and large-scale structure of matter in the universe [4]. Typically, a fraction of the kinetic energy is dissipated in each collision, leading to interparticle velocity correlations, a clustering instability [5–9], and in the absence of external energy input, an inelastic collapse [10–15]. The last feature presents an obstacle to long-time simulations, as an infinite number of collisions occur in a finite time. In this Letter, we propose that a freely evolving inelastic gas is asymptotically in the universality class of a completely inelastic, sticky gas. Specifically, the temperature decreases in time as t−2 over an intermediate range, but asymptotically decays as t−2/3. To test this hypothesis, we employ a simulation in which collisions between particles with sufficiently small relative velocities are perfectly elastic. This method allows us to bypass the inelastic collapse and probe the asymptotic regime. We consider N identical point particles undergoing inelastic collisions in a one-dimensional periodic system of length L. The particles have typical interparticle spacing x0 = L/N and their typical velocity is v0. We employ dimensionless space and time variables, x → x/x0, and t → tv0/x0, thereby rescaling the ring length to N . Inelastic and momentum conserving collisions are implemented by changing the sign of the relative velocity and reducing its magnitude by a factor r = 1 − 2 , with 0 ≤ r ≤ 1, after each collision. It is convenient to view the particle identities as “exchanged” upon collision, so that in a perfectly elastic collision the particles merely pass through each other, while for a small inelasticity each particle suffers a small deflection. The outcome of a collision between a particle with velocity v and another particle with velocity u is therefore