Inhomogeneous quasi-stationary state of dense fluid of inelastic hard spheres

Abstract

We study closed dense collections of hard spheres that collide inelastically with constant coefficient of normal restitution. We find inhomogeneous states (IS) where the density profile is spatially non-uniform but constant in time. The states are exact solutions of non-linear partial differential equations that describe the coupled distributions of density and temperature when inelastic losses of energy per collision are small. The derivation is performed without modelling the equations' coefficients that are unknown in the dense limit (such as the equation of state), using only their scaling form specific for hard spheres. The IS is exact non-linear state of this many-body system. It captures a fundamental property of inelastic collections of particles: the possibility of preserving non-uniform temperature via the interplay of inelastic cooling and heat conduction, generalizing previous results in the dilute case. We perform numerical simulations to demonstrate that arbitrary initial state evolves to the IS in the limit of long times where the container has the geometry of the channel. The evolution is like gas-liquid transition. The liquid condenses in a vanishing part of the total volume but takes most of the mass of the system. However, the gaseous phase, which mass grows only logarithmically with the system size, is relevant because its fast particles carry most of the energy of the system. Remarkably, the system self-organizes to dissipate no energy: the inelastic decay of energy is a power-law $[1+t/t_c]^{-2}$ where $t_c$ diverges in the thermodynamic limit. This behavior is caused by unusual spatial distribution of particles: on approach to one of the container's walls the density grows inversely with the distance. We discuss the relation of our results to the recently proposed finite-time singularity in other container's geometries.