Clustering in a One-dimensional Inelastic Lattice Gas

TL;DR

This paper investigates a one-dimensional lattice model related to inelastic gases, revealing how shocks and clusters form and evolve, with findings on their independence from certain parameters and their dependence on initial conditions.

Contribution

It introduces a lattice model that captures shock and cluster formation in inelastic gases, providing insights into their microscopic origins and late-time dynamics.

Findings

Shocks form independently of the coefficient of restitution.

Cluster velocities follow a continuum equation.

Shock locations depend on initial velocity configurations.

Abstract

We analyze a lattice model closely related to the one-dimensional inelastic gas with periodic boundary condition. The one-dimensional inelastic gas tends to form high density clusters of particles with almost the same velocity, separated by regions of low density; plotted as a function of particle indices, the velocities of the gas particles exhibit sharp gradients, which we call shocks. Shocks and clusters are seen to form in the lattice model too, although no true positions of the particles are taken into account. The locations of the shocks in terms of the particle index show remarkable independence on the coefficient of restitution and the sequence of collisions used to update the system, but they do depend on the initial configuration of the particle velocities. We explain the microscopic origin of the shocks. We show that dynamics of the velocity profile inside a cluster satisfies a simple continuum equation, thereby allowing us to study cluster-cluster interactions at late times.