Driven inelastic Maxwell gases

TL;DR

This paper analyzes the inelastic Maxwell model with external driving, deriving exact equations for velocity correlations, and identifies conditions under which the system reaches a steady state or not.

Contribution

It provides exact formulas for the variance and correlation functions in driven inelastic Maxwell gases, clarifying steady state conditions for different driving mechanisms.

Findings

System reaches steady state for $r_w eq -1$

No steady state for $r_w = -1$

Steady state exists for Ornstein-Uhlenbeck driving with $ eq 0$

Abstract

We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities, and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates $\tau_c^-{1}$ and $\tau_w^{-1}$ respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution $r$. The velocity change of a particle with velocity $v$, due to driving, is taken to be $\Delta v=-(1+r_w) v+\eta$, mimicking the collision with a vibrating wall, where $r_w$ the coefficient of restitution of the particle with the "wall" and $\eta$ is Gaussian white noise. The Ornstein-Uhlenbeck driving mechanism given by $\frac{dv}{dt}=-\Gamma v+\eta$ is found to be a special case of the driving modeled as a point process. Using both the continuum and discrete versions we show that while the equations for the one-particle and the two-particle velocity distribution functions do not close, the joint evolution equations of the variance and the two-particle velocity correlation functions close. With the exact formula for the variance we find that, for $r_w\ne-1$, the system goes to a steady state. On the other hand, for $r_w=-1$, the system does not have a steady state. Similarly, the system goes to a steady state for the Ornstein-Uhlenbeck driving with $\Gamma\not=0$, whereas for the purely diffusive driving ($\Gamma=0$), the system does not have a steady state.