A model for slowing particles in random media

TL;DR

This paper introduces a simple model for particles slowing down in a random obstacle field, deriving a kinetic equation with a delta function term to describe the asymptotic behavior as obstacle size shrinks.

Contribution

It provides a new kinetic model for particle slowing in random media with a delta function collision term, extending previous models to include mass conservation.

Findings

Convergence of particle density to a kinetic equation with a delta function in velocity.

The model captures the slowing process with a specific velocity-dependent rate.

Mass conservation is maintained in the asymptotic limit.

Abstract

We present a simple model in dimension $d\geq 2$ for slowing particles in random media, where point particles move in straight lines among and inside spherical identical obstacles with Poisson distributed centres. When crossing an obstacle, a particle is slowed down according to the law $\dot{V}= -\frac{\kappa}{\epsilon} S(|V|) V$, where $V$ is the velocity of the point particle, $\kappa$ is a positive constant, $\epsilon$ is the radius of the obstacle and $S(|V|)$ is a given slowing profile. With this choice, the slowing rate in the obstacles is such that the variation of speed at each crossing is of order $1$. We study the asymptotic limit of the particle system when $\epsilon$ vanishes and the mean free path of the point particles stays finite. We prove the convergence of the point particles density measure to the solution of a kinetic-like equation with a collision term which includes a contribution proportional to a $\delta$ function in $v=0$; this contribution guarantees the conservation of mass for the limit equation.