Diffusivity in multiple scattering systems

TL;DR

This paper investigates how microscopic surface interactions influence the diffusivity of particles in multi-dimensional channels, deriving a relationship between scattering properties and macroscopic diffusion behavior, with exact calculations in 2D cases.

Contribution

It establishes a connection between the scattering operator spectrum and diffusivity, generalizes a central limit theorem for these systems, and computes explicit diffusivity values for certain 2D models.

Findings

Diffusivity relates to the spectrum of the scattering operator P.

Stationary distributions include Maxwell-Boltzmann and Knudsen cosine law.

Exact diffusivity values are computed for specific 2D surface models.

Abstract

We consider random flights of point particles inside $n$-dimensional channels of the form $\mathbb{R}^{k} \times \mathbb{B}^{n-k}$, where $\mathbb{B}^{n-k}$ is a ball of radius $r$ in dimension $n-k$. The particle velocities immediately after each collision with the boundary of the channel comprise a Markov chain with a transition probabilities operator $P$ that is determined by a choice of (billiard-like) random mechanical model of the particle-surface interaction at the "microscopic" scale. Our central concern is the relationship between the scattering properties encoded in $P$ and the constant of diffusivity of a Brownian motion obtained by an appropriate limit of the random flight in the channel. Markov operators obtained in this way are {\em natural} (definition below), which means, in particular, that (1) the (at the surface) Maxwell-Boltzmann velocity distribution with a given surface temperature, when the surface model contains moving parts, or (2) the so-called Knudsen cosine law, when this model is purely geometric, is the stationary distribution of $P$. We show by a suitable generalization of a central limit theorem of Kipnis and Varadhan how the diffusivity is expressed in terms of the spectrum of $P$ and compute, in the case of 2-dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2).