Non-Gaussian diffusion · Lorentz gas

A Light Particle Diffusing Through Heavy Gas

Main claim

A thermal average over the particle speed turns the classical Lorentz-gas picture into a theory for Brownian yet non-Gaussian diffusion at intermediate times. Even in a simple gas, particles with different speeds can collectively produce a spread that is broader than a Gaussian distribution.

Representative simulation of particle diffusion in a binary gas mixture — Representative image from the binary-gas diffusion research theme.

Problem

A very light minority particle in a heavy gas can have a mean-square displacement that grows linearly in time while its displacement distribution remains non-Gaussian. This Brownian-yet-non-Gaussian behavior resembles a Lorentz gas, but the classical model keeps the tracer speed fixed and does not by itself reproduce the observed statistics.

Approach

We first derived the mean-square displacement, non-Gaussian parameter, and displacement distribution of a dilute Lorentz gas for a fixed particle speed using a point-process description of collisions. We then averaged those results over the Maxwell-Boltzmann distribution of initial speeds and compared the theory with kinetic Monte Carlo data for a binary gas.

What we found

A fixed-speed Lorentz gas remains Gaussian at long times and cannot explain the phenomenon. Averaging over the thermal distribution of speeds, however, reproduces Brownian-yet-non-Gaussian diffusion over the intermediate regime relevant to a large mass contrast. The theory also predicts a broad, non-exponential tail in the displacement distribution and clarifies where speed relaxation must eventually enter.

Why it matters

The work builds a direct bridge between a classical Lorentz-gas model and anomalous transport in an ordinary binary gas. It shows that a distribution of particle speeds can generate non-Gaussian statistics without invoking a structurally complex medium, and supplies an analytical starting point for more complete models in which the speed also changes with time.

Keywords

Brownian-yet-non-Gaussian diffusion, Lorentz gas, binary gas, kinetic theory, van Hove function.

Paper

Fumiaki Nakai, Takashi Uneyama, “Brownian yet non-Gaussian diffusion of a light particle in heavy gas: Lorentz-gas-based analysis,” Physical Review E 108, 044129 (2023).

Published article Open-access preprint