Diffusion and Correlations in Lattice Gas Automata

TL;DR

This paper analyzes diffusion in a lattice gas automaton by comparing theoretical spectral density calculations with simulation data, validating the automaton as a microscopic model for fluid diffusion across scales.

Contribution

It introduces a method to isolate diffusive modes in lattice gas automata and compares theoretical predictions with simulations over all wavelengths.

Findings

Spectral functions match classical non-thermal fluid behavior.

Hydrodynamic limit of Boltzmann theory accurately describes long wavelengths.

Boltzmann theory agrees well with simulations at shorter wavelengths.

Abstract

We present an analysis of diffusion in terms of the spontaneous density fluctuations in a non-thermal two-species fluid modeled by a lattice gas automaton. The power spectrum of the density correlation function is computed with statistical mechanical methods, analytically in the hydrodynamic limit, and numerically from a Boltzmann expression for shorter time and space scales. In particular we define an observable -- the weighted difference of the species densities -- whose fluctuation correlations yield the diffusive mode independently of the other modes so that the corresponding power spectrum provides a measure of diffusion dynamics solely. Automaton simulations are performed to obtain measurements of the spectral density over the complete range of wavelengths (from the microscopic scale to the macroscopic scale of the automaton universe). Comparison of the theoretical results with the numerical experiments data yields the following results: (i) the spectral functions of the lattice gas fluctuations are in accordance with those of a classical `non-thermal' fluid; (ii) the Landau-Placzek theory, obtained as the hydrodynamic limit of the Boltzmann theory, describes the spectra correctly in the long wavelength limit; (iii) at shorter wavelengths and at moderate densities the complete Boltzmann theory provides good agreement with the simulation data. These results offer convincing validation of lattice gas automata as a microscopic approach to diffusion phenomena in fluid systems.