Microscopic Theory for Long Range Spatial Correlations in Lattice Gas Automata

TL;DR

This paper develops a microscopic kinetic theory to explain algebraic decay of spatial correlations in lattice gas automata, especially when collision rules violate semi-detailed-balance, revealing long-range correlations with specific decay rates.

Contribution

It provides analytical expressions for long-range correlation functions in lattice gas automata with non semi-detailed-balance collision rules, supported by computer simulations.

Findings

Correlation functions decay as 1/r^2 or 1/r^4 depending on model symmetries.

Analytical predictions match simulation results.

Long-range correlations arise from violation of semi-detailed-balance.

Abstract

Lattice gas automata with collision rules that violate the conditions of semi-detailed-balance exhibit algebraic decay of equal time spatial correlations between fluctuations of conserved densities. This is shown on the basis of a systematic microscopic theory. Analytical expressions for the dominant long range behavior of correlation functions are derived using kinetic theory. We discuss a model of interacting random walkers with x-y anisotropy whose pair correlation function decays as 1/r^2, and an isotropic fluid-type model with momentum correlations decaying as 1/r^2. The pair correlation function for an interacting random walker model with interactions satisfying all symmetries of the square lattice is shown to have 1/r^4 density correlations. Theoretical predictions for the amplitude of the algebraic tails are compared with the results of computer simulations.