Logarithmic superdiffusion in two dimensional driven lattice gases

TL;DR

This paper numerically verifies the mode coupling theory prediction that density fluctuations in two-dimensional driven lattice gases exhibit logarithmic superdiffusion, with the diffusivity diverging as $( ext{ln} t)^{2/3}$, using a novel parallel algorithm.

Contribution

First numerical confirmation of the predicted logarithmic superdiffusion in 2D driven lattice gases, validating mode coupling theory with a new massively parallel simulation approach.

Findings

Diffusivity diverges as $( ext{ln} t)^{2/3}$ in simulations.

Prefactor variation with anisotropy parameter $p$ matches theoretical predictions.

Novel parallel coupling algorithm reduces fluctuations in correlation function estimates.

Abstract

The spreading of density fluctuations in two-dimensional driven diffusive systems is marginally anomalous. Mode coupling theory predicts that the diffusivity in the direction of the drive diverges with time as $(\ln t)^{2/3}$ with a prefactor depending on the macroscopic current-density relation and the diffusion tensor of the fluctuating hydrodynamic field equation. Here we present the first numerical verification of this behavior for a particular version of the two-dimensional asymmetric exclusion process. Particles jump strictly asymmetrically along one of the lattice directions and symmetrically along the other, and an anisotropy parameter $p$ governs the ratio between the two rates. Using a novel massively parallel coupling algorithm that strongly reduces the fluctuations in the numerical estimate of the two-point correlation function, we are able to accurately determine the exponent of the logarithmic correction. In addition, the variation of the prefactor with $p$ provides a stringent test of mode coupling theory.