Variational calculation of transport coefficients in diffusive lattice gases

TL;DR

This paper introduces a variational approximation scheme for calculating diffusion coefficients in diffusive lattice gases, providing accurate bounds and density profiles that align well with simulations.

Contribution

It proposes a finite-dimensional variational method to estimate diffusivity in non-gradient lattice gases, extending analytical tools beyond gradient cases.

Findings

Provides upper bounds for diffusivity close to simulation results

Accurately predicts non-equilibrium density profiles

Demonstrates effectiveness for one-dimensional generalized exclusion processes

Abstract

A diffusive lattice gas is characterized by the diffusion coefficient depending only on the density. The Green-Kubo formula for diffusivity can be represented as a variational formula, but even when the equilibrium properties of a lattice gas are analytically known the diffusion coefficient can be computed only in the exceptional situation when the lattice gas is gradient. In the general case, minimization over an infinite-dimensional space is required. We propose an approximation scheme based on minimizing over finite-dimensional subspaces of functions. The procedure is demonstrated for one-dimensional generalized exclusion processes in which each site can accommodate at most two particles. Our analytical predictions provide upper bounds for the diffusivity that are very close to simulation results throughout the entire density range. We also analyze non-equilibrium density profiles for finite chains coupled to reservoirs. The predictions for the profiles are in excellent agreement with simulations.