Variational calculation of transport coefficients in diffusive lattice gases

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.

14 pages, 11 figures. v2: added remarks and references, replaced figures

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• Short-range and long-range correlations in driven dense colloidal mixtures in narrow pores
• Relaxation and fluctuations of a mass- and dipole-conserving stochastic lattice gas
• Exact large-scale correlations in diffusive systems with general interactions: explicit characterisation without the Dean--Kawasaki equation
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• Macroscopic fluctuation theory of interacting Brownian particles