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.

arXiv comment14 pages, 11 figures. v2: added remarks and references, replaced figures

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• A variational principle for mass transport
• Bulk diffusion in a kinetically constrained lattice gas
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• Motion of active tracer in a lattice gas with cross-shaped particles
• Exact hydrodynamics and onset of phase separation for an active exclusion process
• Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotics
• Hydrodynamic approximations for driven dense colloidal mixtures in narrow pores
• Dynamical patterns and nonreciprocal effective interactions in an active-passive mixture through exact hydrodynamic analysis
• Tracer and current fluctuations in driven diffusive systems
• Short-range and long-range correlations in driven dense colloidal mixtures in narrow pores
• Exact large-scale correlations in diffusive systems with general interactions: explicit characterisation without the Dean--Kawasaki equation
• Relaxation and fluctuations of a mass- and dipole-conserving stochastic lattice gas
• Computation of the self-diffusion coefficient with low-rank tensor methods: application to the simulation of a cross-diffusion system
• Microscopic and macroscopic perspectives on stationary nonequilibrium states
• Macroscopic fluctuation theory of interacting Brownian particles