Bulk diffusion in a kinetically constrained lattice gas

TL;DR

This paper introduces a systematic variational method to approximate the diffusion coefficient in non-gradient lattice gases, demonstrated on the Kob-Andersen model, advancing understanding of microscopic dynamics in constrained systems.

Contribution

It develops a new analytical approximation technique for diffusion coefficients in non-gradient lattice gases using a variational formula, applicable to models like Kob-Andersen.

Findings

Derived upper bounds for the diffusion coefficient.

Applied method successfully to the Kob-Andersen model.

Enhanced understanding of diffusion in kinetically constrained systems.

Abstract

In the hydrodynamic regime, the evolution of a stochastic lattice gas with symmetric hopping rules is described by a diffusion equation with density-dependent diffusion coefficient encapsulating all microscopic details of the dynamics. This diffusion coefficient is, in principle, determined by a Green-Kubo formula. In practice, even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient cannot be computed except when a lattice gas additionally satisfies the gradient condition. We develop a procedure to systematically obtain analytical approximations for the diffusion coefficient for non-gradient lattice gases with known equilibrium. The method relies on a variational formula found by Varadhan and Spohn which is a version of the Green-Kubo formula particularly suitable for diffusive lattice gases. Restricting the variational formula to finite-dimensional sub-spaces allows one to perform the minimization and gives upper bounds for the diffusion coefficient. We apply this approach to a kinetically constrained non-gradient lattice gas, viz. to the Kob-Andersen model on the square lattice.