Void formation in diffusive lattice gases

TL;DR

This paper investigates the probability and most probable density history of spontaneous void formation in diffusive lattice gases, using macroscopic fluctuation theory, and compares equilibrium and non-equilibrium cases.

Contribution

It provides analytical and numerical results for void formation probabilities in diffusive lattice gases, including both annealed and quenched initial conditions, and derives microscopic results for non-interacting particles.

Findings

Void formation probability in annealed case matches Boltzmann-Gibbs formula.

In quenched case, probability is analytically derived for non-interacting walkers.

For small voids, equilibrium probability results are recovered.

Abstract

What is the probability that a macroscopic void will spontaneously arise, at a specified time T, in an initially homogeneous gas? We address this question for diffusive lattice gases, and also determine the most probable density history leading to the void formation. We employ the macroscopic fluctuation theory by Bertini et al and consider both annealed and quenched averaging procedures (the initial condition is allowed to fluctuate in the annealed setting). We show that in the annealed case the void formation probability is given by the equilibrium Boltzmann-Gibbs formula, so the probability is independent of T (and also of the void shape, as only the volume matters). In the quenched case, which is intrinsically non-equilibrium, we evaluate the void formation probability analytically for non-interacting random walkers and probe it numerically for the simple symmetric exclusion process. For voids that are small compared with the diffusion length, the equilibrium result for the void formation probability is recovered. We also re-derive our main results for non-interacting random walkers from an exact microscopic analysis.