On the Mean Residence Time in Stochastic Lattice-Gas Models

TL;DR

This paper rigorously proves a heuristic law relating fluid residence time to influx in stochastic lattice-gas models, providing explicit formulas for specific models like the Ising model and ASEP.

Contribution

It establishes a rigorous foundation for the mean residence time law in stochastic lattice-gas models under mild assumptions.

Findings

Validated the law for general injection, diffusion, and extraction dynamics.

Derived explicit residence time formulas for the Ising model and ASEP.

Confirmed the law's applicability in specific lattice-gas models.

Abstract

A heuristic law widely used in fluid dynamics for steady flows states that the amount of a fluid in a control volume is the product of the fluid influx and the mean time that the particles of the fluid spend in the volume, or mean residence time. We rigorously prove that if the mean residence time is introduced in terms of sample-path averages, then stochastic lattice-gas models with general injection, diffusion, and extraction dynamics verify this law. Only mild assumptions are needed in order to make the particles distinguishable so that their residence time can be unambiguously defined. We use our general result to obtain explicit expressions of the mean residence time for the Ising model on a ring with Glauber + Kawasaki dynamics and for the totally asymmetric simple exclusion process with open boundaries.