Homogenization Approaches to Multiphase Lattice Random Walks

TL;DR

This paper compares various homogenization methods for multiphase lattice random walks, revealing that only the discontinuous parabolic model accurately predicts long-term behavior, highlighting the importance of phase partitioning.

Contribution

It demonstrates that the discontinuous parabolic homogenization approach correctly captures the long-term properties of multiphase lattice random walks, unlike other models.

Findings

Parabolic models from Langevin equations fail to predict long-term behavior.

Discontinuous parabolic model accurately describes the system.

Implications for equilibrium constraints and transport properties.

Abstract

This article analyzes several different homogenization approaches to the long-term properties of multiphase lattice random walks, recently introduced by Giona and Cocco, and characterized by different values of the hopping times and of the distance between neighboring sites in each lattice phase. Both parabolic and hyperbolic models are considered. While all the parabolic models deriving from microscopic Langevin equations driven by Wiener processes fail to predict the long-term hydrodynamic behavior observed in lattice models, the discontinuous parabolic model, in which the phase partition coefficient is a-priori imposed, provides the correct answer. The implications of this result as regards the connection between equilibrium constraints and non-equilibrium transport properties is thoroughly addressed.