Effective medium approximation for lattice random walks with long-range jumps

TL;DR

This paper develops an effective medium approximation for lattice random walks with long-range jumps, providing analytical solutions for anomalous diffusion and validating results with numerical simulations.

Contribution

It introduces a self-consistent EMA approach for disordered lattice walks with long-range jumps, including analytical solutions for power-law cases.

Findings

Analytical expression for anomalous diffusivity.

Good agreement between EMA predictions and simulations.

Identification of Levy-like effective medium in power-law cases.

Abstract

We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective medium approximation (EMA) to find the disorder averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In case of a power-law scaling between transition rates and distance, lattice variants of Levy-ights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions, and demonstrate very good agreement with numerical simulations.