Drift and trapping in biased diffusion on disordered lattices

TL;DR

This paper analyzes the transition from drift to no-drift in biased diffusion on disordered lattices, revealing a logarithmic decay of velocity at critical bias and divergence of relaxation time below it, supported by simulations.

Contribution

It provides a refined theoretical understanding of the transition behavior in biased diffusion on percolation networks, including a new extrapolation form validated by simulations.

Findings

Velocity decreases as 1/log(t) at critical bias B_c

Relaxation time diverges exponentially as B approaches B_c from below

The proposed extrapolation form matches Monte Carlo simulation results

Abstract

We reexamine the theory of transition from drift to no-drift in biased diffusion on percolation networks. We argue that for the bias field B equal to the critical value B_c, the average velocity at large times t decreases to zero as 1/log(t). For B < B_c, the time required to reach the steady-state velocity diverges as exp(const/|B_c-B|). We propose an extrapolation form that describes the behavior of average velocity as a function of time at intermediate time scales. This form is found to have a very good agreement with the results of extensive Monte Carlo simulations on a 3-dimensional site-percolation network and moderate bias.