Critical dynamics of ballistic and Brownian particles in a heterogeneous environment

TL;DR

This study investigates the universal dynamic behavior of tracer particles near the localization transition in disordered media, revealing slow crossover scaling and the influence of heterogeneities on transport properties.

Contribution

The paper demonstrates universality in the long-time dynamics of particles near the transition and characterizes the slow variation of the crossover scaling function.

Findings

Universality holds at long times near the transition.

The crossover scaling function varies extremely slowly over at least 5 decades.

Apparent power laws with varying exponents are observed in experimental time windows.

Abstract

The dynamic properties of a classical tracer particle in a random, disordered medium are investigated close to the localization transition. For Lorentz models obeying Newtonian and diffusive motion at the microscale, we have performed large-scale computer simulations, demonstrating that universality holds at long times in the immediate vicinity of the transition. The scaling function describing the crossover from anomalous transport to diffusive motion is found to vary extremely slowly and spans at least 5 decades in time. To extract the scaling function, one has to allow for the leading universal corrections to scaling. Our findings suggest that apparent power laws with varying exponents generically occur and dominate experimentally accessible time windows as soon as the heterogeneities cover a decade in length scale. We extract the divergent length scales, quantify the spatial heterogeneities in terms of the non-Gaussian parameter, and corroborate our results by a thorough finite-size analysis.