Spatial Correlations Restore Zwanzig's Mean-Field Diffusion Result in Rugged Energy Landscapes

TL;DR

This paper develops a theoretical framework showing how spatial correlations in rugged energy landscapes restore Zwanzig's mean-field diffusion predictions by suppressing extreme trapping events that dominate uncorrelated landscapes.

Contribution

The authors introduce a unified theory explaining the breakdown and restoration of Zwanzig's diffusion result through the role of spatial correlations in energy landscapes.

Findings

Spatial correlations smooth the energy landscape, reducing extreme traps.

Zwanzig's exponential diffusion scaling is recovered with correlations.

Numerical examples show reduced escape times in correlated landscapes.

Abstract

Transport in disordered environments is often controlled not by typical fluctuations but by rare, extreme events that dominate long-time dynamics. In such settings, Zwanzig's classic mean-field theory predicts that energetic roughness reduces the diffusion coefficient by an exponential factor governed solely by the variance of the disorder. However, this prediction breaks down in uncorrelated Gaussian landscapes, where rare but deep multi-site traps dominate transport and lead to a much stronger suppression of diffusion. Here, we present a unified theoretical framework that clarifies both the origin of this breakdown and its resolution. We show that Zwanzig's local averaging can be interpreted as a Gaussian cumulant expansion whose validity is destroyed by uncorrelated disorder through the emergence of extreme trapping events. Introducing Gaussian spatial correlations fundamentally reshapes the landscape: roughness increments become smoother, asymmetric multi-site traps are suppressed, and the statistics of escape pathways are regularized. As a result, Zwanzig's exponential scaling is recovered. We provide an explicit analytical derivation demonstrating how spatial correlations modify trap statistics and restore mean-field diffusion, complemented by illustrative numerical examples showing the dramatic reduction of escape times in correlated landscapes.