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