Diffusion in Rugged Energy Landscapes in the Presence of Spatial Correlations : A Surprising Route to Zwanzig's Mean-Field Prediction

Abstract

Diffusion in rugged free-energy landscapes is central to diverse problems in chemical physics, biomolecular dynamics, polymer transport and numerous disordered systems. Zwanzig's well-known classic mean-field theory predicts that roughness reduces the diffusion coefficient by an exponential factor determined solely by the variance of the disorder. However, the numerical studies of Banerjee, Seki, and Bagchi (BSB) showed that this result fails for uncorrelated Gaussian-distributed site energies because rare but deep three-site traps dominate long-time transport. BSB introduced Gaussian \emph{spatial} correlations - originally developed in astrophysics to model turbulent density fluctuations - and demonstrated that even modest correlations suppress these pathological traps and restore Zwanzig's exponential scaling. Here we present here a unified theoretical framework clarifying (i) why Zwanzig's local averaging, may be viewed as a Gaussian cumulant expansion, could breakdown. In particular, how its validity is destroyed by uncorrelated disorder, and (ii) how Gaussian spatial correlations reshape roughness increments, eliminate asymmetric multi-site traps, and thereby recover mean-field diffusion, and (iii) a derivation showing exactly how Gaussian spatial correlations modify roughness increments, trap statistics, and ultimately the diffusion constant. We also provide explicit numerical triplet examples illustrating the dramatic reduction of escape times produced by spatial correlations.