Diffusion in Rugged Energy Landscapes in the Presence of Spatial Correlations : A Surprising Route to Zwanzig's Mean-Field Prediction
Biman Bagchi
TL;DR
The paper addresses diffusion on rugged energy landscapes and explains why Zwanzig's mean-field prediction $D_{Zw}=D_0\exp(-\beta^2\epsilon^2)$ can fail for uncorrelated disorder due to rare deep multi-site traps. By introducing Gaussian spatial correlations with length $\lambda$ and deriving how they modify the variance of roughness increments $\mathrm{Var}[\Delta\eta] = 2\epsilon^2\left[1-\exp(-a^2/\lambda^2)\right]$, the increment cross-correlation $C$, and the three-site-trap probability $P_{\mathrm{TST}}(\lambda)$, the authors show that correlations suppress extreme traps and restore Zwanzig’s mean-field diffusion. They demonstrate that finite correlation length $\lambda$ suppresses TSTs, reduces trap severity, and restores Zwanzig’s diffusion with $D\approx D_0\exp(-\beta^2\epsilon^2)$. The work is supported by explicit triplet numerical examples and Brownian-dynamics simulations, illustrating dramatic reductions in escape times and highlighting the role of correlated disorder in biomolecular transport and other disordered systems.
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.