Effects of Dynamic Disorder on Diffusion in Rugged Energy Landscapes

TL;DR

This paper develops an analytical theory for diffusion on rugged energy landscapes with dynamic disorder, revealing a crossover from trap-dominated to motional narrowing regimes as fluctuation rates increase.

Contribution

It introduces a minimal, analytically tractable model for diffusion with dynamic disorder, extending previous static landscape theories to include time-fluctuating energies.

Findings

Dynamic disorder causes a crossover from trap-dominated to motional narrowing regimes.

Analytic expression for diffusion constant in terms of mean waiting times.

Numerical results confirm the theoretical crossover behavior.

Abstract

Established theoretical studies of diffusion in rugged (or rough) potential surfaces have largely focused on quenched energy landscapes. Here we study diffusion on a rugged energy landscape in the presence of dynamic disorder, a situation relevant to a wide range of disordered systems, including glasses, disordered solids, and biomolecular transport. For static (quenched) Gaussian disorder, Zwanzig derived a compact mean field expression for the diffusion constant, showing that increasing ruggedness leads to a sharp reduction of diffusive transport. Subsequent work demonstrated that in one-dimensional discrete lattices diffusion is further suppressed by rare but long-lived multi-site traps that lie beyond the mean-field description. In many physical systems, however, the local energy landscape is not frozen but fluctuates in time, there by modifying trap lifetimes and transport properties. In this work we develop a minimal, analytically tractable theory of diffusion on rugged energy landscapes with dynamic disorder by allowing site or barrier energies to fluctuate as dichotomic (telegraph) processes with given amplitude and flipping rate. Using a Kehr type formulation appropriate for discrete hopping processes, we derive an analytic expression for the diffusion constant in terms of mean waiting times. We show that dynamic disorder induces a continuous crossover from quasi-quenched, trap-dominated transport to an annealed, motional narrowing regime as the fluctuation rate increases. Explicit numerical calculations confirm this crossover, interpolating between rare-event-dominated diffusion and Zwanzig mean-field regime.