Effects of Dynamic Disorder on Diffusion in Rugged Energy Landscapes
Biman Bagchi
TL;DR
This work tackles diffusion on rugged energy landscapes in the presence of dynamic disorder, modeling environmental fluctuations as telegraph processes and deriving a closed-form expression for the effective hopping rate that governs long-time transport. By linking a mean-waiting-time framework with a Kubo–Zwanzig stochastic Liouville formulation, the authors show a continuous crossover from quasi-quenched, trap-dominated transport to an annealed, motional-narrowing regime as the fluctuation rate $ u$ increases, with explicit interpolation between the Banerjee–Seki–Bagchi and Zwanzig limits. The approach reveals how dynamic disorder renormalizes trap lifetimes, preserves the essential harmonic-average structure of one-dimensional diffusion, and provides a practical numerical protocol to compute diffusion constants via bond-by-bond statistics and quenched averaging. The results offer a unifying view of diffusion in dynamic landscapes, with implications for glassy dynamics and biomolecular transport, and suggest directions for extending the theory to continuous environmental fluctuations and higher dimensions.
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.