Subdiffusion from competition between multi-exponential friction memory and energy barriers

Abstract

Subdiffusion is a hallmark of complex systems, ranging from protein folding to transport in viscoelastic media. However, despite its pervasiveness, the mechanistic origins of subdiffusion remain contested. Here, we analyze both Markovian and non-Markovian dynamics, in the presence and absence of energy barriers, in order to disentangle the distinct contributions of memory-dependent friction and energy barriers to the emergence of subdiffusive behavior. Focusing on the mean squared displacement (MSD), we develop an analytical framework that connects subdiffusion to multi-scale memory effects in the generalized Langevin equation (GLE), and derive the subdiffusive scaling behavior of the MSD for systems governed by multi-exponential memory kernels. We identify persistence and relaxation timescales that delineate dynamical regimes in which subdiffusion arises from either memory or energy barrier effects. By comparing analytical predictions with simulations, we confirm that memory dominates the overdamped dynamics for barrier heights up to approximately \(2\,k_BT\), a regime recently shown to be relevant for fast-folding proteins. Overall, our results advance the theoretical understanding of anomalous diffusion and provide practical tools that are broadly applicable to fields as diverse as molecular biophysics, polymer physics, and active matter systems.

Subdiffusion in the context of the generalized Langevin equation can arise due to energy barriers, from friction memory or from a combination of both. We derive the power-law scaling for multi-exponential memory functions that directly translates to the subdiffusive scaling in the MSD. This allows us to disentangle contributions from energy barriers and from memory. It turns out that memory governs the subdiffusion for small energy barriers in the order of a few \(k_BT\). For higher energy barriers, the short time dynamics are still dominated by memory and long-time dynamics are governed by a combination of memory effects and energy barrier contributions.