Anomalous diffusion in coupled viscoelastic media: A fractional Langevin equation approach

TL;DR

This paper develops a fractional Langevin equation model for two coupled particles in different viscoelastic media, revealing how memory heterogeneity and coupling lead to transient anomalous diffusion and recovery dynamics.

Contribution

It introduces a novel coupled fractional Langevin equation framework to analyze how differing memory effects influence anomalous diffusion in complex media.

Findings

Recovery dynamics depend on differing memory exponents.

Identifies conditions for transient acceleration of subdiffusive particles.

Provides theoretical insights into biological anomalous diffusion phenomena.

Abstract

Anomalous diffusion often arises in complex environments where viscoelastic or crowded conditions influence particle motion. In many biological and soft-matter systems, distinct components of the medium exhibit unique viscoelastic responses, resulting in time-dependent changes in the observed diffusion exponents. Here, we develop a theoretical model of two particles, each embedded in a distinct viscoelastic medium, and coupled via a harmonic potential. By formulating and solving a system of coupled fractional Langevin equations (FLEs) with memory exponents $0<\alpha<\beta\leq 1$, we uncover rich transient anomalous diffusion phenomena arising from the interplay of memory kernels and bilinear coupling. Notably, we identify recovery dynamics, where a subdiffusive particle ($\alpha$) transiently accelerates and eventually regains its intrinsic long-time mobility. This recovery emerges only when memory exponents differ ($\alpha<\beta$), whereas identical exponents ($\alpha=\beta$) suppress recovery. Our theoretical predictions offer insight into experimentally observed transient anomalous diffusions, such as polymer--protein complexes and cross-linked cytoskeletal networks, highlighting the critical role of memory heterogeneity and mechanical interactions in biological anomalous diffusion.