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

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.