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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mn>0</a:mn> <a:mo>&lt;</a:mo> <a:mi>α</a:mi> <a:mo>&lt;</a:mo> <a:mi>β</a:mi> <a:mo>≤</a:mo> <a:mn>1</a:mn> </a:mrow> </a:math> , 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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mo>(</b:mo> <b:mi>α</b:mi> <b:mo>)</b:mo> </b:math> transiently accelerates and eventually regains its intrinsic long-time mobility. This recovery emerges only when memory exponents differ <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mo>(</c:mo> <c:mi>α</c:mi> <c:mo>&lt;</c:mo> <c:mi>β</c:mi> <c:mo>)</c:mo> </c:mrow> </c:math> , whereas identical exponents <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mo>(</d:mo> <d:mi>α</d:mi> <d:mo>=</d:mo> <d:mi>β</d:mi> <d:mo>)</d:mo> </d:mrow> </d:math> suppress recovery. Our FLE models are further validated in comparison with Langevin dynamics simulations of polymer-based real systems. 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.