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

TL;DR

The paper develops a two-component viscoelastic diffusion model using coupled fractional Langevin equations with memory exponents $0<\alpha<\beta\le 1$, deriving effective decoupled GLEs and revealing a recovery dynamic where the slower subsystem transiently accelerates when linked to the faster one. By analyzing mean square displacements, the authors identify multiple transient scaling regimes, including a four-stage MSD in the strong-coupling limit and a no-recovery regime when $\alpha=\beta$, all governed by interplay between memory kernels and harmonic coupling. Theoretical results are validated against polymer simulations, showing quantitative agreement and illustrating applicability to chromatin dynamics and crosslinked cytoskeletal networks. The work highlights memory-heterogeneity and mechanical interactions as key drivers of transient anomalous diffusion in complex biophysical environments and offers a framework for interpreting time-dependent diffusion exponents observed in experiments.

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 $0<α<β\leq 1$, 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 ($α$) transiently accelerates and eventually regains its intrinsic long-time mobility. This recovery emerges only when memory exponents differ ($α<β$), whereas identical exponents ($α=β$) 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.

Figures (7)

• Figure 1: Schematic illustrations of biological and soft-matter systems modeled by two coupled FLEs. (a) A polymer chain (e.g., chromatin) tethered to a macromolecular complex or protein condensate. (b) A composite cytoskeletal network comprising flexible polymers cross-linked with stiff polymers. (c) A membrane protein embedded within a lipid bilayer, interacting with both trans- and extracellular domains. (d) Our minimal bipartite model of these systems. Two particles, each governed by its own viscoelastic memory kernel characterized by distinct memory exponents ($\alpha$ and $\beta$), are mechanically coupled by a harmonic potential.
• Figure 2: Effective memory kernels in the Laplace domain: $\tilde{K}_\alpha^\mathrm{(eff)}(s)$ [Eq. \ref{['eq:memory_eff']}, upper panel] and $\tilde{K}_\beta^\mathrm{(eff)}(s)$ [Eq. \ref{['eq:kernel_laplace_b_exact']}, lower panel]. In both panels, the dotted and dotted-dashed curves depict $\tilde{K}_\alpha(s)$ and $\tilde{K}_\beta(s)$, respectively, while the dashed lines represent $\tilde{\Phi}_\alpha(s)$ and $\tilde{\Phi}_\beta(s)$ from Eqs. \ref{['eq:transmitted_laplace']} and \ref{['eq:transmitted_laplace2']}. Gray vertical lines indicate $1/\tau_\alpha$, $1/\tau_\beta$, and $1/\tau_c$ shown in Eqs. \ref{['eq:kernel_laplace_a']} and \ref{['eq:kernel_laplace_b']}. We set $\alpha=0.25$, $\beta=0.75$, $\gamma_\alpha=\gamma_\beta=1$, and varied $k$, so $\tau_c$ remains the same for all cases.
• Figure 3: The MSD of the relative displacement $r_{\alpha\beta}$. Solid lines depict the MSDs computed from Eqs. \ref{['eq:kernel_rab']} and \ref{['eq:MSD_general_confined']} for different values of $k$, while the dashed line represents the uncoupled case ($k = 0$) [Eq. \ref{['eq:MSD_rab_free']}]. The dotted and dot-dashed lines correspond to the free subdiffusion [Eq. \ref{['eq:MSD_k=0']}] governed by $K_\alpha(t)$ and $K_\beta(t)$, respectively. Gray vertical lines indicate $\tau_\alpha$, $\tau_\beta$, and $\tau_c$ shown in Eqs. \ref{['eq:MSD_rab_asymp_weak']} and \ref{['eq:MSD_rab_asymp_str']}. We used the parameters, $\alpha=0.25$, $\beta=0.75$, and $\gamma_\alpha=\gamma_\beta=1$, with $k$ varied such that $\tau_c$ remains unchanged for all cases.
• Figure 4: The MSDs and the anomalous exponents in the weak interaction regime ($k<k_c$). (a) MSDs \ref{['eq:MSD_general']} for $x_\alpha(t)$ (solid) and $x_\beta(t)$ (dashed) as a function of time $t$ for two distinct values of $k/k_c$. (b) The time-dependent anomalous exponent \ref{['eq:anomaly_exponent']} derived from the MSDs. Gray vertical lines indicate $\tau_\alpha$ and $\tau_\beta$. We set $\alpha=0.25$, $\beta=0.75$, $k=1$, $\gamma_\alpha=1$, and $k_B T=1$ and varied $\gamma_\beta$ while fixing $\tau_\alpha$ for all cases.
• Figure 5: The MSDs and the anomalous exponents in the strong interaction regime ($k>k_c$). (a) MSDs \ref{['eq:MSD_general']} for $x_\alpha(t)$ (solid) and $x_\beta(t)$ (dashed) as a function of time $t$ for different $k/k_c$. (b) The anomalous exponent \ref{['eq:anomaly_exponent']}. Gray vertical lines represent $\tau_\alpha$, $\tau_\beta$, and $\tau_c$ shown in Eqs. \ref{['eq:MSD_strong_alpha_asymps']} and \ref{['eq:MSD_strong_beta_asymps']}. We set $\alpha=0.25$, $\beta=0.75$, $k=1$, $\gamma_\alpha=1$, $k_B T=1$, and varied $\gamma_\beta$ while fixing $\tau_\alpha$ for all cases.