Active Brownian particles in power-law viscoelastic media

TL;DR

This work models active Brownian particles in power-law viscoelastic media by extending the fractional Langevin equation to include independent memory for translation and rotation. Using Caputo-type derivatives with memory kernels $K_T(t) ∝ t^{-\alpha_T}$ and $K_R(t) ∝ t^{-\alpha_R}$, the authors obtain analytical solutions and develop a robust L1 discretization scheme to simulate the dynamics, validated against analytical results for the mean squared displacement (MSD) and orientation correlations. A Gaussian closure yields a tractable autocorrelation for the self-propulsion direction, revealing that memory ($\alpha_R<1$) prolongs orientation persistence via $\tau_p = (2/D_R)^{1/\alpha_R} \Gamma(1+1/\alpha_R)$, with $\alpha_R = 2 - 2 H_R$ linking to fluctuation-dissipation. The MSD exhibits multiple diffusion regimes—short-time subdiffusion $t^{\alpha_T}$, a ballistic-like persistence, a mid-time superdiffusive $t^{2-\alpha_R}$ phase, and long-time normal diffusion with a renormalized coefficient $D_l$—whose crossover times depend on both $\alpha_T$ and $\alpha_R$. Overall, the study uncovers a rich dynamical landscape for active particles in scale-free media and provides a versatile framework for exploring transport in gels, polymer networks, and crowded cellular environments.

Abstract

Many active particles are embedded in environments that exhibit viscoelastic properties. An important class of such media lacks a single characteristic relaxation timescale when subjected to a time-dependent stress. Rather, the stress response spans a broad continuum of timescales, a behavior naturally described by a scale-free, fractal-like power-law relaxation modulus. Using a generalization of the fractional Langevin equation, we investigate an active Brownian particle embedded in a power-law viscoelastic environment with translational and rotational dynamics governed by independent fractional orders. We solve the model analytically, develop a numerical scheme to validate the theoretical predictions, and provide tools that can be used in further studies. A rich variety of diffusion regimes emerges, which modify the intermediate-time behavior of the mean squared displacement. Notably, we find that the competition between translational and rotational contributions favors a superdiffusive persistence over the standard ballistic motion, and over-stretches its characteristic timescale, fundamentally altering the standard relation between persistence and propulsion in active matter.

Figures (3)

• Figure 1: Dynamics of the angular component of an active Brownian particle for different fractional orders $\alpha_R$. Analytical expressions are shown as solid brown lines, while colored markers represent sampled points from numerical simulations with $95\%$ confidence bands. (a) Angular MSD in units of $\textrm{rad}^2$. The inset illustrates the thermalization of the numerical solutions toward the overdamped limit for $\alpha_R=0.2$ (orange) and $\alpha_R=0.7$ (yellow), using a time step $h=0.005$. (b) Autocorrelation function of the self-propulsion direction. All curves are normalized by the rotational timescale $\tau_R = (\eta_R/k_BT)^{1/\alpha_R}$.
• Figure 2: Active diffusion regimes for $\alpha\equiv\alpha_R=\alpha_T$. (a)-(c) MSD for $\upsilon\tau_R/L=20$ (solid orange line) and $\upsilon\tau_R/L=\infty$ (solid navy) for $\alpha={0.7,0.5,0.3}$, respectively. Bottom-right insets show the comparison between analytical and numerical results, while top-left insets present a zoom into the $t^{2-\alpha_R}$ superdiffusive regime. Crossover times between diffusion regimes are indicated by grey lines using the convention: $\tau_{\alpha_T}$-solid, $\tau_{2}$-dashed, $\tau_{P}$-dotted and $\tau_l$-dotted-dashed. (d) Summary of the diffusion regimes. (e) Crossover times as a function of $\alpha$. (f) Representative particle trajectories for each $\alpha$. All times-scales are in units of $\tau_R$, and all spatial coordinates are normalized by the characteristic length $\upsilon\tau_R$.
• Figure 3: Diffusion regimes for decoupled translational and rotational orders. Analytical MSDs are presented in solid colored lines while numerical results are shown in purple in the figure insets. The relevant timescales $\tau_{\alpha_T}$ and $\tau_p$ are represented, respectively, in vertical solid and dotted lines, following the color convention of the MSD. (a) MSD for fixed $\alpha_T=0.7$. (b) MSD for $\alpha_R=0.7$. Times are normalized by $\tau_R$ and spatial variables by the characteristic length $\upsilon\tau_R$.