Enhanced particle diffusion in fluctuating binary environments

Abstract

We investigate single-particle diffusion in a two-state Langevin model where the friction coefficient randomly switches between low-friction (liquid-like) and high-friction (glassy-like) states. The dynamics are governed by the ratio between the friction switching time $τ$ and the intrinsic velocity relaxation time $τ_0$. For fast switching ($τ/τ_0 \lesssim 1$) the motion is homogeneous and Brownian, whereas for slow switching ($τ/τ_0 \gg 1$) the particle exhibits intermittent dynamics and an enhanced diffusion coefficient. Analysis of the single-particle overlap function $Q(t)$ and the dynamic susceptibility $χ_4(t)$ reveals decoupling of the diffusion coefficient from the average friction upon cooling, which coincides with increasing temporal dynamic heterogeneity. This minimal model provides a transparent framework for understanding single-particle transport in media with fluctuating local mobility, including supercooled liquids and phase-separated soft materials.

Figures (4)

• Figure 1: Representative particle trajectories from simulations of the two–state Langevin model, color–coded by the instantaneous diffusion coefficient. (a) Fast–switching regime, $\tau/\tau_0 \approx 1$, corresponding to effectively homogeneous Brownian motion. (b) Slow–switching regime, $\tau/\tau_0 \gg 1$, where the particle exhibits extended displacements during long low–friction intervals, leading to temporally heterogeneous dynamics and enhanced diffusion. All trajectories are shown over the same total time $t_{\max}=100$ and simulation parameters $T=1$, $\gamma_\mathrm{l}=1$, and $\gamma_\mathrm{g}=100$.
• Figure 2: (a) Mean–squared displacement $\langle \Delta r^2(t) \rangle$ for different ratios $\tau/\tau_0$, illustrating the crossover from homogeneous to heterogeneous dynamics. (b) Normalized transport coefficients: open circles show $D/(k_B T\,\mu)$ testing the Einstein relation, while open squares show $D\,\langle \gamma \rangle/(k_B T)$ testing the relation $\mu = 1/\langle \gamma \rangle$. The solid line is calculated based on Eq. \ref{['eq:diffusion_enhancement']}. The Einstein relation holds for all $\tau/\tau_0$, whereas $\mu = 1/\langle \gamma \rangle$ fails in the slow–switching regime, leading to the observed enhancement of $D/D_0$.
• Figure 3: (a) Temperature dependence of the friction coefficients for the liquid–like (Arrhenius) and glass–like (VFT) states, together with their mean value $\langle\gamma(T)\rangle = \tfrac{1}{2}(\gamma_\mathrm{l}+\gamma_\mathrm{g})$. (b) Diffusion coefficient $D$ as a function of reduced temperature $T_0/T$ for fast ($\tau/\tau_0 \lesssim 1$, red circles) and slow ($\tau/\tau_0 \gg 1$, blue squares) switching regimes. The dashed line shows the relation $D_0 = k_B T / \langle\gamma(T)\rangle$ and the solid line is calculated based on Eq. \ref{['eq:diffusion_enhancement']}. Deviations in the slow–switching limit reflect the growing dominance of the high–friction state and the onset of heterogeneous dynamics.
• Figure 4: Analysis of dynamical heterogeneity at fixed switching time $\tau=100$. (a) Single-particle overlap function $Q(t)$ for several temperatures, showing slower decay and a pronounced plateau upon cooling. (b) Four-point susceptibility $\chi_4(t)$ with an increasing peak height and shift to longer times at lower $T$, reflecting enhanced temporal intermittency. (c) Comparison of relaxation times $\tau_\alpha$ (red circles) from $Q(t)$ and $\tau_{\chi_4}$ (blue squares) from the $\chi_4(t)$ peak, showing similar temperature dependence. (d) The $\mu = 1/\langle \gamma \rangle$ deviation quantified via $D\tau_\alpha$ (red circles), $D\tau_{\chi_4}$ (blue squares), and $D\langle\gamma\rangle/T$ (green triangles, right axis). All three quantities increase below $T_0/T \approx 0.7$, indicating enhanced diffusion and heterogeneous dynamics in the slow-switching regime.