Crossover dynamics and non-Gaussian fluctuations in inertial active chains

TL;DR

This work develops an analytically tractable framework for inertial active chains in one dimension, capturing the interplay of inertia, persistence, and harmonic interactions. Using a Green's-function approach, it derives MSCV and MSD, reveals six intermediate dynamical regimes with explicit crossovers, and shows steady-state, inertia-sensitive velocity statistics. Beyond second moments, the paper uncovers non-Gaussian fluctuations in ABPs through excess kurtosis and full distributions, contrasting with AOUPs, and demonstrates robust data collapses across temporal regimes. The results provide experimentally testable signatures of inertia in active matter and connect microscopic particle dynamics to multiparticle interactions in a transparent, model-compatible setting.

Abstract

We study the dynamics of inertial active particles in a one-dimensional chain with harmonic nearest-neighbor interactions, highlighting the interplay of persistence, interaction, and inertial timescales. Using a Green's function approach, we derive the mean-squared displacement (MSD) and mean-squared change in velocity (MSCV), revealing multiple crossovers between ballistic, diffusive, and subdiffusive regimes and providing analytic expressions for scaling coefficients and crossover times. Non-Gaussian deviations in active Brownian particles are captured through excess kurtosis, reflecting heavy-tailed, finite-support, or bimodal distributions that evolve systematically over time. Time-dependent probability distributions exhibit distinct data collapses within different temporal regimes, confirming the robustness of the scaling behavior. Overall, this framework connects multiparticle interactions to microscopic dynamics, revealing experimentally accessible signatures of inertia in active matter.

Figures (10)

• Figure 1: Time evolution of MSCV ($a,\,c$) and MSD ($b,\, d$) for different parameter sets $(\tau_m, \tau_k, \tau_a)$: ($i$) (0.01, 10, 10), ($ii$) (10, 100, 0.01), ($iii$) (10, 0.01, 10), ($iv$) (0.01, 100, 0.01), ($v$) (0.01, 5, 100), and ($vi$) (10, 0.01, 0.1). Solid lines show numerical integrations of Eqs. (\ref{['bulk_msv_full_form']}) and (\ref{['bulk_msd_full_form']}), respectively, for MSCV and MSD, and points denote simulations at fixed activity $v_0 = 1$. Arrows indicate crossover times, and panels ($c,d$) illustrate corresponding dynamical scalings.
• Figure 2: ($a$) Variation of the steady-state MSCV $\langle \Delta v^2 \rangle_{\rm ss}$ as a function of inertia $\tau_m$ for a few values of $\tau_k$ at $\tau_a = 1$. ($b$) variation of $\Delta^v_{ss}$ as a function of interaction time $\tau_k$ for a few values of $\tau_m$ at $\tau_a = 1$ and ($c$) variation of $\Delta^v_{ss}$ as a function of persistent time $\tau_a$ for a few values of $\tau_{m}$ and $\tau_k$. The symbols denote the data points from the simulation, and the solid line is the plot of steady state MSCV presented in Eq.\ref{['eq:ss_mscv']}. The activity was kept fixed as $v_0 = 1$.
• Figure 3: Velocity autocorrelation and spatial velocity correlations. ($a$) Normalized velocity autocorrelation $C_{vv}(t)/C_{vv}(0)$ and ($b$) normalized equal-time spatial velocity correlation $C_{l,l+\Delta r}^v(0)/C_{l,l}^v(0)$ as a function of scaled distance $\Delta r/N = (m - N/2)/N$. Parameter sets $(\tau_m, \tau_k, \tau_a)$ are (I) (10, 0.01, 0.1), (II) (0.1, 0.01, 0.1), (III) (0.1, 0.01, 10), (IV) (0.1, 10, 0.1), (V) (0.1, 10, 10), (VI) (0.1, 0.02, 0.1), (VII) (10, 0.1, 10), and (VIII) (0.1, 0.1, 10), with fixed activity $v_0 = 1$. Oscillatory decay in ($a$) occurs at large inertia, while small inertia yields monotonic decay. In ($b$), the correlation length increases for weak interaction, small inertia, or large persistence. Points denote simulation results and solid lines represent numerical integration of Eq. (\ref{['eq:corrv_comp']}). Simulations and integrations in ($b$) were performed with $N = 64$ beads, whereas panel ($a$) used $N=256$.
• Figure 4: Excess kurtosis for velocity and displacement, ${\cal K}_{\Delta v}$ ($a$) and ${\cal K}_{\Delta x}$ ($b$), versus time for inertia values given in the legend of panel ($b$). We use $v_{\rm abp} = \sqrt{2}$. Solid lines correspond to $\tau_{\rm abp} = 0.05$, $\tau_k = 20$, while dashed lines correspond to $\tau_{\rm abp} = 50$, $\tau_k = 1$. Insets ($i$) -- ($iv$) show the probability distributions of velocity changes, and insets ($v$) -- ($vi$) those of displacements. Insets ($i$), ($iv$), ($v$), and ($vi$) correspond to the blue-curve parameters: $\tau_{\rm abp} = 0.05$, $\tau_k = 20$, $\tau_m = 0.01$, while ($ii$) and ($iii$) correspond to the red-curve parameters with $\tau_m = 1$ and identical remaining parameters. Insets ($i$), ($ii$), and ($v$) are evaluated at $t = 0.001$, and ($iii$), ($iv$), and ($vi$) at $t = 100$.
• Figure 5: ($a$)–($c$) Time evolution and scaling collapse of $P(\Delta v, t)$. Insets show data collapse with $P(\Delta v, t) = t^{-\mu} f_{\Delta v}(\Delta v/t^\mu)$: $\mu = 1$ (ballistic, $a$), $1/2$ (diffusive, $b$), $1/4$ (sub-diffusive, $c$). Parameters $v_{abp}=\sqrt{2}$, $\tau_m=0.01$, $\tau_k=5$, $\tau_a=100$ are the same as for line plot ($v$) in Fig. \ref{['fig:msd_msv']}($a$) and ($b$).