Physics of active polymers: scaling analysis via a compounding formula

Abstract

Active polymeric systems exhibit a rich spectrum of non-equilibrium phenomena arising from stochastic forces that explicitly break detailed balance. Despite the rapid growth of experimental and numerical studies, analytical progress remains limited. To date, theoretical understanding relies largely on variants of the active Rouse model, whose formal solutions, though exact, are often obscured by summations over Rouse modes and therefore provide limited direct physical insight. In this work, we develop a transparent scaling theory that captures the tagged-monomer mean-squared displacement (MSD) in active polymers through a compounding formula: the MSD of a monomer in the chain is expressed as that of an isolated active particle, modulated by a connectivity factor encoding tension propagation along the polymer backbone. This approach isolates the role of activity from that of polymer connectivity and reveals the emergent dynamical regimes in a physically intuitive manner. We test the scaling predictions against exact calculations for a broad class of generalized active polymer models driven by diverse noise statistics. The agreement demonstrates the robustness of the scaling framework across microscopic details. Our results provide a simple and extensible theoretical structure that can be applied to complex and analytically intractable active polymer systems, thereby offering a unifying perspective on non-equilibrium polymer dynamics.

Figures (7)

• Figure 1: In an active bath at steady state $m_A$ monomers (Eq. \ref{['m_A']}) are dynamically correlated, which leads to two different scaling relations for transient and steady state dynamics, Eqs. \ref{['MSD_CF_prediction_tr']} and \ref{['MSD_CF_prediction_ss']}. This correlation can be quantified from the calculation of displacement correlations reported in Sec. \ref{['sec:H']}, showing that all monomers in the block $m_A$ perform an averaged correlated motion (black arrow).
• Figure 2: Normal mode analysis for tagged monomer MSD for an active Ornstein-Uhlenbeck noise. (a) Steady-state MSD (Eq. \ref{['MSD_ss_A']}) for the generalized Rouse model with $\eta=1.75$. (b) Transient MSD (Eq. \ref{['MSD_tr_A']}) for $\eta=1.8$. The parameter $\tau_A=100$ and $\tau_0=1$ were used. The sums are extended to two $N$ (number of normal modes) corresponding to polymers of different lengths. The MSD saturates at the Rouse time $\tau_R$ as Eqs. \ref{['MSD_ss_A']} and \ref{['MSD_tr_A']} do not take into account the center of mass motion. As $\tau_R \sim N^{\eta}$ the saturation is visible for the two shorter $N$. The dashed lines are the predictions of the compounding formula approach, Eqs. \ref{['MSD_CF_prediction_tr']} and \ref{['MSD_CF_prediction_ss']}.
• Figure 3: Comparison between the MSD for the real space (thin lines) and normal mode (thick lines) for the AOU noise, showing the distinct scaling regimes for the steady state and transient cases.
• Figure 4: Plot of the short time scale behavior of the displacement correlator for thermal equilibrium dynamics $H^{(th)}(n,\tau)$ vs. $\tau$ for three values of $n$. This quantity for small $\tau$ is given by Eq. \ref{['H_th_1']}. At a time scale $\tau^*$ the displacement of monomers within a separation $n \leq 10$ start showing correlated motion (blue curve) while those with separation $n \geq 20$ are weakly correlated (green curve). Inset: The number of monomers which at time $\tau$ perform correlated motion grows as $m(\tau) \sim \tau^{1/2}$, according to the tension propagation mechanism (see text).
• Figure 5: Plot of the steady state displacement correlation $H_{ss}(n,\tau)$ vs. $\tau$ for a few values of $n$. This quantity is calculated numerically from Eq. \ref{['Hss']} using $g(u)=\exp(-u/\tau_A)$. The values used are $\tau_0=1$ and $\tau_A=100$ (dot-dashed line). The dashed lines show the $\sim \tau^2$ and $\sim \sqrt{\tau}$ behavior expected at short ($\tau \ll \tau_A$) and long ($\tau \gg \tau_A$) time scales.