Confinement-Induced Delay in Chiral Active Brownian Particles

TL;DR

The paper addresses how harmonic confinement induces dynamical delays in two-dimensional chiral Active Brownian Particles, revealing a confinement-induced delay function that breaks time-reversal symmetry in an overdamped setting. It develops analytical expressions for mean positions, MSD, orientation correlations, and a delay function, complemented by simulations, to map how trap strength and rotational noise shape the dynamics. A central finding is the steady-state radius $R_{\text{st}}= v_0/\sqrt{(\mu k)^2+\Omega^2}$ and the transition of the stationary distribution from ring-like to localized as confinement strengthens, illustrating how activity and confinement jointly sculpt nonequilibrium steady states. The results offer quantitative observables for experiments and guide understanding of directed transport and irreversibility in confined active systems.

Abstract

We investigate the interplay between chirality and confinement in harmonically trapped active particles. The circular character of chiral motion combines with the radial symmetry of the potential to create distinctive non-equilibrium behavior. Chirality induces oscillatory cross-correlations between positional components that vanish in the absence of torque while the harmonic potential generates a finite delay between orientation and velocity - a signature of time-reversal symmetry breaking distinct from inertial delay mechanisms. The delay function exhibits characteristic temporal evolution with depth and persistence controlled by trap strength and rotational noise. The stationary probability distribution displays strongly non-Maxwellian characteristics, transitioning from broad annuli to compact localized peaks as confinement increases with the distribution radius governed by the competition between chiral propulsion and trap strength. These features emerge from the interplay between chiral swimming and the restoring force of the trap, revealing how confinement and activity jointly shape particle dynamics and transport properties in nonequilibrium steady states.

Figures (9)

• Figure 1: Simulated trajectories of a chiral active Brownian particle. Panels (a, b) correspond to a weak trap ($k=0.08$) and (c, d) to a strong trap ($k=1.0$). The angular velocity is $\Omega = +2.87$ for (a, c) and $\Omega = -2.87$ for (b, d). The particle's path is the solid line, the orientation vector $\hat{\mathbf{n}}$ is shown by red arrows, and the instantaneous velocity $\mathbf{v}$ is shown by green arrows. Fixed simulation parameters are $v_0=1.0$, $D_r=0.005$, and $D_t=0.02$.
• Figure 2: Mean radial position $\langle r(t)\rangle$ of a chiral active particle in a harmonic trap. Solid lines denote theory, and symbols denote simulation obtained from the model directly. Panels show variation with (a) $\Omega$, (b) $k$, (c) $D_r$, and (d) $v_0$. Note: we compute the magnitude of the mean position vector, $|\langle r(t)\rangle|$ not the mean of the instantaneous magnitude, $\langle |r(t)|\rangle$.
• Figure 3: Mean square displacement $\langle r^2(t) \rangle$ for active Brownian particles with harmonic confinement and external torque. (a) Variation with confinement strength $k$ at fixed angular velocity $\Omega = 2$. (b) Variation with angular velocity $\Omega$ at fixed confinement strength $k = 0.1$. Other parameters: $v_0 = 1$, $D_t = 0.02$, $D_r = 0.001$.
• Figure 4: Mean squared displacement (MSD) of active Brownian particles from simulations (symbols) and theoretical predictions (black lines). (a) MSD for varying angular velocities $\Omega$. (b) MSD for varying rotational diffusion coefficients $D_r$. Insets show the corresponding effective diffusion coefficient $D_{\mathrm{eff}}$ as a function of $\Omega$ (left) and $D_r$ (right). The fixed parameters are $v_0 = 1.5$ and $D_t = 0.05$.
• Figure 5: Comparison of analytical (black lines) and simulated (markers) cross-correlation functions $\langle x(t)y(t) \rangle$ for a chiral active Brownian particle. (a) Effect of chirality $\Omega = 0.5\,, 1\,, 2$ at fixed trap stiffness $\kappa = 0.5$. (b) Effect of trap stiffness $\kappa = 0, 0.5, 2$ at fixed chirality $\Omega = 1$. Other parameters: $v_0 = 1$, $D_r = 0.002$, $\mu = 1$.