Characterizing exact dynamics of a trapped active Brownian particle under torque in two and three dimensions

Abstract

The interplay of chirality, self-propulsion, and spatial confinement generates striking non-equilibrium fluctuations whose higher-order statistics carry information about the dynamics and shape of the position distribution. Here, we present an exact analytical framework, based on a Laplace-transform solution of the Fokker-Planck equation, for the transient dynamics of a chiral active Brownian particle in a harmonic trap, in both two and three dimensions. We obtain closed-form expressions for all time-dependent moments up to fourth order, enabling a complete characterization of the excess kurtosis throughout the transient and steady-state regimes. In two dimensions, the excess kurtosis exhibits a damped oscillatory response with multiple re-entrant crossovers, evolving from negative values that reflect active off-centered ring-like position distributions to positive values characteristic of heavy-tailed fluctuations. This damped oscillatory excess kurtosis appears both for free and harmonic confinement, although increasing the trapping stiffness progressively suppresses it, and the positive excess kurtosis eventually vanishes at sufficiently high stiffness. In contrast, in three dimensions, the excess kurtosis remains negative, indicating a robustly active non-Gaussian state characterized by half-ring-like to band-like position distributions in the two-dimensional plane spanned by the torque axis and its normal radial direction. Our results demonstrate how chirality, propulsion, and confinement, together with dimensionality, shape transient dynamics, while providing experimentally accessible signatures of confined chiral active dynamics.

Figures (8)

• Figure 1: Schematic of an active Brownian particle under torque in a harmonic trap. The particle self-propels with active speed $v_0$ along its orientation vector $\mathbf{\hat{u}}$, possesses intrinsic chirality or experiences an applied torque $\omega$ that rotates its orientation, and is confined by a trap of stiffness $k$. The combined action of torque, propulsion, and confinement generates a characteristic circular trajectory in two dimensions or a spiral trajectory in three dimensions within the trapping potential.
• Figure 2: Mean-squared displacement (MSD) of a chiral active Brownian particle (CABP) in a harmonic trap in two dimensions (2d) as a function of time for (a) chirality $\Omega=1,10,100$ with activity $\text{Pe}=100$ and harmonic trap stiffness $\beta=1$, (b) $\text{Pe}=10, 100, 1000$ with $\Omega=100$ and $\beta=1$, and (c) $\beta=0.1,1,10$ with ${\rm{Pe}}=100$ and $\Omega=100$. Steady state MSD as a function of (a) $\Omega$ for $\beta=0.1,1,10$ with ${\rm{Pe}}=100$, (b) ${\rm{Pe}}$ for $\Omega=1,10,100$ with $\beta=1$, (c) $\beta$ for $\Omega=1,10,100,1000$ with ${\rm{Pe}}=100$. The lines in (a), (b) and (c) are plots of Eq. (\ref{['eq:msd_2d_dimensionless']}) and the points are from simulations. The lines in (d), (e), and (f) are plots of Eq. \ref{['eq:msd_2d_st']}.
• Figure 3: Excess kurtosis of a chiral active Brownian particle (CABP) in a harmonic trap in two dimensions (2d) as a function of time for (a) chirality $\Omega=1,10,100, 1000$ with activity $\text{Pe}=100$ and harmonic trap stiffness $\beta=1$, (b) $\text{Pe}=10, 100, 1000$ with $\Omega=100$ and $\beta=1$, and (c) $\beta=0.1,1,10$ with ${\rm{Pe}}=100$ and $\Omega=100$. The lines in (a), (b) and (c) are plots of Eq. (\ref{['ex_kurt_2d']}) and the points are from simulations. (d) Three representative points with negative, positive, and negative excess kurtosis are selected to examine the corresponding position distributions in the $x-y$ plane (f-h) and in the radial direction (j-l). (e) Analytic orientation autocorrelation function $\langle{\hat{\textbf{u}}}\cdot{\hat{\textbf{u}}}_0\rangle=e^{- \tilde{t}}\cos (\Omega \tilde{t})$(see detailed derivation in Appendix-\ref{['app-A']}) showing oscillations that occur in phase with the excess kurtosis. (f,h) Bimodal position distributions associated with the negative excess kurtosis active states (see (i) and (iii) in (d)), with the corresponding radial distributions in (j) and (l) exhibiting clear deviations from the Gaussian profile obtained from the MSD(Eq. \ref{['eq:msd_2d_dimensionless']}). (g) Unimodal position distribution associated with the positive excess kurtosis state (see (ii) in (d)), with the corresponding radial distribution in (k) displaying a weak heavy-tailed deviation from the Gaussian(see inset of (k)).
• Figure 4: Mean-squared displacement (MSD) of an active Brownian particle under torque in a harmonic trap in three dimensions (3d) as a function of time for (a) chirality $\Omega=1,10,100$ with activity $\text{Pe}=100$ and harmonic trap stiffness $\beta=1$, (b) $\text{Pe}=10, 100, 1000$ with $\Omega=100$ and $\beta=1$, and (c) $\beta=0.1,1,10$ with ${\rm{Pe}}=100$ and $\Omega=100$. Steady state MSD as a function of (a) $\Omega$ for $\beta=0.1,1,10$ with ${\rm{Pe}}=100$, (b) ${\rm{Pe}}$ for $\Omega=1,10,100$ with $\beta=1$, (c) $\beta$ for $\Omega=1,10,100,1000$ with ${\rm{Pe}}=100$. The lines in (a), (b) and (c) are plots of Eq. (\ref{['eq:msd_3d_dimensionless']}) and the points are from simulations. The lines in (d), (e), and (f) are plots of Eq. (\ref{['eq:msd_3d_st_dimensionless']}).
• Figure 5: Excess kurtosis of an active Brownian particle under torque in a harmonic trap in three dimensions (3d) as a function of time for (a) torque $\Omega=1,10,100$ with activity $\text{Pe}=100$ and harmonic trap stiffness $\beta=1$, (b) $\text{Pe}=10, 100, 1000$ with $\Omega=100$ and $\beta=1$, and (c) $\beta=0.1,1,10$ with ${\rm{Pe}}=100$ and $\Omega=100$. Lines represent analytic predictions, while points denote simulation results. (d,e) Two-dimensional projections of the position distribution in the $(\tilde{r}_{\perp}, \tilde{z})$ plane(where $\tilde{r}_{\perp} = \sqrt{\tilde{x}^{2} + \tilde{y}^{2}}$) at four different time snapshots in the active regimes($\tilde{\cal{K}}<0$) for weak torque $\Omega = 1$ (d) and strong torque $\Omega = 100$ (e) with $\mathrm{Pe} = 100$ and $\beta = 1$. In the active regime, the distribution evolves from an initial half-ring–like structure to a long-time off-centered half ring at weak torque, and to band-like structures at strong torque, reflecting how increasing rotational precession suppresses radial spreading and enhances anisotropic active motion.