Jerky chiral active particles

TL;DR

This paper introduces jerky chiral active Brownian particles (jcABPs) by adding jerk to the translational dynamics of a chiral ABP, and derives exact analytic results for the Green's function, mean displacement, and mean-squared displacement across limiting cases. The authors show that jerk induces transient oscillations and modifies the classic circular swimming, causing the long-time trajectory to become a nontrivial spiral whose radius depends on the jerk, friction, and chirality timescales. With finite chirality and rotational noise, the mean trajectory emerges as a superposition of two decaying oscillatory modes, producing complex spirals that can resemble spira mirabilis or damped Lissajous patterns, depending on parameter values. These findings reveal memory-induced dynamical richness in active matter and lay groundwork for extending to many-body and continuum descriptions with delayed responses.

Abstract

We introduce jerky chiral active Brownian particles (jcABPs), a generalization of conventional chiral active Brownian particles (cABPs) subjected to jerk, the time derivative of acceleration, and analytically derive their mean displacement and mean squared displacement (MSD). Our results show that jerk induces anomalous fluctuations and oscillatory behavior on the standard circular swimming of chiral active particles. The interplay of jerk, chirality and persistence produces a family of mean trajectories including damped and exploding Lissajous patterns alongside the well-known spira mirabilis (logarithmic spirals). Our work on jerky chiral active particles opens a new route to explore rich dynamical effects in active matter.

Figures (8)

• Figure 1: Mean displacement $\langle x(t) \rangle$ in units of the persistence length $l_P$ as a function of the reduced time $t/\tau_P$ of a jerky active particle without chirality. The mean displacement grows as $t^3$ at short times and saturates to $l_P$ for $t\to\infty$, in agreement with the limits of our exact result provided in equation (\ref{['mean_wo_chirality']}). (a) The parameter values used are $\tau_J=0.2 \tau_P$, and $\tau_F=0.28 \tau_P$ corresponding to $\alpha \approx 2.55/\tau_P$. Here, $\tau_P>2\tau_J$ and there is a smooth exponential relaxation of the mean at large times. (b) The parameter values used are $\tau_J=20 \tau_P$, and $\tau_F= \tau_P$ corresponding to $\alpha \approx 1/\tau_P$. Here, $\tau_P<2\tau_J$ and we observe oscillations.
• Figure 2: Mean squared displacement in units of $l_P^2$ as a function of the reduced time $t/\tau_P$ of a jerky active particle without chirality given in equation (\ref{['msd_zero_omega']}). The value $D=0.5$ is held fixed for both plots. The MSD crosses from $\mathrm{MSD}(t)\propto t^5$ at very short times to $\mathrm{MSD}(t)\propto t$ at large times. (a) The parameter values used are $\tau_J=0.2 \tau_P$, and $\tau_F=0.28 \tau_P$ corresponding to $\alpha \approx 2.55/\tau_P$. There is a smooth exponential relaxation of the MSD at large times. (b) The parameter values used are $\tau_J=20 \tau_P$, and $\tau_F=0.2 \tau_P$ corresponding to $\alpha \approx 1/\tau_P$. We observe oscillations superimposed on exponential relaxation.
• Figure 3: Mean displacement (in units of $v_0 \tau_C$) of a jerky active particle with chirality, but in the absence of rotational noise (i.e., $D_r=0$), provided in equation (\ref{['mean_zeroDr']}). (a) The timescales are $\tau_J=\tau_C,~\tau_F=2\sqrt{2}\tau_C$ corresponding to $\alpha \approx i0.35/\tau_C$. (b) The timescales are $\tau_J=6 \tau_C,~\tau_F=2\sqrt{2}\tau_C$ corresponding to $\alpha \approx 0.34/\tau_C$. (c) The timescales are $\tau_J=10\tau_C,~\tau_F=2\tau_C$ corresponding to $\alpha \approx 0.5/\tau_C$. (d) The timescales are $\tau_J=10\tau_C,~\tau_F=\tau_C$ corresponding to $\alpha \approx 1/\tau_C$. Note that this is a close to resonance situation where $|\mathrm{Re}[\omega_j]|\approx 1/\tau_C$. (e) The timescales are $\tau_J=0.2\tau_C,~\tau_F=0.2 \tau_C$ corresponding to $\alpha \approx 4.3/\tau_C$.
• Figure 4: Mean squared displacement (in units of ${(v_0 \tau_C)}^2$) as a function of $t/\tau_C$ of a jerky active particle with chirality, but in the absence of rotational noise (i.e., $D_r=0$), provided in equation (\ref{['msd_zero_Dr']}). The value $D=0.5$ is kept fixed for both plots. The MSD crosses from $\mathrm{MSD}(t)\propto t^5$ at very short times to $\mathrm{MSD}(t)\propto t$ at large times. (a) The parameter values used are $\tau_J=0.2 \tau_C$, and $\tau_F=0.2 \tau_C$ corresponding to $\alpha \approx 4.3/\tau_C$. (b) The parameter values used are $\tau_J=10 \tau_C$, and $\tau_F=0.4 \tau_C$ corresponding to $\alpha \approx 2.5/\tau_C$.
• Figure 5: Mean displacement including initial relaxation when $\tau_P \approx 2 \tau_J$: Mean displacement in units of $l_P$ as a function of the reduced time $t/\tau_P$ of a jerky active particle with chirality and rotational noise. The parameter values $\tau_P=10,~D=0.5,~v_0=0.05$ and $\tau_C=0.05 \tau_P$ are fixed for all plots. Plots in the same row correspond to same value of $\tau_J$, but $\tau_F$ is varied. (a) $\tau_J=\tau_P, \tau_F=0.001\tau_P$. (b) $\tau_J=\tau_P, \tau_F=0.005\tau_P$. (c) $\tau_J=\tau_P, \tau_F=0.01\tau_P$. (d) $\tau_J=\tau_P, \tau_F=0.2\tau_P$. (e) $\tau_J=0.5\tau_P, \tau_F=0.001\tau_P$. (f) $\tau_J=0.5\tau_P, \tau_F=0.005\tau_P$. (g) $\tau_J=0.5\tau_P, \tau_F=0.01\tau_P$. (h) $\tau_J=0.5\tau_P, \tau_F=0.2\tau_P$. We observe disortions and beats on the spirals.