Jerky Chiral Active Brownian Particles

• jcABPs are chiral active Brownian particles enhanced by a third-order (jerk) term that transforms standard circular motion into complex Lissajous and spiral trajectories.
• The model defines multiple timescales—including jerk-to-inertia, frictional, orientational persistence, and chiral rotation—that determine whether translational relaxation is oscillatory or exponential.
• Analytical results show a t³ mean displacement and t⁵ mean squared displacement at short times, while long-time diffusion remains independent of the jerk and inertia contributions.

Searching arXiv for recent and baseline papers on jerky chiral active Brownian particles and adjacent chiral-ABP models. First, I’ll look for the explicit jcABP paper and then fetch neighboring references on smooth chiral ABPs, inertial chiral ABPs, and transiently chiral/jump-driven models. Jerky chiral active Brownian particles (jcABPs) are a higher-derivative extension of two-dimensional chiral active Brownian particles in which translational motion is governed not only by friction and inertia but also by jerk, the time derivative of acceleration. In the explicit jcABP construction, the standard circular swimming of a chiral active particle is modified by a third-order translational equation, producing anomalous fluctuations, oscillatory corrections to mean motion, and a family of mean trajectories that includes damped and exploding Lissajous patterns together with the spira mirabilis of noisy chiral swimmers (Jose et al., 25 Aug 2025).

1. Minimal jcABP model

The defining stochastic equations are

$\lambda \dddot{\vec r}(t)+m\ddot{\vec r}(t)+\gamma \dot{\vec r}(t) =\gamma v_0 \hat n(t)+\sqrt{2D\,\gamma^2}\,\vec\eta(t), \qquad \dot\theta(t)=\omega_0+\sqrt{2D_r}\,\eta_r(t),$

with $\vec r(t)=(x(t),y(t))$, $\hat n(t)=(\cos\theta,\sin\theta)$, mass $m$, friction $\gamma$, jerk coefficient $\lambda$, propulsion speed $v_0$, translational diffusion coefficient $D$, rotational diffusion coefficient $D_r$, and imposed chiral angular velocity $\omega_0$ (Jose et al., 25 Aug 2025). In this formulation, jerk enters linearly through $\vec r(t)=(x(t),y(t))$0, while chirality remains encoded in the standard rotational drift-noise equation for $\vec r(t)=(x(t),y(t))$1.

The characteristic times are

$\vec r(t)=(x(t),y(t))$2

These respectively set the jerk-to-inertia timescale, a frictional timescale, orientational persistence, and the chiral rotation period scale (Jose et al., 25 Aug 2025). Natural units are the persistence time $\vec r(t)=(x(t),y(t))$3 and persistence length $\vec r(t)=(x(t),y(t))$4, and the standard initial condition used in the exact analysis is

$\vec r(t)=(x(t),y(t))$5

The model interpolates between several familiar active-particle limits. When $\vec r(t)=(x(t),y(t))$6, the jerk term disappears and one recovers the underdamped active Brownian or chiral active Brownian particle with inertia $\vec r(t)=(x(t),y(t))$7. When $\vec r(t)=(x(t),y(t))$8 as well, the translational dynamics becomes overdamped and reduces to the standard ABP/cABP. When $\vec r(t)=(x(t),y(t))$9, the model becomes jerky but achiral (Jose et al., 25 Aug 2025).

2. Linear response structure and translational modes

Because the translational equation is linear, the exact solution is organized by a scalar Green’s function $\hat n(t)=(\cos\theta,\sin\theta)$0 satisfying

$\hat n(t)=(\cos\theta,\sin\theta)$1

with Fourier transform

$\hat n(t)=(\cos\theta,\sin\theta)$2

The nonzero poles are

$\hat n(t)=(\cos\theta,\sin\theta)$3

and the frequency parameter

$\hat n(t)=(\cos\theta,\sin\theta)$4

controls whether translational relaxation is oscillatory or purely exponential (Jose et al., 25 Aug 2025).

Oscillatory transients occur when

$\hat n(t)=(\cos\theta,\sin\theta)$5

so that $\hat n(t)=(\cos\theta,\sin\theta)$6 is real. If $\hat n(t)=(\cos\theta,\sin\theta)$7 is imaginary, the jerk-induced translational mode decays without oscillation. For positive $\hat n(t)=(\cos\theta,\sin\theta)$8, the imaginary parts of the poles are negative and the response remains bounded. If signs are changed, notably $\hat n(t)=(\cos\theta,\sin\theta)$9, unstable exponentially growing modes appear; this is the origin of the “exploding” branch of jcABP mean trajectories (Jose et al., 25 Aug 2025).

The exact position is the convolution

$m$0

This linear-response structure is central: chirality enters through the orientation kernel, while jerk enters through $m$1. Their interference produces the characteristic mixed spiral–Lissajous morphology of jcABP motion (Jose et al., 25 Aug 2025).

3. Mean trajectories and fluctuation laws

In the nonchiral limit, the mean drift is along the initial heading and acquires a jerk-specific damped oscillatory correction,

$m$2

Its short-time behavior is

$m$3

showing that jerk replaces the usual initial ballistic trend by a $m$4 law for the mean displacement (Jose et al., 25 Aug 2025).

The nonchiral mean-squared displacement has the striking short-time asymptotic

$m$5

while at long times

$m$6

Thus the short-time fluctuation law is anomalously steep, $m$7, whereas the long-time diffusion coefficient loses memory of both jerk and inertia in the achiral case (Jose et al., 25 Aug 2025).

With finite chirality and rotational noise, the mean displacement decomposes into two damped modes,

$m$8

$m$9

with spiral center

$\gamma$0

The first mode is the standard cABP logarithmic spiral; the second is a jerk-induced damped oscillatory mode. Because generally $\gamma$1 and $\gamma$2, the second mode is not circular but Lissajous-like (Jose et al., 25 Aug 2025).

This decomposition yields the trajectory taxonomy reported for jcABPs. Standard spira mirabilis is recovered asymptotically when $\gamma$3, because the jerk mode decays faster. Damped Lissajous patterns occur when $\gamma$4, so the jerk mode is longer-lived. Beat-like and lobed patterns appear when $\gamma$5 is comparable to $\gamma$6. Exploding Lissajous patterns arise for $\gamma$7, where the jerk-induced envelope $\gamma$8 grows rather than decays (Jose et al., 25 Aug 2025).

For finite chirality, the long-time diffusion coefficient is

$\gamma$9

A notable result is that this coefficient does not depend on jerk or inertia, even though transient trajectories and short-time MSD exponents are dramatically altered (Jose et al., 25 Aug 2025).

4. Neighboring formulations of “jerky” chiral motion

The explicit jcABP is only one member of a broader family of chiral-active models with delayed, intermittent, or higher-order kinematics. A useful distinction is between explicit translational jerk, finite velocity relaxation, jump-driven reorientation, and overdamped phase lag.

Model family	Defining mechanism	Distinctive consequence
Explicit jcABP	Third-order translational equation with $\lambda$0	$\lambda$1 MSD, jerk-induced oscillatory mode, Lissajous and spira-mirabilis interference
Inertial cABP	Underdamped translational velocity relaxation	VACF factorizes into inertial and chiral envelopes; long-time diffusion equals overdamped cABP
Transiently chiral active particles	Poisson tumbles with diffusing reorientation angle $\lambda$2	Intermittent chirality, handedness over $\lambda$3 tumbles, fixed-$\lambda$4 transport effects
Confined overdamped cABP	Harmonic-trap-induced lag between orientation and velocity	Finite delay without inertia or jerk

In the inertial two-dimensional cABP,

$\lambda$5

the velocity autocorrelation factorizes into an inertial envelope and a chiral envelope, a nonzero perpendicular velocity component measures inertial lag, and the long-time positional diffusion equals the overdamped cABP value, independent of mass (Pattanayak et al., 23 Nov 2025). This is closely related to jcABPs if “jerky” is understood as finite velocity relaxation, but it is not the explicit third-derivative model.

A different neighboring construction introduces transiently chiral active particles as “active Brownian particles that undergo tumbles via a diffusing reorientation angle,” producing a jump-diffusion angular process with temporary handedness rather than a fixed $\lambda$6 (Britton et al., 2 Jul 2025). By contrast, the explicit jcABP keeps the continuous cABP orientation dynamics and places the higher-order structure in the translational sector.

A common misconception is that any orientation–velocity lag implies inertial or jerky dynamics. The trapped overdamped cABP literature shows that a finite orientation–velocity delay can arise in a purely overdamped model solely because of harmonic confinement; the delay function

$\lambda$7

is already nonzero without translational inertia or jerk (Barman, 17 Oct 2025).

For nonchiral but explicit jerk, a complementary baseline is the jerk-active-particle theory with active Ornstein–Uhlenbeck propulsion, where giant superballistic MSD exponents $\lambda$8, $\lambda$9, $v_0$0, and $v_0$1 arise from competition between jerk, inertia, damping, and activity (Löwen, 11 Jul 2025). This suggests that anomalously high transient exponents are a generic signature of jerk-dominated transport, even before chirality is introduced.

5. Smooth-chirality baselines and many-body reference problems

The explicit jcABP theory is a single-particle theory. For collective behavior, the natural reference point is the smooth chiral ABP or Brownian circle-swimmer literature. In interacting two-dimensional circle swimmers with

$v_0$2

a continuum reduction gives

$v_0$3

and a spinodal criterion $v_0$4 for motility-induced phase separation (MIPS). The critical point is reported as

$v_0$5

Increasing chirality suppresses MIPS by shortening effective persistence and shifts the MIPS region toward higher $v_0$6 and higher $v_0$7 (Bickmann et al., 2020).

At stronger torque, the many-body cABP literature reports an interruption of conventional MIPS by a dynamical clustering state. In overdamped repulsive circle active Brownian particles, sufficiently large $v_0$8 produces a finite-wavelength instability, multiple dynamically reconfiguring clusters, and a nonvanishing circulating current; the paper argues that no equilibrium-like phase-separation theory can be constructed for chiral active colloids even with tiny active torque (Ma et al., 2021).

Dense chiral fluids near arrest display an additional mechanism, “hammering,” in which rapidly spinning particles repeatedly collide with the same cage neighbors and fluidize an otherwise glassy state. The large-persistence regime can then split into collective swirling/fluidization at low spin and practically frozen absorbing behavior at high spin (Debets et al., 2022). Because the cited explicit jcABP work does not treat interactions, these smooth-cABP results are best interpreted as baseline limits. This suggests that future jcABP many-body theories should test whether translational jerk enhances, suppresses, or qualitatively reorganizes MIPS suppression, dynamical clustering, and hammering.

6. Conceptual status, extensions, and experimental relevance

The explicit jcABP is best defined narrowly: a chiral ABP whose translational dynamics contains a third-order derivative term $v_0$9 (Jose et al., 25 Aug 2025). Two broader usages of “jerky chiral motion” appear in adjacent literature but are not equivalent. One is renewal-style interruption of circular motion through stochastic position–orientation resetting, which produces piecewise chiral segments and a criterion $D$0 for oscillatory versus non-oscillatory orientation correlations (Shee, 17 Aug 2025). Another is experimentally programmed active motion combining light-modulated propulsion and magnetic steering, which realizes straight runs, circular motion, polygons, Lévy walks, run-and-tumble dynamics, nested polygons, and on-demand switching between modes within a single colloidal platform (Raghavendra et al., 29 Apr 2026). These systems are close to jcABP phenomenology at the trajectory level, but they are control protocols rather than the explicit jerk equation.

The conceptual boundaries are therefore sharp. A jcABP is not merely an inertial cABP, because translational inertia introduces finite velocity relaxation without an explicit jerk variable. It is not merely a jump-driven transiently chiral particle, because Poissonian tumbles with a diffusing reorientation angle act in orientation space rather than through $D$1. And it is not merely a confined cABP with delay, because confinement alone can generate orientation–velocity lag without higher-order kinematics (Pattanayak et al., 23 Nov 2025, Britton et al., 2 Jul 2025, Barman, 17 Oct 2025).

Within those boundaries, the cited literature supports a compact characterization. jcABPs provide a tractable higher-derivative generalization of chiral active Brownian motion in which jerk leaves long-time diffusion unchanged in the analyzed single-particle settings but profoundly reshapes transient transport: mean displacement begins as $D$2, MSD as $D$3, deterministic circular motion acquires jerk-dependent radius and phase, and noisy chiral spirals are deformed by a second damped oscillatory mode into Lissajous-like patterns (Jose et al., 25 Aug 2025). The smooth-circle-swimmer, inertial-cABP, transiently chiral, resetting, and confined-delay literatures then supply the natural neighboring limits against which jcABP behavior can be classified and extended (Bickmann et al., 2020, Pattanayak et al., 23 Nov 2025, Britton et al., 2 Jul 2025, Shee, 17 Aug 2025, Barman, 17 Oct 2025).