Chiral Active Brownian Particle (cABP)

• cABP is defined as an active particle model with a constant intrinsic angular drift that transforms straight runs into circular or helical paths.
• It demonstrates unique confinement effects, including oscillatory orientation memory and non-Boltzmann stationary states due to geometric influences.
• Interactions among cABPs lead to modified phase behavior—from suppression of motility-induced phase separation to emergent dynamical clustering and crystallization.

A chiral Active Brownian Particle (cABP), also written CABP in parts of the literature, is an active Brownian particle whose propulsion direction undergoes a constant intrinsic angular drift in addition to rotational diffusion. In two dimensions this minimal modification turns persistent straight runs into circular or spiral-like motion, while in three dimensions a constant torque vector produces helical trajectories (Ma et al., 2021). Across the literature represented here, the cABP serves as a canonical “circle swimmer” model for studying how broken mirror symmetry reshapes single-particle transport, confinement, nonequilibrium steady states, and collective organization from motility-induced phase separation to crystallization and glassy dynamics (Sevilla, 2016).

1. Minimal model and defining kinematics

In a standard two-dimensional formulation, particle $i$ has position $\mathbf r_i$, orientation angle $\theta_i$, and propulsion direction

$\mathbf e_i=(\cos\theta_i,\sin\theta_i).$

Its overdamped dynamics is written as

$$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$

so that $\omega_0$ acts as a constant active torque and a free particle follows a circular trajectory of radius $R=v_0/\omega_0$ in the deterministic dilute limit (Ma et al., 2021). A closely related notation writes the orientation equation as

$$\dot{\hat{\mathbf e}}_i=\left(\Omega \hat{\mathbf z}+\sqrt{\frac{2}{\tau_p}}\,\boldsymbol{\eta}_i\right)\times \hat{\mathbf e}_i,$$

with the same orbital radius $R=v_0/\Omega$ and persistence time $\tau_p$ (Jeong et al., 25 Jun 2025).

Several dimensionless ratios organize the 2D problem. The interacting-circle-swimmer literature uses $\mathbf r_i$0 as a direct measure of chirality relative to orientational decorrelation, together with rotational Péclet number $\mathbf r_i$1 and area fraction $\mathbf r_i$2 (Ma et al., 2021). In single-particle formulations, the same competition appears as $\mathbf r_i$3, or equivalently through the orbit radius $\mathbf r_i$4, the persistence length $\mathbf r_i$5, and trap or confinement timescales (Barman, 17 Oct 2025).

In three dimensions, chirality is encoded by a constant torque vector $\mathbf r_i$6 entering the orientational noise sector, with overdamped translational motion

$\mathbf r_i$7

and $\mathbf r_i$8 (Sevilla, 2016). This generalization is not a simple uplift of the planar model: the torque rotates the orientation about a fixed axis, so the natural deterministic trajectory is helical rather than circular.

A recurring structural point is that chirality does not merely add angular noise. In 2D it produces the orientational correlator $\mathbf r_i$9, so orientational memory becomes oscillatory rather than purely decaying (Barman, 17 Oct 2025). This oscillatory persistence is the seed for the current loops, resonance phenomena, and distribution-shape oscillations that distinguish cABPs from achiral ABPs.

2. Single-particle transport, confinement, and distribution shape

Under isotropic harmonic confinement, a 2D cABP obeys

$\theta_i$0

and develops several effects absent in the achiral or unconfined overdamped problem: oscillatory $\theta_i$1–$\theta_i$2 cross-correlations, a finite orientation–velocity delay despite the absence of inertia, and strongly non-Boltzmann stationary states (Barman, 17 Oct 2025). The deterministic steady orbit radius is

$\theta_i$3

which interpolates between broad annular distributions under weak confinement and compact central peaks under strong confinement (Barman, 17 Oct 2025).

For more general external potentials, radial symmetry and anisotropy lead to qualitatively different roles for chirality. In radially symmetric traps, chirality acts as an additional effective confinement mechanism that reduces spatial fluctuations and can be interpreted as lowering an effective temperature. In elliptic harmonic traps, by contrast, chirality combines with anisotropy to produce non-Maxwell-Boltzmann stationary distributions, nonzero cross-correlations $\theta_i$4, and parity breaking of the positional distribution that vanishes when either chirality is removed or radial symmetry is restored (Caprini et al., 2023). A useful implication is that confinement does not merely localize cABPs; it can convert handed propulsion into static signatures such as tilted covariance ellipses.

Exact transient moment theory sharpens this picture. For a trapped cABP in both 2D and 3D, closed-form expressions are available for all time-dependent moments up to fourth order, which makes it possible to track the excess kurtosis throughout the approach to steady state (Pattanayak et al., 27 Mar 2026). In 2D, the excess kurtosis shows a damped oscillatory response with multiple re-entrant sign changes: negative values correspond to active off-centered ring-like distributions, whereas positive values indicate weak heavy-tailed fluctuations. Increasing the trap stiffness suppresses these oscillations and can eliminate the positive-kurtosis windows (Pattanayak et al., 27 Mar 2026). In 3D, the excess kurtosis remains negative, and the position distribution is described instead as half-ring-like to band-like in the plane spanned by the torque axis and the normal radial direction (Pattanayak et al., 27 Mar 2026).

First-passage observables display an equally characteristic chirality dependence. In confined one- and two-dimensional domains, the mean first passage time can be non-monotonic as a function of chirality, with an optimal intermediate value minimizing escape time; for large activity the optimum occurs approximately at $\theta_i$5, meaning that the free-space curvature radius becomes comparable to the confinement scale (Iyaniwura et al., 21 May 2026). The high-chirality asymptotics show that rapid spinning drives the cABP toward passive-like escape laws, but with boundary-layer corrections that remain orientation dependent (Iyaniwura et al., 21 May 2026).

3. Interacting cABPs and collective phases

For interacting repulsive disks, chirality first appears as a suppression of the persistence-controlled self-trapping mechanism that underlies motility-induced phase separation (MIPS). A predictive field theory for Brownian circle swimmers shows that, at second order in gradients, chirality maps onto an effective enhancement of rotational diffusion,

$\theta_i$6

which shifts the MIPS spinodal to higher activity while leaving the predicted critical density $\theta_i$7 unchanged (Bickmann et al., 2020). This mapping captures the statement that angular propulsion weakens persistence-driven accumulation.

At stronger chirality, however, suppression is not the whole story. In two-dimensional cABPs with repulsive WCA interactions, sufficiently large $\theta_i$8 generates a nonequilibrium steady state of dynamical clustering that interrupts conventional MIPS. The coarse-grained polarization equation acquires the chiral rotation term $\theta_i$9, the homogeneous state develops a finite-wavelength type-II instability for $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$0, and the resulting steady current has nonzero curl, so the clustered state consists of finite clusters that continually merge, split, and decay rather than coarsening into a single macroscopic droplet (Ma et al., 2021). In this formulation, chirality does not merely renormalize persistence; it creates circulating currents that destabilize dense domains.

At lower densities, the same intrinsic rotation can stabilize crystalline order in a restricted nonequilibrium window. Two-dimensional cABPs with purely repulsive WCA interactions crystallize at area fractions such as $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$1 and $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$2, well below the equilibrium hard-disk melting density $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$3, provided the orbital radius is long enough to align circulating particles but short enough that neighboring orbits avoid destructive collisions (Jeong et al., 25 Jun 2025). The crystal is most naturally diagnosed through the centers of the circular orbits,

$\mathbf e_i=(\cos\theta_i,\sin\theta_i).$4

and the lower transition is reported to be consistent with a KTHNY-like fluid $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$5 hexatic $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$6 solid sequence, while the upper re-entrant melting appears more abrupt because collisions destroy the underlying circular-motion degree of freedom itself (Jeong et al., 25 Jun 2025).

In dense glass-forming mixtures, chirality again changes the mechanism rather than only the timescale. Interacting cABPs exhibit three dynamical regimes as the persistence time is varied at fixed spinning frequency, including an intermediate re-entrant regime governed by a “hammering” mechanism in which rapidly spinning particles trapped in cages repeatedly strike the same neighbors and gradually remodel those cages (Debets et al., 2022). At low chirality and large persistence, the same system instead develops long-ranged velocity correlations and vortex-like collective motion; at very high chirality and persistence it can approach an absorbing state of local circular or elliptical cage motion (Debets et al., 2022).

4. Geometry, obstacles, resetting, and controlled transport

One of the most distinctive features of cABPs is their sensitivity to geometric confinement. In two-dimensional lattices of disk obstacles, the effective diffusivity of chiral active particles depends strongly on obstacle symmetry and on the ratio of orbital radius to lattice scale, whereas the comparable achiral ABP does not show the same sensitivity. The same framework yields a reentrant directional-locking effect under an external field and chirality-selective separation in a parallelogram lattice without mirror symmetry, which separates clockwise and counterclockwise particles by giving them different effective diffusivities (Chan et al., 2023). A plausible implication is that environmental geometry can act as a chirality-sensitive sensor or sorter without requiring intrinsically chiral obstacles.

External torque in asymmetric channels introduces an additional resonance mechanism. For gravitactic cABPs in a two-dimensional asymmetric periodic channel, the effective longitudinal diffusion coefficient develops a pronounced peak when the external torque $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$7 approaches the intrinsic angular velocity $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$8, and the resonance is accompanied by accumulation near the upper-left corner of the channel cell (Khatri et al., 19 Sep 2025). The effect is strongest at low rotational diffusion, weakens as $\mathbf e_i=(\cos\theta_i,\sin\theta_i).$9 becomes too large, and depends strongly on the aspect ratio of the channel bottlenecks (Khatri et al., 19 Sep 2025).

Resetting provides a different route to steering. In a 2D cABP with Poissonian resetting of both position and orientation to fixed initial values, the steady-state mean-squared displacement is

$$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$0

and becomes non-monotonic in rotational diffusion when resets are infrequent compared with chiral rotation (Shee, 17 Aug 2025). The same model uses the steady-state excess kurtosis and the orientation autocorrelation to define three dynamical states—an activity-dominated chiral state and two resetting-dominated states with and without chirality—and identifies the crossover line

$$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$1

as the boundary between oscillatory and monotonic orientation memory (Shee, 17 Aug 2025).

These control mechanisms connect directly to search and escape problems. The mean-first-passage analysis summarized above shows that chirality can either hinder or accelerate escape depending on geometry, initial orientation, and whether one or multiple exits are present (Iyaniwura et al., 21 May 2026). The recurring theme is that curvature is beneficial when it redirects otherwise trapped trajectories, but detrimental once it averages propulsion into passive-like wandering.

5. Generalizations of the cABP framework

The cABP has become a reference point for several controlled extensions that preserve the core idea of intrinsic rotational drift while modifying the translational or orientational sector.

Variant	Defining modification	Representative consequence
Inertial cABP	Finite translational mass $$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$2 with overdamped orientation	Long-time positional diffusion remains the overdamped cABP value, independent of mass (Pattanayak et al., 23 Nov 2025)
Chirality-reversing ABP	Angular velocity sign flips at rate $$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$3	Four dynamical regimes; effective ABP or effective cABP behavior on intermediate timescales (Das et al., 2023)
Jerky cABP	Third-order translational dynamics with jerk term $$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$4	Short-time MSD scales as $$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$5; mean trajectories include damped and exploding Lissajous patterns (Jose et al., 25 Aug 2025)
Vision-based chiral iABP	Polar alignment plus non-reciprocal visual steering	Phase diagram with spinners, vortices, ripples, worm-like swarms, rotary clusters, and irregular aggregates (Bhaskar et al., 4 Jan 2026)

In the inertial extension, exact time-resolved moment theory shows that the velocity autocorrelation decomposes into an inertial envelope and a chiral envelope, that the steady mean-squared velocity defines a kinetic temperature, and that the violation of a modified fluctuation–dissipation relation vanishes both at large mass and at large chirality (Pattanayak et al., 23 Nov 2025). The positional long-time diffusion coefficient nevertheless remains

$$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$6

which is precisely the overdamped cABP value in the dimensionless variables of that work (Pattanayak et al., 23 Nov 2025).

The chirality-reversing model introduces a telegraph process $$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$7 into

$$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$8

and yields four dynamical regimes controlled by the competing timescales $$\dot{\mathbf r}_i = -\mu\sum_{j\neq i}\nabla_i V(|\mathbf r_i-\mathbf r_j|) +v_0\mathbf e_i +\sqrt{2D_t}\,\boldsymbol{\xi}_i, \qquad \dot{\theta}_i = \omega_0+\sqrt{2D_r}\,\nu_i,$$9 and $\omega_0$0. In the fast-reversal window the process reduces to an effective ABP with

$\omega_0$1

whereas in the slow-reversal window it behaves as an effective cABP in its long-time diffusive regime (Das et al., 2023).

The jerky cABP keeps the standard chiral orientation dynamics but replaces the translational equation with

$\omega_0$2

This produces anomalous short-time transport,

$\omega_0$3

while the long-time effective diffusion remains the usual chiral active value (Jose et al., 25 Aug 2025). The authors’ trajectory classification into spira mirabilis, damped Lissajous, and exploding Lissajous patterns is a particularly explicit illustration of how added memory modifies but does not erase the cABP backbone (Jose et al., 25 Aug 2025).

6. Conceptual issues, mappings, and scope

A persistent conceptual theme in the cABP literature is whether chirality can be reduced to a renormalized rotational diffusion. The second-order field theory for circle swimmers answers “partly yes”: at that level, the density dynamics is mapped by $\omega_0$4, which usefully explains the suppression of MIPS (Bickmann et al., 2020). The strongly chiral collective theory answers “not in general”: higher-order and finite-wavelength effects generate transverse polarization response, circulating currents with nonzero curl, and nonequilibrium steady states that cannot be written as gradient flows of a scalar free energy (Ma et al., 2021). The two statements are therefore compatible, but they apply to different levels of description.

A related issue is the comparison with other active models. Under harmonic confinement, chiral active Ornstein–Uhlenbeck particles and cABPs can share the same qualitative reduction of positional variance and the same chirality-induced cross-correlations in anisotropic traps, yet the cABP remains non-Gaussian in situations where the AOUP is exactly Gaussian (Caprini et al., 2023). Likewise, in three dimensions an ensemble of chiral active particles with uniformly distributed random chirality axes can exhibit “anomalous, yet Brownian” diffusion: the mean-squared displacement is linear in time, but the positional distribution remains non-Gaussian as diagnosed by the kurtosis (Sevilla, 2016).

The supplied record (Muzzeddu et al., 2022) is explicitly described as a REVTeX template rather than a usable scientific source, so it does not contribute substantive cABP results. Within the remaining literature, the cABP emerges as a minimal but remarkably extensible nonequilibrium model: a self-propelled particle with intrinsic angular drift whose consequences range from orbit-center crystallization and current-driven clustering to geometry-sensitive transport, confinement-induced delay, and exact higher-order signatures of non-Gaussian steady states.