Chiral Active Brownian Particles

• Chiral active Brownian particles are self-propelled entities with a constant angular drift that induces circular trajectories, setting them apart from conventional active particles.
• Their deterministic rotation converts persistent linear motion into circularized trajectories, substantially tuning effective diffusivity and response to confinement.
• Chirality drives novel collective phenomena, such as dynamic clustering, low-density crystallization, and topologically protected boundary transport.

Chiral active Brownian particles are active Brownian particles whose orientational dynamics contains a deterministic angular drift in addition to rotational diffusion, so that isolated trajectories are circular in two dimensions rather than persistent straight runs. In the minimal overdamped formulation, a particle self-propels at fixed speed along an internal orientation while a constant angular velocity breaks mirror symmetry and time-reversal symmetry at the level of trajectories. Recent work has established that this apparently simple modification reorganizes transport, confinement response, collective pattern formation, phase behavior, and the mechanics of composite objects, including polymers, passive inclusions, and boundary-bound states (Sevilla, 2016, Jeong et al., 25 Jun 2025, Anand, 4 Mar 2025).

1. Definition and minimal stochastic description

A standard two-dimensional chiral active Brownian particle is specified by a position $\mathbf r=(x,y)$ and an orientation $\hat{\mathbf n}=(\cos\theta,\sin\theta)$. A common overdamped form is

$$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$

or, in the notation used for polymers of chiral active Brownian particles,

$$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$

The defining distinction from ordinary active Brownian particles is the constant drift term $\Omega_0$ or $\omega$, which causes isolated particles to execute circular swimming rather than persistent straight runs (Khatri et al., 19 Sep 2025, Anand, 4 Mar 2025).

For an isolated deterministic swimmer, the natural geometric scale is the circular radius

$R=\frac{v_0}{\Omega},$

or, in dimensionless notation frequently used in recent studies,

$$R_{\omega_0}=\frac{v_0}{\omega_0},\qquad R_\Omega=\frac{v_0}{|\Omega|}.$$

The orientational persistence time remains controlled by rotational decorrelation, but chirality converts the usual persistent random walk into a circularized persistence process (Khatri et al., 2022, Khatri et al., 19 Sep 2025).

Several dimensionless parameters recur across the literature. A common reduced chirality is

$\Omega=\frac{\omega}{D_{\rm rot}}$

or equivalently $\omega/D_r$, while activity is often parameterized by a Péclet number such as

$\hat{\mathbf n}=(\cos\theta,\sin\theta)$0

These ratios compare deterministic turning to orientational diffusion and active propulsion to diffusive relaxation, respectively (Anand, 4 Mar 2025, Levis et al., 2017).

The three-dimensional extension replaces the scalar angular drift by a torque vector $\hat{\mathbf n}=(\cos\theta,\sin\theta)$1, and the joint distribution in position and orientation can be expanded in spherical harmonics. In that setting, the positional density exhibits a ballistic-to-diffusive crossover, fixed-axis chirality suppresses effective spreading, and random-axis chirality can yield “anomalous, yet Brownian, diffusion,” namely linear mean-squared displacement together with a non-Gaussian positional distribution (Sevilla, 2016).

2. Free-space transport, orientational memory, and exact single-particle results

In free space, chirality suppresses long-time translational diffusion because persistent propulsion is continuously redirected into circular motion. For a two-dimensional chiral active Brownian particle with rotational diffusion $\hat{\mathbf n}=(\cos\theta,\sin\theta)$2, one widely used free-space result is

$\hat{\mathbf n}=(\cos\theta,\sin\theta)$3

with $\hat{\mathbf n}=(\cos\theta,\sin\theta)$4, while an equivalent form appearing in reset and trap studies is

$\hat{\mathbf n}=(\cos\theta,\sin\theta)$5

Both expressions make the same point: increasing $\hat{\mathbf n}=(\cos\theta,\sin\theta)$6 reduces the active contribution to diffusion (Khatri et al., 2022, Barman, 17 Oct 2025).

The orientational autocorrelation of a free chiral active Brownian particle is oscillatory rather than purely decaying,

$\hat{\mathbf n}=(\cos\theta,\sin\theta)$7

and this oscillatory memory propagates into the positional statistics. In three dimensions, the marginal positional distribution evolves from a shell-like short-time form toward a Gaussian long-time form, and the kurtosis oscillates between the Gaussian value and the value associated with a spherical-shell distribution when chirality is strong (Sevilla, 2016, Barman, 17 Oct 2025).

Several recent works have extended the single-particle framework without abandoning the chiral Brownian core. Position-orientation resetting produces a nonequilibrium steady state for a free-space chiral active Brownian particle, with exact steady-state mean-squared displacement

$\hat{\mathbf n}=(\cos\theta,\sin\theta)$8

and an effective diffusion

$\hat{\mathbf n}=(\cos\theta,\sin\theta)$9

In that model, the steady-state mean-squared displacement is non-monotonic in $$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$0 when resets are infrequent compared to chiral rotation, and the condition

$$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$1

marks the boundary between oscillatory and monotonic orientation autocorrelation (Shee, 17 Aug 2025).

A different theoretical extension replaces permanent chirality by a diffusing tumble angle. The resulting transiently chiral active particle admits an exact Doi–Peliti field theory and reproduces ordinary active Brownian behavior in observables averaged over the reorientation angle, while conditioned observables reveal finite-time chirality through memory of successive tumbles (Britton et al., 2 Jul 2025). Another extension adds a jerk term to the translational dynamics,

$$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$2

leaving the angular dynamics chiral but introducing higher-order translational memory. In that model, short-time motion becomes super-ballistic, the mean displacement starts as $$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$3, and the mean-squared displacement starts as $$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$4, while the long-time diffusion coefficient remains the standard chiral one (Jose et al., 25 Aug 2025).

3. Confinement, external fields, and geometry-sensitive transport

Confinement converts the intrinsic turning of chiral active Brownian particles into highly geometry-dependent transport. In a Poiseuille flow between parallel walls, the translational and angular dynamics

$$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$5

show that chirality and shear can either reinforce or cancel. The reported consequence is tunable upstream drift, sign reversal of the average velocity, and a chirality-induced enhancement peak in the effective diffusion coefficient, in contrast to the diffusion suppression found for nonchiral particles in the same flow (Khatri et al., 2022).

In an asymmetric periodic triangular channel with gravitactic torque, the angular dynamics

$$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$6

has a deterministic locking transition at $$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$7. Near that point, low rotational diffusion and moderate intrinsic chirality produce resonant enhancement of longitudinal diffusion, accompanied by strong accumulation near the upper-left corner of the channel. The diffusion peak disappears for larger $$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$8 or larger $$\dot{\mathbf r}(t)=V_0\hat{\mathbf n}(t)+\sqrt{2D_t}\,\boldsymbol{\xi}(t),\qquad \dot{\theta}(t)=\Omega_0+\sqrt{2D_r}\,\zeta(t),$$9, showing that this resonance is confinement-mediated rather than a generic free-space consequence of chirality (Khatri et al., 19 Sep 2025).

Obstacle lattices provide a different form of geometric selectivity. In two-dimensional lattices of disk obstacles, chiral active particles were found to be sensitive reporters of lattice structure in a way absent for achiral active particles: the effective diffusivity depends strongly on whether the obstacle array is square or triangular, directional locking under an external field becomes reentrant, and a mirror-asymmetric parallelogram lattice separates clockwise from counterclockwise particles (Chan et al., 2023). A plausible implication is that the orbital radius $$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$0 acts as an additional geometric selector, so that commensurability between free circular motion and obstacle spacing becomes a transport control parameter.

Harmonic confinement produces a distinct nonequilibrium effect. For an overdamped trapped chiral active Brownian particle,

$$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$1

the mean-squared displacement saturates, the stationary position distribution becomes non-Boltzmann, and chirality induces oscillatory positional cross-correlations. Most notably, the trap generates a finite delay between orientation and velocity even without inertia, quantified by

$$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$2

This delay vanishes in the free overdamped case and therefore isolates a confinement-induced breaking of time-reversal symmetry distinct from inertial lag (Barman, 17 Oct 2025).

First-passage properties also become nontrivial under confinement. In intervals and disks with absorbing boundaries, the mean first passage time of a chiral active Brownian particle can be monotonic or non-monotonic in the dimensionless chirality

$$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$3

depending on geometry and initial orientation. The reported high-chirality asymptotics recover passive Brownian escape at leading order, while many finite-$$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$4 cases exhibit an optimal intermediate chirality near $$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$5 that minimizes escape time (Iyaniwura et al., 21 May 2026).

4. Collective phases, phase separation, and dense chiral matter

Interacting chiral active Brownian particles support a broad range of collective states. One of the clearest departures from ordinary active Brownian particles is the interruption of motility-induced phase separation by intrinsic rotation. In repulsive two-dimensional circle swimmers with constant torque, dynamic mean-field theory predicts two instabilities: a MIPS-like long-wavelength instability for weak chirality and a finite-wavelength instability for strong chirality. In simulations, sufficiently strong torque replaces macroscopic gas–liquid-like separation by a dynamical clustering state of many finite clusters that continuously merge, split, and decay (Ma et al., 2021).

When excluded volume is combined with explicit alignment, chirality reorganizes flocking into two canonical regimes. Slow rotation yields rotating macro-clusters or macro-drops whose size scales with system size; fast rotation yields micro-flock patterns, namely finite-sized rotating, phase-synchronized clusters with self-limited size. With steric repulsion, both states survive but become more fluctuating in shape and size, and the characteristic micro-flock length remains proportional to the single-swimmer radius, $$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$6 (Levis et al., 2017).

Chirality can also generate ordering rather than only frustrate it. In a minimal two-dimensional model of repulsive chiral active Brownian particles, circular motion with the right orbital radius produces crystallization at densities $$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$7 and $$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$8, well below the equilibrium hard-disk melting density cited as $$\gamma \dot{\mathbf r}_i(t)= -\nabla_i U(t)+\boldsymbol{\eta}_i^{\rm transl}(t)+F_a\hat{\mathbf u}_i(t),\qquad \dot{\theta}_i(t)=\eta_i^{\rm rot}(t)+\omega.$$9. The ordering window is reentrant in the orbital radius $\Omega_0$0: too small $\Omega_0$1 gives insufficient effective packing, intermediate $\Omega_0$2 yields hexatic/crystalline order, and too large $\Omega_0$3 destroys the orbit compatibility needed for the ordered state (Jeong et al., 25 Jun 2025).

At high density, chirality remains dynamically relevant rather than being washed out by caging. In a Kob–Andersen binary mixture of dense chiral active Brownian particles, the long-time diffusion coefficient as a function of $\Omega_0$4 shows an initial nonmonotonic regime, a pronounced re-entrant fluidization, and a large-persistence regime that either develops collective swirling or approaches an absorbing state. The most distinctive mechanism identified there is “hammering”: once the persistence time exceeds the spinning period, a particle performs repeated circular or elliptical caged motion and repeatedly collides with the same neighbor, gradually remodeling the cage until escape occurs (Debets et al., 2022).

Adding anisotropic interactions enriches the phase diagram further. A monolayer of chiral active particles with permanent dipoles and propulsion parallel to the dipole axis exhibits percolated networks, string fluids, two types of vortices, vortices with phase separation, micro-flocking, and macro-flocking at a density $\Omega_0$5, below the usual MIPS threshold of simple active Brownian particles. The authors attribute this to the combined effects of chirality, strong dipolar coupling, and ring- or chain-based motifs, and they show that truncating long-range dipolar interactions destroys the vortices (Liao et al., 2021). A different extension, combining chirality with non-reciprocal visual steering and polar alignment, produces spinners, vortices, ripple loops, worm-like swarms, rotary clusters, and irregular aggregates; in that setting, high chirality yields dilute phases, while moderate to low chirality yields cohesive but dynamic patterns (Bhaskar et al., 4 Jan 2026).

5. Polymers, passive inclusions, and boundary-induced organization

Chiral active Brownian particles also act as active building blocks for composite objects. A flexible open bead-spring polymer in which every monomer is a chiral active Brownian particle develops a distinctive two-spiral folded state in two dimensions. The chain compacts strongly, the bond correlation becomes oscillatory, the two ends form spirals with opposite handedness, and the total spiral content increases with $\Omega_0$6 and $\Omega_0$7 over an intermediate regime before decreasing again at very high activity. The end-to-end correlation becomes oscillatory, signaling coherent whole-chain rotation, and the measured rotation frequency follows

$\Omega_0$8

In the most folded intermediate-$\Omega_0$9 regime, the size scaling can become as compact as

$\omega$0

These results isolate a specifically chiral active-Brownian route to spiral folding that differs from tangentially propelled filaments (Anand, 4 Mar 2025).

A passive circular inclusion immersed in a bath of chiral active Brownian particles can rotate persistently even though the inclusion is geometrically symmetric and non-motile. In the reported model, the torque is generated by angularly biased collisions and a nonuniform angular distribution of particle contacts around the inclusion. The mean angular velocity is nonmonotonic in chirality: it rises sharply with $\omega$1, reaches a maximum at intermediate $\omega$2, and then declines as surface residence times become too short. Two regimes were identified: a low-chirality density-gradient-dominated regime and a high-chirality impact-frequency-dominated regime (Puitandy et al., 17 Jul 2025).

Boundary transport reveals a sharp distinction between ordinary and generalized chiral active Brownian particles. Simple chiral active Brownian particles can support steady edge currents near walls, but those currents were shown not to imply boundary-induced transport along straight boundaries. By contrast, doubly chiral active Brownian particles—particles with both intrinsic angular velocity and translation-rotation coupling—can exhibit robust topologically protected transport along boundaries without backscattering at corners when the two sources of chirality have opposite sign and the translation-rotation coupling dominates: $\omega$3 The paper further shows that this boundary mode differs from the edge behavior of simple chiral active Brownian particles and from that of chiral active rods or self-aligning chiral active Brownian particles, which backscatter at corners (Edwards et al., 7 Jul 2026). This suggests that intrinsic angular drift alone is insufficient for protected edge transport, whereas competing chiral couplings can generate it.

6. Theoretical scope, misconceptions, and open problems

A persistent misconception is that chirality merely renormalizes rotational diffusion. The literature summarized above argues against that view. In confinement, chirality can induce resonant diffusion near a locking transition, boundary-selective accumulation, geometry-sensitive effective diffusivity, and optimal escape times (Khatri et al., 19 Sep 2025, Chan et al., 2023, Iyaniwura et al., 21 May 2026). In dense systems, it can interrupt standard MIPS, fluidize glasses through hammering, or promote low-density crystallization through orbital packing (Ma et al., 2021, Debets et al., 2022, Jeong et al., 25 Jun 2025). In composite objects, it can drive whole-body rotation, boundary modes, or end-localized spiral folding (Puitandy et al., 17 Jul 2025, Edwards et al., 7 Jul 2026, Anand, 4 Mar 2025). This suggests that chirality is not a perturbative decoration of active Brownian motion but a structural control parameter.

Another recurring theme is that many results depend on the mechanism by which chirality couples to the environment. External shear can cancel intrinsic rotation near a wall; gravitactic torque can compete with intrinsic turning; harmonic trapping can create a velocity-orientation delay; resetting can interrupt loops and create nonequilibrium steady states; and translation-rotation coupling can convert bulk chirality into protected edge transport (Khatri et al., 2022, Khatri et al., 19 Sep 2025, Barman, 17 Oct 2025, Shee, 17 Aug 2025, Edwards et al., 7 Jul 2026). A plausible implication is that “chiral active Brownian particle” is best understood as a family of models sharing the same kinematic core but differing sharply in transport consequences once external torques, boundaries, or composite degrees of freedom are added.

The main limitations are also consistent across recent work. Many models are strictly two-dimensional, overdamped, and dry; hydrodynamic interactions are usually neglected; boundaries are often idealized by sliding-reflecting rules or simple harmonic forces; and several prominent findings remain empirical simulation results rather than analytically derived laws. Open questions identified in the recent literature include the robustness of two-spiral polymer states to stiffness and hydrodynamics, analytical explanations of the $\omega$4 polymer rotation scaling, the role of heterogeneous chirality and alignment interactions, asymptotics for narrow-escape problems in more complex geometries, and many-body theories for topologically protected boundary transport (Anand, 4 Mar 2025, Iyaniwura et al., 21 May 2026, Edwards et al., 7 Jul 2026).

Taken together, these developments define chiral active Brownian particles as a broad and technically precise class of nonequilibrium matter in which constant angular drift reshapes active transport across scales: from single-particle diffusion and first-passage statistics, to confinement-induced resonance and geometry sensing, to collective clustering, crystallization, and glassy fluidization, and finally to the behavior of polymers, passive inclusions, and boundary-bound modes (Sevilla, 2016, Jeong et al., 25 Jun 2025, Anand, 4 Mar 2025).