Active Potts Particles: Dynamics & Pattern Formation

• Active Potts particles are active-matter models where discrete Potts states control propulsion or internal cyclic dynamics.
• They exhibit diverse nonequilibrium behaviors, including flocking, phase separation, and spiral or target wave formation, depending on the interaction rules.
• Hydrodynamic analyses reveal that activity mechanism, occupancy constraints, and state-dependent diffusion lead to varied universality classes and pattern transitions.

Active Potts particles are active-matter models in which a discrete Potts state is promoted from a passive label to a dynamical variable that controls either propulsion direction or an internally driven nonequilibrium cycle. In one major realization, particles on a lattice carry a state $\sigma\in\{1,\dots,q\}$, align ferromagnetically with nearby particles, and hop preferentially along the lattice direction encoded by $\sigma$, producing flocking, liquid–gas coexistence, and orientation-selective dense bands (Chatterjee et al., 2019, Mangeat et al., 2020). In a second realization, Potts spins on fixed lattice sites are driven by cyclic flip energies whose sum around a closed loop is nonzero, so global detailed balance is broken and long-lived spiral, target, stripe, and homogeneous-cycling modes arise (Noguchi et al., 2023, Noguchi, 2024). A third line of work adds excluded volume, occupancy restrictions, or purely Brownian motion, thereby isolating how crowding, jamming, and conserved transport modify Potts ordering and its universality (Karmakar et al., 2022, Chen et al., 6 Mar 2025).

1. Core architectures of active Potts particles

The phrase “active Potts particles” is used in at least two technically distinct senses. In motile models, the Potts state is an orientation variable that simultaneously determines self-propulsion direction and participates in short-range alignment. In internally driven models, the Potts state is a chemical or conformational coordinate on a fixed lattice, and activity is implemented by nonconservative cyclic flip energies. Both constructions retain discrete $q$-state symmetry, but they produce different classes of nonequilibrium phenomena: flocking and phase separation in the first case, and pattern-forming reaction-like waves in the second (Chatterjee et al., 2019, Noguchi, 2024).

Realization	Microscopic state	Activity mechanism
Active Potts model (APM)	Lattice particles with $\sigma\in\{1,\dots,q\}$	Biased hopping along the direction encoded by $\sigma$
Restricted / excluded-volume APM	Same, with MPS constraint or soft repulsion	Density-dependent mobility plus biased hopping
Active cyclic Potts model	Potts spins $s\in\{0,\dots,q-1\}$ on fixed sites	Cyclic flip energies with nonzero loop sum
Brownian $q=2$ Potts baseline	Moving particles with Ising/Potts state	Brownian motion without feedback from state to motion

For motile models, the canonical microscopic ingredients are an on-site ferromagnetic Potts interaction and biased diffusion. On a site $i$ with local occupancy $\rho_i$ and state occupations $n_i^\sigma$, the local Hamiltonian is

$\sigma$0

and the hopping rate of a particle in state $\sigma$1 toward direction $\sigma$2 is

$\sigma$3

For cyclic models, by contrast, activity enters through

$\sigma$4

with $\sigma$5 but a nonzero cyclic sum, so there is a steady current in state space rather than propulsion in real space (Mangeat et al., 2020, Noguchi, 2024).

2. Self-propelled Potts particles and flocking

The minimal self-propelled realization is the $\sigma$6-state active Potts model on a lattice without exclusion. For $\sigma$7 on the square lattice, the internal states correspond to right, up, left, and down; for $\sigma$8, the natural geometry is the triangular lattice with six lattice directions (Chatterjee et al., 2019, Mangeat et al., 2020). The local Potts magnetization in direction $\sigma$9 can be written as

$q$0

and the coarse-grained polarization is

$q$1

Flocking is then the spontaneous emergence of a polar liquid in which one state dominates globally and particles move coherently in the associated direction.

In these models, flocking is not described as a simple homogeneous order–disorder transition. Instead, it is organized by a liquid–gas picture with three regimes: a disordered homogeneous gas, an ordered homogeneous polar liquid, and a coexistence region where a dense polar stripe moves in a dilute disordered background (Chatterjee et al., 2019, Mangeat et al., 2020). The coexistence densities $q$2 and $q$3 are essentially independent of the global mean density at fixed control parameters; increasing $q$4 changes the fraction of the system occupied by the dense stripe rather than its internal density. This is the same structural interpretation used for the active Ising model and Vicsek-like flocking, but with discrete $q$5 symmetry and explicit lattice anisotropy.

For $q$6, the dense stripe itself undergoes a genuinely new reorientation transition. At low self-propulsion bias $q$7, the ordered dense phase forms a transversal band, meaning that the dense stripe is elongated perpendicular to the mean particle motion. Above a threshold $q$8, the stripe reorients into a longitudinal lane, with motion parallel to its long axis; in the $q$9 model, the orientational observable $\sigma\in\{1,\dots,q\}$0 jumps discontinuously from a value near $\sigma\in\{1,\dots,q\}$1 to a value close to $\sigma\in\{1,\dots,q\}$2, and the threshold is essentially independent of $\sigma\in\{1,\dots,q\}$3 (Chatterjee et al., 2019). The broader $\sigma\in\{1,\dots,q\}$4 and $\sigma\in\{1,\dots,q\}$5 hydrodynamic theory reaches the same conclusion: the phase-separated state switches from transversal band motion to longitudinal lane formation as $\sigma\in\{1,\dots,q\}$6 increases, a phenomenon absent in the Vicsek model and in the $\sigma\in\{1,\dots,q\}$7 active Ising case (Mangeat et al., 2020).

The mechanism is tied to anisotropic transport. In the hydrodynamics, the longitudinal and transverse diffusion coefficients relative to the internal direction are

$\sigma\in\{1,\dots,q\}$8

while the effective drift is

$\sigma\in\{1,\dots,q\}$9

As $\sigma$0, $\sigma$1 approaches zero, so transverse relaxation becomes ineffective and longitudinal lanes are stabilized (Mangeat et al., 2020). This suggests that the reorientation is a direct consequence of discrete directional symmetry plus propulsion-induced diffusion anisotropy, rather than a generic property of polar active matter.

3. Exclusion, jamming, and hard-core realizations

Adding occupancy constraints changes the phenomenology qualitatively. In the restricted active Potts model, hopping into a target site depends on its occupancy, either through a hard cutoff $\sigma$2 or through a soft on-site repulsion $\sigma$3. For $\sigma$4, the system reduces to an active lattice gas with no meaningful on-site alignment; the relevant control parameter is the Péclet number

$\sigma$5

and the homogeneous gas becomes unstable only above $\sigma$6 (Karmakar et al., 2022). The resulting dense phases are jammed MIPS structures rather than flocks: small clusters, diagonal system-spanning bands, and high-density solids with a square void. In this limit, activity and exclusion alone generate motility-induced phase separation and kinetic arrest.

When $\sigma$7 or soft repulsion is used, alignment re-enters and one recovers both flocking and jamming. The restricted model then supports gas, polar liquids, phase-separated moving bands or lanes, and dense gridlocked clusters in which locally ordered subdomains block each other’s motion (Karmakar et al., 2022). Low temperature, high activity, high density, and strong exclusion favor the jammed phases; increasing temperature, decreasing activity, or increasing the allowed occupancy delays or melts jamming. The corresponding hydrodynamic equations keep the same state-resolved structure as the unrestricted APM but replace the bare mobility by a density-dependent factor $\sigma$8, such as $\sigma$9, $s\in\{0,\dots,q-1\}$0, or $s\in\{0,\dots,q-1\}$1, depending on the restriction protocol.

A different hard-core implementation is the three-state active lattice gas on the triangular lattice, where each occupied site carries one of three orientations separated by $s\in\{0,\dots,q-1\}$2, dynamics proceeds via a noisy majority rule over a 19-site neighborhood, and hopping is allowed only if the target site is vacant (Rosembach et al., 2023). At full occupancy, motion is impossible and the dynamics becomes a 3-state majority-vote model with Potts permutation symmetry. Finite-size scaling then gives

$s\in\{0,\dots,q-1\}$3

consistent with the 2D 3-state Potts universality class. At low density, however, the order–disorder transition is discontinuous: there is strong dependence on the initial condition, hysteresis, an abrupt drop in the global polarization $s\in\{0,\dots,q-1\}$4, and a negative Binder cumulant near the transition (Rosembach et al., 2023). Hard one-particle-per-site exclusion therefore frustrates flocking and promotes condensed, low-mobility structures such as immobile bands and traffic jams.

4. Internally driven cyclic Potts particles

A second major tradition treats activity as internal cycling on a fixed lattice. Here each site carries a Potts state $s\in\{0,\dots,q-1\}$5, nearest-neighbor contact energies favor local phase separation, and nonequilibrium enters through directed flip energies $s\in\{0,\dots,q-1\}$6 around a state cycle (Noguchi et al., 2023, Noguchi, 2024). In the three-state version, the symmetric choice $s\in\{0,\dots,q-1\}$7 yields a cycling energy $s\in\{0,\dots,q-1\}$8. The resulting dynamics interpolates between equilibrium Potts behavior at $s\in\{0,\dots,q-1\}$9 and far-from-equilibrium steady states at $q=2$0.

The basic long-time modes are homogeneous cycling (HC) and spiral waves (SW). In HC, one state covers almost the entire lattice, but the dominant state changes cyclically via nucleation and growth. In SW, all three states coexist and triple junctions act as spiral cores. For large systems, the transition from HC to SW is discontinuous, whereas the reverse transition is absent on accessible timescales; for small systems, HC and SW can coexist temporally (Noguchi et al., 2023). The transition can be rationalized by comparing the lifetime of a homogeneous phase to the time needed for a nucleated domain to grow across the system: once nested nucleation becomes common before a new homogeneous phase fills the system, three-phase contacts and spiral cores proliferate. A continuum theory with coarse-grained densities $q=2$1 reproduces traveling biphasic bands and supports the same competition between nucleation-driven cycling and wave formation (Noguchi et al., 2023, Noguchi et al., 2024).

The four-state cyclic model enlarges the pattern set. Under symmetric four-state cycling, low flipping energy yields HC4, while high flipping energy yields a quad-phase coexistence mode Q in which all four states coexist spatially (Noguchi, 2024). Because diagonal states do not interconvert directly, interfaces between them are only indirectly active and move slowly; this produces long-lived diagonal two-phase coexistence modes $q=2$2 and $q=2$3 under asymmetric driving. When a three-state cycle is added on top of the four-state cycle, competition between cycles produces HC3, non-cycling single-phase states $q=2$4, and, at larger drive, spiral-wave modes built from the three-state sector (Noguchi, 2024). More general flip networks with multiple identical loops extend this mechanism: overlapping three-state cycles favor simultaneous families of spiral waves at high flip energy, while overlapping four-state cycles tend instead to preserve single-state dominance with slow domain dynamics (Noguchi, 1 Dec 2025).

These internally active Potts models are not self-propelled in real space, but they are active in the strict nonequilibrium sense: local detailed balance is respected pairwise, yet the cyclic sum of flip energies is nonzero, so each site behaves as a microscopic engine that dissipates energy while cycling through internal states (Noguchi, 2024, Noguchi, 1 Dec 2025). This suggests a natural interpretation in terms of catalytic surfaces, membrane transport, and chemically driven conformational cycles, all of which are explicitly identified as motivating contexts in the three-state and four-state studies (Noguchi et al., 2023, Noguchi et al., 2024).

5. Higher-state, factorizable, and six-state active Potts dynamics

For larger $q=2$5, especially factorizable values such as $q=2$6, the nonequilibrium dynamics becomes more hierarchical. In the square-lattice study of $q=2$7, standard contact energies yield homogeneous cycling of the $q=2$8 homogeneous phases at low flipping energy and a $q=2$9-state coexisting wave mode at high flipping energy for all tested $i$0 (Noguchi, 30 Jun 2025). For factorizable $i$1, however, dynamic modes with skipping states appear. The $i$2 case is the clearest: depending on $i$3 and $i$4, the system exhibits W3 spiral waves of three dominant states, HC3 cycling among three homogeneous phases, M3 mixed phases of three states, and M2W3 modes in which three types of two-state mixed domains form spiral waves (Noguchi, 30 Jun 2025). Although three states can coexist spatially under thermal equilibrium, the scaling exponents at the transition to the W3 mode are modified from the equilibrium values, whereas the HC3–M3 transition remains close to the equilibrium 3-state Potts case.

The dedicated six-state study resolves the geometry of wave modes further. Under weak repulsion at nonflip contacts, the system supports disordered six-state waves; under stronger repulsion, it forms spiral waves of six states; with still stronger repulsion, target waves and stripe waves appear (Noguchi, 24 Apr 2026). The same work shows that spiral waves built from the even triplet $i$5 or the odd triplet $i$6 are not all equivalent: diagonal repulsion produces forward waves $i$7, whereas attractive nonflip contacts produce backward waves $i$8. The transition between even- and odd-dominated waves is first-order for both forward and backward cases (Noguchi, 24 Apr 2026).

Coarsening toward these active wave states is itself structured. Across $i$9 to $\rho_i$0, when the final state is a spiral-wave state, the correlation length follows the Lifshitz–Allen–Cahn law $\rho_i$1 until it saturates at the characteristic wavelength of the waves (Noguchi, 22 Sep 2025). For nonspiral waves, the growth rate is transiently enhanced before saturation, so the effective coarsening exponent rises above the equilibrium value. The same study reports that this dynamic behavior is insensitive to the lattice type, square or hexagonal, and to the update rule, Metropolis or Glauber (Noguchi, 22 Sep 2025). Taken together, these results indicate that the onset of active wave motion does not simply replace equilibrium coarsening; it reorganizes it into a growth process toward a non-static attractor.

6. Hydrodynamics, universality, and conceptual scope

A recurring theme across the field is that hydrodynamic closure is possible but model-dependent. For motile APMs, a refined mean-field treatment of state-resolved densities $\rho_i$2 yields continuum equations with anisotropic diffusion, advection along $\rho_i$3, and nonlinear alignment terms of the form

$\rho_i$4

(Mangeat et al., 2020). The fluctuation-induced $\rho_i$5 term is essential: omitting it yields only homogeneous solutions, whereas retaining it stabilizes phase-separated bands and lanes (Chatterjee et al., 2019). For restricted models, the same structure persists with a density-dependent mobility factor $\rho_i$6, and the resulting hydrodynamics reproduces MIPS, band-to-lane reorientation, and jammed states (Karmakar et al., 2022). For the hard-core three-state discrete Vicsek model, by contrast, a systematic coarse-grained hydrodynamic theory remains open (Rosembach et al., 2023).

Universality is correspondingly stratified. At full occupancy in the three-state excluded-volume model, the transition is continuous and in the 3-state Potts universality class (Rosembach et al., 2023). In the $\rho_i$7 and $\rho_i$8 motile APM at $\rho_i$9, the density-driven transition is first-order and not in the equilibrium Potts universality class (Chatterjee et al., 2019, Mangeat et al., 2020). In cyclic models with factorizable $n_i^\sigma$0, some transitions remain close to equilibrium Potts values, whereas transitions associated with the onset of wave modes exhibit modified exponents (Noguchi, 30 Jun 2025). A useful passive baseline is the Brownian $n_i^\sigma$1 Potts model, where particles diffuse but internal states do not feed back on motion: its static critical behavior remains that of the 2D Ising model, the off-lattice dynamics falls into Model C, and the on-lattice case is Model A with an additional irrelevant conserved Model B noise (Chen et al., 6 Mar 2025). This suggests that the phrase “active Potts particles” does not denote a single universality class but a family of nonequilibrium constructions whose critical and pattern-forming behavior depends on whether activity enters through propulsion, exclusion-mediated motility, or cyclic internal currents.

Several directions recur across the literature. Motile APM studies explicitly point to $n_i^\sigma$2, the $n_i^\sigma$3 limit, off-lattice versions, and couplings to excluded volume as natural generalizations (Chatterjee et al., 2019). Excluded-volume work emphasizes that relaxing hard one-particle-per-site constraints or using soft-core interactions is more favorable for sustained flocking (Rosembach et al., 2023). Cyclic-spin studies motivate extensions to more than four states, competing cycles, deformable or curved substrates, and experimental realizations on catalytic surfaces or membranes (Noguchi et al., 2024, Noguchi, 1 Dec 2025). Taken together, these results establish active Potts particles as a broad nonequilibrium framework in which discrete internal symmetry, local interactions, and active driving can be tuned independently to generate flocking, MIPS, jamming, homogeneous cycling, spiral and target waves, stripe modes, and factorized multi-state dynamics.