Active Potts Models in Nonequilibrium Systems

• Active Potts models are nonequilibrium extensions of classic Potts systems that incorporate cyclic or biased spin flips to break detailed balance on multisite lattices.
• They exhibit diverse dynamical regimes, including homogeneous cycling, spiral waves, phase coexistence, and jamming transitions by tuning drive and interaction parameters.
• These models bridge concepts from statistical mechanics, reaction–diffusion systems, and active matter, offering insights into pattern formation and critical phenomena.

Active Potts models generalize classical equilibrium Potts systems by introducing nonequilibrium driving, typically implemented as cyclic or biased single-spin flips that globally break detailed balance on a lattice of multisite Potts variables. These models yield a broad range of far-from-equilibrium spatiotemporal phenomena—including homogeneous cycling, spiral waves, domain coarsening, band formation, phase coexistence, and novel jamming regimes—by interpolating smoothly between classical statistical mechanics and reaction–diffusion-type pattern formation. The field now encompasses both minimal cyclic-flip lattice models and multi-agent flocking matter models that couple internal Potts symmetry to self-propulsion and alignment.

1. Model Definitions and Dynamical Protocols

Active Potts models are defined on a $d$-dimensional lattice (most commonly two-dimensional, periodic, with $N=L^2$ sites), with discrete, $q$-valued Potts spins $s_i \in \{0, 1, \ldots, q-1\}$ or particles carrying an internal Potts state. There are two principal classes:

• Cyclic single-spin-flip models: At each time step, a site $i$ proposes a state change $s_i \rightarrow s_i'$ (typically to $s_i \pm 1 \pmod{q}$), with a transition probability constructed from an interaction Hamiltonian $H_{\mathrm{int}}$ (e.g., $-J \sum_{\langle ij \rangle} \delta_{s_i, s_j}$) and a nonreciprocal "active" flip energy $h_{s, s'} = -h_{s', s}$ such that $\sum_{k=0}^{q-1} h_{k, k+1} \neq 0$. The move is accepted with Metropolis or Glauber probability $\min[1, e^{-\Delta H}]$, where $\Delta H = \Delta H_{\mathrm{int}} - h_{s, s'}$ (Noguchi et al., 2023).
• Active Potts matter models: On-site ferromagnetic interactions favor local alignment; self-propulsion is encoded through biased hopping depending on internal state (e.g., protected directions labeled by Potts states), giving rise to global flocking and density modulations. Anisotropic hopping rates $$W_{\mathrm{hop}}(\sigma \rightarrow p) = D[1 + \epsilon (q \delta_{\sigma, p} - 1)/(q-1)]$$ encode the activity (Chatterjee et al., 2019, Mangeat et al., 2020).

Variants include restrictions on particle occupancy (single/hard/soft-core rules), and multi-cycle flip networks (octahedral, cubic, antiprism) which allow competition among several cyclic flip loops (Noguchi, 1 Dec 2025).

2. Phase Behavior and Emergent Dynamical Modes

Active Potts models organize a rich taxonomy of dynamical regimes and nonequilibrium phases, determined primarily by the amplitude of cyclic drive $h$, interaction strength $J$, particle density, and network/flip-graph topology:

2.1. Cyclic Single-Spin-Flip Models (Minimal Active Potts)

• Homogeneous-Cycling (HC) mode: For small $h\ll J$, the system remains almost uniformly in a single Potts state for long intervals; nucleation events trigger a stochastic global switch to the next Potts state in the cycle. Residence time $$\tau_{\mathrm{one}}\sim\exp(\mathrm{const}\times J)/N$$ decays exponentially with increasing $h$ (Noguchi et al., 2023, Noguchi, 21 Sep 2024).
• Multi-phase Wave (Wq, Q) mode: For large $h$, nucleation barriers are overcome and concurrent fronts develop, leading to coexistence of $q$ states separated by ballistically moving interfaces. For $q=3$, spiral waves dominate; for $q\geq4$, layered traveling waves or mosaics of domains form, depending on flip and contact energies (Noguchi et al., 2023, Noguchi, 21 Sep 2024, Noguchi, 30 Jun 2025).
• Diagonal/Mixed/Skipping-State Modes: Factorizable $q$ (e.g., $q=6$) allows for robust skipping-state spiral domains (e.g., W3: three-state spiral among $\{0,2,4\}$ or $\{1,3,5\}$); $q=4$ supports diagonal two-state cycling (HC2) and mixed phases (Noguchi, 30 Jun 2025).

2.2. Active Matter and Flocking APMs

• Gas, Liquid, Band/Lane Coexistence: Varying density and bias yields uniform disordered "gas," ordered "liquid" (full local alignment), or coexistence with either transverse bands or longitudinal lanes. Transverse–longitudinal reorientation is a nontrivial effect for $q\ge4$, controlled by anisotropic diffusion (Chatterjee et al., 2019, Mangeat et al., 2020).
• Motility-Induced Phase Separation and Jamming: Occupancy-restricted APMs generate arrested, jammed clusters through a motility-induced first-order phase separation (MIPS). The transition lines, binodals, and jamming thresholds are analytically tractable (Karmakar et al., 2022, Rosembach et al., 2023).

2.3. Multi-cycle Competition

• Networks of overlapping $3$- or $4$-state flip cycles yield rich coexistence phenomena: with multiple 3-cycles, all spiral wave types can coexist (W4, W6, W8) for high $h$; at intermediate $h$, only a subset of spiral wave types dominate or switch stochastically. For 4-cycle-dominated graphs (e.g., cubes), only single-state (homogeneous) phases are stable, punctuated by rare domain incursions (Noguchi, 1 Dec 2025).

3. Mesoscopic and Continuum Theories

The nonequilibrium lattice dynamics admit a coarse-grained description in terms of reaction–diffusion field theories for local state densities $\rho_i(\mathbf{r}, t)$. The key structural elements are:

• Nonreciprocal Reaction Terms: For minimal models, the continuum equations take the form

$$\partial_t \rho_i = D\nabla^2 \rho_i + f_i(\{\rho\}, h)$$

with nonlinear $f_i$ including a cyclic bias proportional to $h$ (Noguchi et al., 2023, Noguchi, 21 Sep 2024).

• Free-Energy Functionals: Generalized free-energy densities combine entropy of mixing, self-energies, and quadratic interaction and gradient terms, with the nonequilibrium bias entering via nonreciprocal single-site fields (Noguchi et al., 3 Jul 2024).
• Linear Stability and Instabilities: Uniform mixed states undergo Hopf and finite-$k$ instabilities as $h$ crosses a threshold. Analytical results for the growth rates and pattern selection rely on the structure of the flip terms and the competition with nucleation rates (Noguchi et al., 2023, Noguchi, 22 Sep 2025).
• Hydrodynamic Flocking APMs: In active flocking models, coupled PDEs for densities and polarization (local magnetization) fields describe the evolution,

$$\partial_t \rho_\sigma = D_\| \partial_\|^2 \rho_\sigma + D_\perp \partial_\perp^2 \rho_\sigma - v \partial_\| \rho_\sigma + \sum_{\sigma'\neq\sigma}F_{\sigma,\sigma'}(\rho)(\rho_\sigma-\rho_{\sigma'})$$

capturing band-to-lane reorientation, motility-induced phase separation, and jamming (Chatterjee et al., 2019, Karmakar et al., 2022).

4. Finite-Size Scaling, Pattern Formation, and Transitions

• Transition Classification: The HC–wave (or HC–spiral) transition is continuous for small systems but becomes sharply discontinuous, with pronounced hysteresis and phase coexistence for large $N$; the threshold $h_c^+$ vanishes as $N \rightarrow \infty$ (spiral/wave phase dominates for any $h>0$) (Noguchi et al., 2023).
• Coarsening Dynamics: Statistical measures such as the correlation length $r_{\mathrm{cr}}$ and mean cluster size $S(t)$ display $t^{1/2}$ scaling consistent with Allen–Cahn (curvature-driven) dynamics until saturation to the steady-state wavelength. Nonspiral wave regimes show transient enhancements of the coarsening exponent (Noguchi, 22 Sep 2025).
• Order Parameters & Binder Cumulants: For $n$-fold symmetric phases, $R_n = \langle s_n \rangle_t$ (with $s_n(t) = (1/N)|\sum_j e^{2\pi i n s_j/q}|$), susceptibilities $\chi_n$, and Binder parameters $U_n$ are used to locate and characterize phase boundaries. Second-order transitions in mixed/skipping-state modes yield exponents close to equilibrium Potts universality; dynamical transitions involving spiral waves exhibit altered correlation-length exponents (Noguchi, 30 Jun 2025).
• Hysteresis and Metastability: For sufficiently large $N$, return from the spiral/wave phase to homogeneous cycling is exponentially rare, leading to the robust hysteresis observed both in minimal cyclic models and in occupancy-restricted flocking APMs with jamming (Noguchi et al., 2023, Karmakar et al., 2022).
• Resilience to Protocol Details: The dynamic and scaling behavior are robust to choice of lattice geometry (square or hexagonal) and update rule (Metropolis or Glauber) (Noguchi, 22 Sep 2025).

5. Role of Flip Networks, Symmetry, and Factorization

• Flip-Cycle Topology: The combinatorics of allowed state transitions (the "flip-graph") tightly control the number and nature of coexisting wave types and spiral modes. Embedding multiple three-state cycles yields coexistence of multiple spiral-wave types; exclusive four-state cycles favor homogeneous dominance (Noguchi, 1 Dec 2025).
• Factorizable $q$: For $q$ divisible by smaller integers, coherent skipping-state patterns emerge (e.g., at $q=6$ both three-state spiral waves and mixed pairs of states are stabilized, depending on additional contact energies) (Noguchi, 30 Jun 2025, Noguchi, 22 Sep 2025).
• Symmetry and Transition Universality: While dynamical pattern-forming transitions involving propagating waves are sensitive to nonequilibrium drive, purely static mixed or diagonal phase transitions retain equilibrium universality class exponents (e.g., three-state Potts exponents for $q=6$ skipping states) (Noguchi, 30 Jun 2025).

6. Broader Implications and Connections

Active Potts models connect multiple domains:

• Reaction–diffusion systems: The cyclic flip dynamics provide a lattice realization of nonequilibrium reaction cycles; spiral waves mirror those in excitable chemical or biological media (Noguchi et al., 2023, Noguchi et al., 3 Jul 2024).
• Active matter and flocking: Discrete heading models interpolate between active Ising ($q=2$), Vicsek-like flocks, and continuous-symmetry Toner–Tu hydrodynamics. Motility-induced jamming and band–lane transitions are direct consequences of the interplay between internal state symmetry and translational activity (Chatterjee et al., 2019, Karmakar et al., 2022, Rosembach et al., 2023).
• Rock–paper–scissors (RPS) and evolutionary games: The RPS dynamics correspond to $q=3$ active Potts with cyclic drive; higher-order cycles extend RPS-type extinction and coexistence phenomena to richer pattern sets (Noguchi et al., 2023, Noguchi, 1 Dec 2025).
• Critical phenomena: Activity modifies correlation-length exponents and introduces new nonequilibrium pattern universality classes, especially in modes with dynamical wave propagation (Noguchi, 30 Jun 2025, Noguchi, 22 Sep 2025).

A plausible implication is that by tuning activity, interaction structure, and flip topology, active Potts models provide a minimal but flexible theoretical framework to engineer or analyze rich far-from-equilibrium pattern selection, dynamic symmetry breaking, and jamming transitions in synthetic or biological lattices.