Flocking with a $q$-fold discrete symmetry: Band-to-lane transition in the active Potts model

We study the $q$-state active Potts model (APM) on a two-dimensional lattice in which self-propelled particles have $q$ internal states corresponding to the $q$ directions of motion. A local alignment rule inspired by the ferromagnetic $q$-state Potts model and self-propulsion via biased diffusion according to the internal particle states elicits collective motion at high densities and low noise. We formulate a coarse-grained hydrodynamic theory with which we compute the phase diagrams of the APM for $q=4$ and $q=6$ and analyze the flocking dynamics in the coexistence region, where the high-density (polar liquid) phase forms a fluctuating stripe of coherently moving particles on the background of the low-density (gas) phase. A reorientation transition of the phase-separated profiles from transversal band motion to longitudinal lane formation is found, which is absent in the Vicsek model and the active Ising model. The origin of this reorientation transition is revealed by a stability analysis: for large velocities the transverse diffusivity approaches zero and stabilizes lanes. Computer simulations corroborate the analytical predictions of the flocking and reorientation transitions and validate the phase diagrams of the APM.

Figure 1

Schematic of APM showing biased hopping rates to a nearest-neighbor lattice site for (a) $q=4$ and (b) $q=6$ states, with rates $D(1+\epsilon )$ in the favored direction and $D[1-\epsilon /(q-1\left)\right]$ in other directions [more than one particle is allowed on a site but not shown in (a) and (b) due to clarity of representation]. (c) Spin flips (i.e., directional changes) are performed locally (i.e., without moving the particle) with a rate $W_{\mathrm{flip}}(\sigma ,{\sigma }^{\prime })$ for a particle changing direction from state $\sigma$ to state ${\sigma }^{\prime }$.

Figure 2

Triangular mesh grids taken to integrate the hydrodynamic equation [Eq. (17)] for $q=4$ (left) and $q=6$ (right) are represented here with $\mathcal{N}=20$ vertices on each border.

Figure 3

Stationary profiles [(a),(c),(e),(g)] and snapshots of the bands [(b),(d)] and lanes [(f),(h)] (red and blue denote the liquid and gas phases, respectively) of the APM. [(a),(b)] Band motion for $q=4$, $\beta =0.75$, and $\epsilon =0.5$ where ${\rho }_{\mathrm{gas}}=1.10$ and ${\rho }_{\mathrm{liq}}=1.69$, and [(c),(d)] for $q=6$, $\beta =0.65$, and $\epsilon =0.5$ where ${\rho }_{\mathrm{gas}}=1.30$ and ${\rho }_{\mathrm{liq}}=1.75$. [(e),(f)] Lane formation for $q=4$, $\beta =0.75$, and $\epsilon =2.5$ where ${\rho }_{\mathrm{gas}}=0.815$ and ${\rho }_{\mathrm{liq}}=2.40$, and [(g),(h)] for $q=6$, $\beta =0.65$, and $\epsilon =4.5$ where ${\rho }_{\mathrm{gas}}=1.04$ and ${\rho }_{\mathrm{liq}}=2.11$. The reorientation transition occurs for $q=4$ at $\epsilon =2.0$ between (b) and (f) plotted for ${\rho }_0=1.33$, and for $q=6$ at $\epsilon =3.8$ between (d) and (h) plotted for ${\rho }_0=1.5$. The linear system size is $L=50$, and the numerical parameters are $\mathrm{\Delta }t=0.2$ and $\mathcal{N}=100$.

Figure 4

Phase diagrams of the $q$-state APM: (a) Temperature-density phase diagram for $q=4$ and $\epsilon =2.5$ and (c) for $q=6$ and $\epsilon =4$. (b) Velocity-density phase diagram for $q=4$ and $\beta =0.75$ and (d) for $q=6$ and $\beta =0.65$. The linear system size is $L=100$, and the numerical parameters are $\mathrm{\Delta }t=0.1$ and $\mathcal{N}=200$. The stationary state is reached for the final time $t_{\max }=1000$. In all phases diagrams the regions where transverse band motion and longitudinal lane formation are stable within the phase-separated domain $(\mathrm{G}+\mathrm{L}\mathrm{)}$ are shown.

Figure 5

Normalized magnetization $m_0/{\rho }_0$ of the homogeneous solutions for $q=4$ and $\beta =0.75$ in (a) and $q=6$ and $\beta =0.65$ in (b) as a function of the average density ${\rho }_0$. We show the analytical solutions given by Eq. (28) and the stability of these solutions analyzed in Secs. 4a and 4b for the disordered and ordered solutions, respectively. The square symbols represent the stationary state of the numerical solution for $L=100$ and the numerical parameters $\mathrm{\Delta }t=0.2$ and $\mathcal{N}=200$, starting from a phase-separated profile at $\epsilon =0$ for which the stationary state is homogeneous for all density values.

Figure 6

[(a),(b)] Stability diagrams for $q=4$ and $\beta =0.75$ and [(d),(e)] for $q=6$ and $\beta =0.65$. In (a) and (d), the domains where a perturbation along $x$ (${\lambda }_{\parallel }<0$) or along $y$ (${\lambda }_{\perp }<0$) is stable, are displayed. The two stability domains are separated, respectively, by the spinodals ${\phi }_{\mathrm{liq}}^{\parallel }$ and ${\phi }_{\mathrm{liq}}^{\perp }$. In (b) and (e), these analytical domains are compared to numerical solutions, starting with a small perturbation around the ordered homogeneous solution. Disks: both perturbations vanish, up triangles: only perturbation along $x$ grows, right triangles: only perturbation along $y$ grows, squares: both perturbations grow. (c) ${\epsilon }_{*}$ value as a function of the temperature $T={\beta }^{-1}$ for $q=4$. The same quantity is shown in (f) for $q=6$. Transverse and longitudinal stripes are stable, respectively, below and above this line. The plotted asymptotic expressions correspond to Eqs. (47, 48) in (c) and Eqs. (51)–(52) in (f).

Figure 7

Dynamics of the $q=4$-state APM for $\beta =0.75$, ${\rho }_0=1.33$ and $L=50$, and the numerical parameters $\mathrm{\Delta }t=0.2$ and $\mathcal{N}=100$. (a) The reorientation transition starting from an unstable longitudinal lane at $\epsilon =0.5<{\epsilon }_{*}$ transforming into a stable transverse band in the stationary state. (b) Opposite situation starting from an unstable transverse band at $\epsilon =2.5>{\epsilon }_{*}$ transforming into a stable longitudinal lane in the stationary state.

Figure 8

[(a)–(d)] The three different phases of the four-state APM for $\epsilon =0.9$. In (a), a disordered gaseous state evolves at $\beta =0.5$, ${\rho }_0=2$ while, in (c) an ordered liquid phase found at $\beta =0.95$, ${\rho }_0=6$. (b) At intermediate $\beta =0.8$, ${\rho }_0=3$, a stable liquid-gas coexistence phase is observed. (d) Stationary snapshot at $t={10}^4$, corresponding to the data in (b), showing a transversely moving band with $\sigma =3$. [(e)–(h)] Phase-separated density profiles of liquid (${\rho }_{\mathrm{liq}}$) and gaseous (${\rho }_{\mathrm{gas}}$) phases for four-state APM at a fixed $\beta$ with increasing initial average ${\rho }_0$. (e) Density profiles for $\beta =0.5$ and $\epsilon =2.1$, with phase-separated snapshot in (f) for ${\rho }_0=8$. (g) Density profiles for $\beta =0.5$ and $\epsilon =2.7$, and corresponding snapshot for ${\rho }_0=8$ is shown in (h). Arrows and color bars in (d), (f), and (h), respectively, represent direction of particles within the liquid stripes and on-site particle density.

Figure 9

[(a),(b)] Phase diagrams of the $q=4$-state APM. (a) Temperature ($T$) versus density (${\rho }_0$) phase diagram for fixed $\epsilon =2.1$, where ${\rho }_{\mathrm{gas}}$ and ${\rho }_{\mathrm{liq}}$ separate the three phases, gas (G), gas-liquid coexistence $(\mathrm{G}+\mathrm{L})$, and liquid (L). The dotted line in the $\mathrm{G}+\mathrm{L}$ region indicates $\epsilon =0$ transition density (${\rho }_{*}$) line as a function of the temperature. (b) $\epsilon$ versus ${\rho }_0$ phase diagram for $\beta =0.9$. The black dotted lines in (a) and (b) represent the reorientation transitions.

Figure 10

Characterization of phase transition in the four-state APM for $\epsilon =0$ and $\beta =0.6$. (a) Maximal magnetization $m_{\max }$ versus ${\rho }_0$ for lattice size $L=32$, 64, 96, and 128 are shown. The sudden jump in the magnetization signals a possible first-order phase transition. The phase transition is further corroborated by $U_4$ in (b), where the critical density ${\rho }_{*}=4.19\pm 0.01$ is estimated from the intersection of the data for various $L$. In (a) and (b), the point of transition is marked by the dotted lines. (c) Susceptibility ($\chi$) versus ${\rho }_0$ shows a discontinuous peak around ${\rho }_{*}$.

Figure 11

[(a)–(d)] The three phases of six-state APM for $\epsilon =2.5$. (a) Gas phase ($\beta =0.5$, ${\rho }_0=1$), (b) gas-liquid coexistence phase ($\beta =0.6$, ${\rho }_0=3$), and (c) liquid phase ($\beta =0.9$, ${\rho }_0=6$). The corresponding snapshot to (b) is shown in (d) where on site particle density is depicted in the color bar shown. [(e)–(h)] Phase separated density profiles for six-state APM with $\beta =0.7$ and increasing initial average ${\rho }_0$. (e) Density profiles for $\epsilon =2.5$, and phase-separated snapshot in (f) for ${\rho }_0=2.5$. (g) Density profiles for $\epsilon =4.5$, and corresponding snapshot for ${\rho }_0=2.5$ is shown in (h). Color bar represents on-site particle density.

Figure 12

Phase diagrams of the $q=6$-state APM, (a) $T$ versus ${\rho }_0$ for $\epsilon =3$ and (b) $\epsilon$ versus ${\rho }_0$ for $\beta =0.6$. ${\rho }_{\mathrm{gas}}$ and ${\rho }_{\mathrm{liq}}$ are the coexistence lines which respectively identify the densities of gas and liquid of the phase-separated profiles. Analogous to Fig. 9, ${\rho }_{*}$ in the $\mathrm{G}+\mathrm{L}$ region of (a) indicates $\epsilon =0$ transition densities. The reorientation transitions from transverse to longitudinal motion of the liquid stripes as a function of $T$ and $\epsilon$ are indicated by black dotted lines in (a) and (b), respectively.

Figure 13

Analogous to Fig. 10 but for $q=6$. (a) $m_{\max }$ versus ${\rho }_0$ for $L=32$, 64, and 96. (b) Quantification of ${\rho }_{*}$ from $U_4$ versus ${\rho }_0$, where ${\rho }_{*}=2.60\pm 0.02$. The dotted line in (b) mark the point of intersection of various $L$. The signature of a first-order phase transition in (a) and (b) are further validated in (c) where susceptibility ($\chi$) against ${\rho }_0$ shows a discontinuous peak around ${\rho }_{*}$.

Figure 14

Evolution of the variance $\langle {m_i^{\sigma }}^2\rangle -\langle {m_i^{\sigma }\rangle }^2$ of the magnetization $m_{\sigma }$ with the mean population $\langle {\rho }_i\rangle ={\rho }_0$ in the disordered phase ($q=4$ and $\beta =0.4$). The relation is linear and identical for all states $\sigma$.