Jamming and flocking in the restricted active Potts model

We study the active Potts model with either site occupancy restriction or on-site repulsion to explore jamming and kinetic arrest in a flocking model. The incorporation of such volume exclusion features leads to a surprisingly rich variety of self-organized spatial patterns. While bands and lanes of moving particles commonly occur without or under weak volume exclusion, strong volume exclusion along with low temperature, high activity, and large particle density facilitates jams due to motility-induced phase separation. Through several phase diagrams, we identify the phase boundaries separating the jammed and free-flowing phases and study the transition between these phases which provide us with both qualitative and quantitative predictions of how jamming might be delayed or dissolved. We further formulate and analyze a hydrodynamic theory for the restricted APM which predicts various features of the microscopic model.

Figure 1

(above) Sketch of the $q=4$ rAPM showing biased hopping (solid bent arrow) to a neighboring lattice site with rate $D(1+\epsilon )$ and unbiased hopping (faint bent arrow) along the remaining directions with rate $D(1-\epsilon /3)$. Hard-core restriction is represented by a Heaviside step function $\mathrm{\Theta }(\mathrm{MPS}-{\rho }_i),$ and soft-core repulsion is represented by the probability $exp(-2\beta U{\rho }_i)$. Red, green, blue, and black circles represent particles of state $\sigma =\{1,2,3,4\}$, respectively. (below) Illustration of on-site flipping of a particle from $\sigma =3$ to $\sigma =1$ with flipping rate $W_{\mathrm{flip}}$.

Figure 2

Time evolution snapshots of the rAPM with MPS = 1 displaying MIPS. The right boundary of the diagonal high-density band is due to particles of state $\sigma =3$ (blue) and $\sigma =4$ (black), and the left boundary is due to particles of state $\sigma =1$ (red) and $\sigma =2$ (green). Parameters: Pe = 50 and ${\rho }_0=0.3$.

Figure 3

$\mathrm{Pe}-{\rho }_0$ phase diagram (for $\gamma =10$ and varying $\epsilon$) for MPS = 1 illustrated by snapshots at time $t={10}^5$. As density increases, the system undergoes a transition from the disordered gas phase to a MIPS state [31] for $\mathrm{Pe}>8$. At the intermediate densities, we observe solid diagonal bands where a small section of the band (Pe = 50, ${\rho }_0=0.3$) is magnified to provide a detailed structure of the domain.

Figure 4

(a) $\mathrm{Pe}-{\rho }_0$ phase diagram for MPS=1 for the snapshots in Fig. 3. (b) MIPS phase diagram (for $\epsilon =0.6$, 1.5, and 2.4) showing that the binodals ${\rho }_{\mathrm{low}}$ and ${\rho }_{\mathrm{high}}$ are independent of $\epsilon$. The critical Pe is estimated as ${\mathrm{Pe}}_c\simeq 8$ above which the phase separation occurs.

Figure 5

(a), (c) Steady-state snapshots of the four-state rAPM with $\mathrm{MPS}>1$ in the $T-\mathrm{MPS}$ plane. (a) For $\epsilon =0.9$, an increase in $\mathrm{MPS}$ induces a transition from the jammed MIPS state to the liquid phase at lower $T$ and to the coexistence region with transversely propagating liquid band at a higher $T$. (c) Snapshots for $\epsilon =2.7$ where a jam-to-lane transition occurs at large $\mathrm{MPS}$ and higher $T$. Here the coexistence band is longitudinal. Legend: red ($\sigma =1$): right; green ($\sigma =2$): up; blue ($\sigma =3$): left; black ($\sigma =4$): down; white: empty. Arrows indicate the direction of motion. (b), (d) $T-\mathrm{MPS}$ phase diagrams showing the four phases of rAPM where phase transition happens from the jammed state to the free-flowing state as either MPS or $T$ is increased. Parameter: ${\rho }_0=4$.

Figure 6

(a), (c) Steady-state snapshots for $\mathrm{MPS}>1$ in the $T-{\rho }_0$ plane for (a) MPS $=15$ and $\epsilon =0.9$ and (c) MPS $=60$ and $\epsilon =2.4$. (b), (d) $T-{\rho }_0$ phase diagrams of the rAPM with hard-core restriction for (b) $\epsilon =0.9$ and (d) $\epsilon =2.4$. In (b) the liquid phase appears for intermediate densities but makes a transition to the jammed MIPS state at higher densities, whereas in (d), at large activity, jamming appears early and the liquid phase is observed only when the temperature is moderately high.

Figure 7

(a) Steady-state snapshots and (b) $\epsilon -{\rho }_0$ phase diagram of the rAPM with MPS = 20 for $T=1.42$ ($\beta =0.7$). (a) An increase in ${\rho }_0$ and $\epsilon$ drives the system towards a jammed state. (b) The phase diagram resembles the unrestricted APM [26, 27] for $\epsilon <2$ but breaks down at larger ${\rho }_0$ and $\epsilon$. The reorientation transition (transverse to longitudinal band motion) happens at $\epsilon \sim 2.1$ and a liquid-gas phase transition takes place at ${\rho }_0\sim 3.5$ for $\epsilon =0$ [26, 27].

Figure 8

Normalized maximal magnetization $m_{\max }$ for the rAPM with $\mathrm{MPS}>1$ as a function of $T, {\rho }_0$, and $\epsilon$ in (a)–(c), respectively. $m_{\max }$ of the coexistence region is larger than the jammed phase. For the disordered gas phase, $m_{\max }=0$ and $m_{\max }\simeq 1$ for the liquid phase.

Figure 9

(a), (c) Steady-state snapshots for the rAPM with soft-core repulsion in the $T-U$ plane for (a) $\epsilon =0.9$ and (c) $\epsilon =2.7$ with fixed ${\rho }_0=4$. (b), (d) $T-U$ phase diagrams for the parameters of (a) and (c), respectively. Transition to jamming happens at low $T$ and large $U$ with the fact that high propulsion facilitates jamming.

Figure 10

(a), (c) Snapshots of the rAPM with soft-core repulsion illustrating the different self-organized patterns as a function of $T$ and ${\rho }_0$ for (a) small $\epsilon =0.9$ ($U=0.07$) and (c) large $\epsilon =2.7$ ($U=0.02$). The snapshots signify that low temperature, high density, and large activity facilitate jamming. (b), (d) $T\text{−}{\rho }_0$ phase diagrams of the rAPM with soft-core repulsion for parameters of (a) and (c), respectively. Apart from the usual gas and liquid binodals observed in the unrestricted APM, we observe a jammed phase adjoining both coexistence and liquid phases at large densities and low temperatures.

Figure 11

(a) Steady-state snapshots of the rAPM with soft-core repulsion in the $\epsilon -{\rho }_0$ plane for $T=1.42$ ($\beta =0.7$) and $U=0.07$. (b) The soft-core $\epsilon -{\rho }_0$ phase diagram qualitatively resembles the hard-core phase diagram of Fig. 7 showing that the jamming threshold decreases with density and activity.

Figure 12

(a)–(c) Normalized maximal magnetization $m_{\max }$ of the rAPM with soft-core repulsion as a function of $T, \epsilon$, and $U$, respectively. A small nonzero $m_{\max }$ is indicative of a jammed phase, and a comparatively high $m_{\max }<1$ characterizes a coexistence region.

Figure 13

(a) MIPS density profile for $\mathrm{MPS}=1, {\rho }_0=0.75$ and $\mathrm{Pe}=10$, obtained numerically with $\mathrm{FreeFEM}++$ for $\mathcal{N}=100, \mathrm{\Delta }t=0.01$, and $t_{\max }=1000$. The density of the two disordered phases is ${\rho }_{\mathrm{low}}\simeq 0.391$ and ${\rho }_{\mathrm{high}}\simeq 0.941$. (b) Velocity-density phase diagram for $\mathrm{MPS}=1$, computed with the numerical solutions of Eq. (17), in a square domain of linear size $L=50$, for three values of $\epsilon$. The dotted line shows the spinodals given by Eq. (18), and the solid line represents the theoretical value of the binodals calculated in Appendix pp6.

Figure 14

(a) Jammed density profile for the hard-core rAPM with $\mathrm{MPS}=5$, for $\beta =0.75, {\rho }_0=2, \epsilon =2.7$, obtained numerically with $\mathrm{FreeFEM}++$ for $\mathcal{N}=75, \mathrm{\Delta }t=0.02$, and $t_{\max }=1000$. (b) Velocity-density phase diagram for the hard-core rAPM, computed with the numerical solutions of Eq. (19), for $\beta =0.75$ and $\mathrm{MPS}=6$ in a square domain of linear size $L=50$.

Figure 15

(a) Jammed density profile for the soft-core rAPM with $U=1$, for $\beta =0.75, {\rho }_0=2, \epsilon =2.7$, obtained numerically with $\mathrm{FreeFEM}++$ for $\mathcal{N}=75, \mathrm{\Delta }t=0.1$, and $t_{\max }=5000$. (b) Velocity-density phase diagram for the soft-core rAPM, computed with the numerical solutions of Eq. (22), for $\beta =0.75$ and $U=0.25$ in a square domain of linear size $L=50$.

Figure 16

MSD $R^2\left(t\right)$ vs $t$ (on a log-log scale) for several $\beta$. (a) Unrestricted APM and (b) rAPM with hard-core repulsion.

Figure 17

Morphological characteristics and transformation of jammed clusters in the rAPM with soft-core interactions as a function of temperature. Parameters: $U=0.16, \epsilon =2.7$, and ${\rho }_0=4$.

Figure 18

Jammed clusters in the active clock model and Vicsek model with soft-core repulsion: (a) four-state ACM [28] ($\beta =0.9, {\rho }_0=3, \overline{\epsilon }=0.9$, and $U=0.05$) and (b) Vicsek model [19] (noise $\eta =0.2, {\rho }_0=2, v_0=0.5$, and $U=0.07$). Colorbar represents particle orientation, which for four-state ACM particles can have only four discrete orientations: 0, $\pi /2, \pi$, and $3\pi /2$, whereas for VM, a particle can have any orientation in $[-\pi ,\pi ]$.

Figure 19

$U-{\rho }_0$ phase diagram of the rAPM with soft-core repulsion with snapshots at $t={10}^5$ for $\epsilon =0$ and $\beta =0.7$. For small $U$, similar to the unrestricted APM, we observe a phase transition from a disordered gaseous phase to an ordered liquid phase as density is increased. Due to $q=4$, a steady-state liquid phase can be formed by any of the four states, and the color code for the liquid phase is red ($q=1$), green ($q=2$), blue ($q=3$), and black ($q=4$). Arrows indicate the direction of motion. At large $U$, particles can segregate only locally into small clusters forming a orientationally disordered phase at large spatial scales.

Figure 20

Spinodals and binodals of the rAPM with MPS=1. (a) The dotted lines show the spinodal lines given by Eq. (F9), and the solid lines show the binodals calculated from the relations $g_0\left({\rho }_{\mathrm{high}}\right)=g_0\left({\rho }_{\mathrm{low}}\right)$ and $h_0\left({\rho }_{\mathrm{high}}\right)=h_0\left({\rho }_{\mathrm{low}}\right)$, with Eqs. (F25) and (F26). The dash-dotted lines display the asymptotic value of the binodals at large Péclet number, given by Eqs. (F28) and (F29). (b) Relative error between the binodal densities and their asymptotic values at large Péclet number.

Figure 21

(a), (b) Velocity-density stability diagram of the rAPM with hard-core repulsion for the disordered and ordered homogeneous solution for $\beta =0.75$ for $\mathrm{MPS}=2$. (a) The stability region of the disordered solution is plotted in blue, according to the eigenvalue ${\lambda }_{\mathrm{gas}}$ given by Eq. (G11). (b) The stability region of the ordered solution is plotted in blue, according to the eigenvalues ${\lambda }_{\parallel }$ and ${\lambda }_{\perp }$ given by Eqs. (G26) and (G27), respectively. The ordered solution is unstable only under longitudinal perturbations (${\lambda }_{\parallel }>0$) in the light blue region, and under transverse perturbations (${\lambda }_{\perp }>0$) in the light red region. (c) ${\epsilon }_{*}$ value for which the reorientation transition occurs in the rAPM with $\mathrm{MPS}>$, as a function of MPS for several temperatures $T={\beta }^{-1}$.

Figure 22

(a), (b) Velocity-density stability diagram of the rAPM with soft-core repulsion for the disordered and ordered homogeneous solution for $\beta =0.75$ and $U=0.5$. (a) The stability region of the disordered solution is plotted in blue, according to the eigenvalue ${\lambda }_{\mathrm{gas}}$ given by Eq. (H11). (b) The stability region of the ordered solution is plotted in blue, according to the eigenvalues ${\lambda }_{\parallel }$ and ${\lambda }_{\perp }$ given by Eqs. (H26) and (H27), respectively. The ordered solution is only unstable under longitudinal perturbations (${\lambda }_{\parallel }>0$) in the light blue region, and under transverse perturbations (${\lambda }_{\perp }>0$) in the light red region. (c) ${\epsilon }_{*}$ value for which the reorientation transition occurs, as a function of $U$ for several temperatures $T={\beta }^{-1}$.