Spatiotemporal Patterns in Active Four-State Potts Models

Before simulating the lattice Potts models, we investigate the flip dynamics in homogeneous mixed states. It corresponds to a lattice model with no interactions between neighboring sites or with strong mixing, like in a continuous flow stirred tank reactor (CSTR) in chemical reaction.Hohmann and J. Müller (1996) Each site flips independently, and hence, the density $\rho_{i}$ of the $s=i$ state is followed by the equation,

where $w_{ij}$ is the flip rate from $s=i$ to $s=j$ and $\sum_{i}\rho_{i}=1$. We consider two types of flip routes as shown in Fig. 1a and b. In the first case (Fig. 1a), each state can flip only to successive states ($s=k$ to $s=[k-1]$ or $s=[k+1]$), where $[k^{\prime}]=k^{\prime}\mathrm{\ mod\ }4$, i.e., $w_{02}=w_{20}=w_{13}=w_{31}=0$. We consider the condition of $w_{k[k+1]}\geq w_{[k+1]k}>0$ for $k\in[0,3]$, and therefore, each site exhibits a cyclic change in the four states on average as $s=0\to 1\to 2\to 3\to 0$. In the second case (Fig. 1b), diagonal flips between $s=0$ and $s=2$ are additionally allowed ($w_{20}\geq w_{02}>0$ and $w_{13}=w_{31}=0$). Each site can exhibit the cyclic change in the three states ($s=0\to 1\to 2\to 0$) as well as that in the four states. Therefore, we refer to them as four-state cyclic condition and three- and four-state cyclic condition, respectively. Chemical reaction on a substrate is an example of possible realizations of these four states. $s=0$ is the unoccupied site, a reactant molecule binds on the site at $s=1$, and the bound molecule changes its structure by sequential reactions as $s=1\to 2\to 3$ (see Fig. S1). The final product unbinds from the substrate in the four cycle ($s=3\to 0$) and the intermediate product unbinds in the three cycle ($s=2\to 0$).

We use the Metropolis rate $w_{ij}=\min[1,\exp(h_{ij})]$ and $h_{ji}=-h_{ij}$ for available flips to ensure that the flips between two states obey the local detailed balance condition. The thermal energy $k_{\mathrm{B}}T$ is normalized to unity and $h_{k[k+1]}\geq 0$. The steady state is obtained by solving the simultaneous equations of $d\rho_{i}/dt=0$. Under the four-state cyclic condition, the steady state is given by

where $h_{\mathrm{cyc}}=h_{01}+h_{12}+h_{23}+h_{30}$. This is similar to that of the three-state cyclic condition reported in Ref. 33. For $h_{\mathrm{cyc}}=0$, Eq. (2) provides the detailed balance relation $\rho_{k}/\rho_{[k-1]}=e^{h_{[k-1]k}}$ in thermal equilibrium. However, for $h_{\mathrm{cyc}}\neq 0$, the density ratio deviates from it as shown in Fig. 1c. Forward flips occur more frequently than the backward flips and the flow rate $q_{k[k+1]}=w_{k[k+1]}\rho_{k}-w_{[k+1]k}\rho_{[k+1]}$ increases with increasing $h_{30}$, and $q_{\mathrm{f}}=q_{01}=q_{12}=q_{23}=q_{30}$ in the steady state (see Fig. 1d).

Under the three- and four-state cyclic condition, the cyclic flips in the three states ($s=0\to 1\to 2\to 0$) increase with increasing $h_{20}$ (see Fig. 1e and f). The total flow $q_{12}=q_{23}+q_{20}$ also increases, while the flow $q_{23}=q_{30}$ through the four states decreases.

We consider the Potts model of a two-dimensional (2D) square lattice consisting of $N$ sites. The nearest neighboring sites have contact energy $-J$ if they are in the same state and $0$ if they are in different states. We use $J=2$ to induce phase separation between different states. The dynamics of the Potts model under the four-state cyclic condition (Fig. 1a) are presented below.

We first consider the symmetric condition, $h_{01}=h_{12}=h_{23}=h_{30}=h$ (Figs. 2–4). At low $h$, the lattice is dominantly occupied by one of the states most of the time, and the dominant phases cyclically change as $s=0\to 1\to 2\to 3\to 0$. Conversely, at high $h$, the domains of the four states spatially coexist. We call these two modes HC4 (homogeneous cycling of the four phases) and Q (quad-phase coexistence), respectively. At medium $h$, these two modes temporally coexist, as shown in Figs. 2 and 3b and Movie S1. These behaviors are similar to those observed in the three-state Potts model Noguchi et al. (2024); however, there are clear differences: i) The spatial coexistence of the four states does not form stable spiral waves unlike that in the three-state Potts model; three domain boundaries can stably meet at one point in the 2D space, whereas four boundaries do not. Thus, the domain boundary of successive states ($s=k$ and $[k+1]$) moves in the direction from the $s=[k+1]$ to $s=k$ domains, but the resultant waves do not have a center. Since the diagonal states ($s=0$ and $2$ or $s=1$ and $3$) do not flip to each other, the domain boundary of the diagonal states does not move ballistically but exhibits slow diffusion through the nucleation of the other states at the boundary (the diagonal boundaries look stopping in Movie S1). ii) Spatial coexistence of the diagonal-state domains. Once the domains of $s=k$ state stochastically disappear ($s=3$ at $t=4900$ in Fig. 2); the $s=[k-1]$ state spreads to the entire space. During the spreading, the domain of the diagonal state $s=[k+1]$ has a long lifetime (see the snapshot at $t=6000$ in Fig. 2 and the state fraction at $t\simeq 500~{}000$ in Fig. 3b). The circular domains of the diagonal state shrink slowly, reducing the boundary energy. Note that a quasi-straight domain boundary connected to itself through a periodic boundary has a longer lifetime, since it has no preferred direction. When the domains of the other states grow before spreading out, the system goes back to the Q mode (see the state fraction at $t\simeq 110~{}000$ in Fig. 3a). iii) In the HC4 mode, the sequence of the dominant phases is occasionally skipped via the nucleation of the next state during the domain growth (see Figs. 3c and 4). For example, the formation of an $s=3$ domain at the boundary of a circular $s=2$ domain results in the $s=3$ dominant phase (see Fig. 4a and Movie S2). The target pattern of $s=3$ and $s=2$ domains is formed in the $s=1$ phase, since the growing $s=2$ domain surrounds the unmoving domain boundary between $s=1$ and $s=3$. In contrast, the contact of the three domain boundaries results in the formation of spiral waves in the three-state Potts model. Thus, the existence of this inactive domain boundary is the origin of the dynamics different from those in the three-state Potts model.

To quantitatively clarify the mode transitions, we calculate the time fraction of each phase (see Fig. 3d). The lattice is considered to be covered by one phase at $N_{s}/N>0.98$ for $s\in[0,3]$, and $n$ phases spatially coexist when $n$ states satisfy $N_{s}/N>0.05$. As $h$ increases, the time fraction of one-phase existence decreases and that of the four-phase coexistence increases; they correspond to the fraction of the HC4 and Q modes, respectively. Hence, we consider the dominant mode to be the HC4 (Q) mode when the one-phase fraction is larger (smaller) than the four-phase fraction (see the solid lines and upper arrows in Fig. 3d). As the system size $N$ increases, the transition between the HC4 and Q modes occurs at lower $h$ (see the dashed lines in Fig. 3d and the solid line in Fig. S2). This is because the nucleation occurs more frequently ($\propto N$) and the complete disappearance of one of the states occurs less frequently in larger systems.

Since the nucleation also occurs more frequently at higher $h$, the mean lifetime $\tau_{1}$ of the one-phase decreases exponentially with increasing $h$ (see Fig. 3e). The mean lifetime $\tau_{2}$ of the two-phase coexistence decreases more slowly and becomes longer than $\tau_{1}$ at high $h$ in the Q mode (compare two lines in Fig. 3e); hence, a two-phase coexistence temporally appears in the Q mode as shown in Fig. 3a. The cycling of states is considerably faster in the Q mode than that in the HC4 mode, since the domain boundaries, in which the flips occur without an energy penalty, constantly exist in the Q mode (see Fig. 3f).

To examine the effects of asymmetric flip energies, we varied $h_{30}$ while keeping the other parameters constant at $h_{01}=h_{12}=h_{23}=h$ (see Fig. 5). At high or low $h_{30}/h$, one of the states dominates the entire system and does not cyclically change to other phases, like the dominant phases in thermal equilibrium. We call this steady state E_k mode, where $s=k$ is the dominant state. We distinguish the HC4 and E_k modes using the probability distribution of $N_{s}/N$ (see Fig. S3). In the HC4 mode, all of states have a peak at $N_{s}/N\simeq 1$. When the peak of one state at $N_{s}/N\simeq 1$ does not exist or is too low (less than ten times of the local minimum close to $N_{s}/N=1$), we consider it as the E_k mode, where the $s=k$ state has the highest peak. Further, the coexistence of the diagonal two phases appears between the Q and E_k modes, which we call D_k[k+2] mode ($k=0$ or $1$, see Fig. 5a and b and Fig. S3c). We consider that the system is in the D_k[k+2] mode when the time fraction of the two-phase coexistence of the $s=k$ and $s=[k+2]$ states is larger than those of the one-phase and four-phase (see Fig. 5e). The D_k[k+2] modes do not appear in the three-state Potts model, whereas the E_k modes do.

At low $h$, the HC4 mode changes into the E₂ mode with increasing $h_{30}$ (see Fig. 5c). This is different from that in the homogeneously mixed system, in which $N_{0}$ increases and $N_{3}$ decreases (see Fig. 1c). This dependency is similar to that of the three-state Potts model Noguchi and Fournier (2024) (see the next subsection), but the mechanism is different. The lifetime $\tau_{1}$ of the $s=3$ dominant phase decreases with increasing $h_{30}$, whereas those of the other phases are almost constant at $h_{30}/h<1.4$ (see Fig. S4b). The transition from the $s=k$ to $s=[k+1]$ dominant phases is caused by nucleation and growth. At high $h_{30}$, an $s=3$ domain in the $s=2$ phase changes into an $s=0$ domain through the nucleation in the $s=3$ domain (see Fig. S4c). When the resultant $s=0$ domain in the $s=2$ phase is circular, it slowly shrinks, and the system returns back to the $s=2$ phase; therefore, the cycling fails. This returning dynamics occurs more frequently at higher $h_{30}$. Thus, the fraction of the $s=2$ phase increases, and the E₂ mode is eventually formed (see Fig. S4a).

At high $h$, the mode sequentially changes from E₃, to D₁₃, Q, D₀₂, and E₂ with increasing $h_{30}$ (see Fig. 5c–e). With increasing system size, the region of each mode remains almost unchanged but the mode boundaries become sharper (narrower regions of mode coexistence), as shown in Fig. 5e. The flow rate $q_{\mathrm{f}}$ is maximized in the Q mode at $h_{30}/h=1$ (see Fig. 5f). In the D₁₃ and D₀₂ modes, $s=1$ and $s=0$ states form smaller domains that shrink slowly, respectively. New domains are formed by the nucleation and growth of the other states ($s=0$ and $2$ for D₁₃ and $s=1$ and $3$ for D₀₂), as shown in Fig. S5 and Movie S3.

We consider the Potts model for the three- and four-state cyclic condition (Fig. 1b). We fix the four-state cyclic condition as $h_{01}=h_{12}=h_{23}=h_{30}=h$ and vary $h_{20}$ of the three-state cycle. As expected, the homogeneous-cycling mode of the three states (HC3, $s=0\to 1\to 2\to 0$) and spiral-wave mode (SW) appear with increasing $h_{20}$ (see Fig. 6). In the SW mode, the domains of the three states ($s=0$, $1$, and $2$) rotate around the contact point of the three domains (see Fig. 6a and Movie S4). We consider the system to be in the SW mode if the time fraction of the coexistence of $s=0$, $1$, and $2$ phases is larger than those of the others (see Fig. 7). The flow through the three states ($q_{20}$ for $s=0\to 1\to 2\to 0$) is faster than that through the four states ($q_{23}$ for $s=0\to 1\to 2\to 3\to 0$) in the SW mode (see Fig. 7c and f). When the one-phase fraction is the largest, if the $s=0$, $s=1$, and $s=2$ states have peaks at $N_{S}/N\simeq 1$ but $s=3$ does not, it is in the HC3 mode (see Fig. S6).

At low $h$, the HC4 mode changes into the HC3 mode, and subsequently to the E₁ mode, with an increase in $h_{20}$ (see Figs. 6b and S7). The E₁ mode is formed due to an increase in the lifetime of the $s=1$ dominant phase (see Fig. S7b) Noguchi and Fournier (2024); let us consider a cluster of two sites of $s=2$ in the $s=1$ phase. There are two ways to reduce the cluster size. One is the direct flip of from $s=2$ to $s=1$ with the rate $\min[1,\exp(2J-h_{12})]$, since the number of the contacts between $s=1$ and $2$ sites is reduced from six to four. The other is the two-step flip ($s=2\to 0\to 1$) with the rates $\min[1,\exp(-J+h_{20})]$ and $\min[1,\exp(3J+h_{01})]$ for the first (rate-limiting) and second steps, respectively. The latter indirect flip proceeds more frequently with increasing $h_{20}$, and hence the $s=1$ phase becomes the dominant phase rather than the $s=0$ phase. In contrast to the mode transition from HC4 to E_k, the nucleation is suppressed during the three-state cycling.