Dynamics in Active Cyclic Potts Model
Noguchi Group
Spatiotemporal patterns, such as traveling waves, have been observed in nonequilibrium active systems. Many phenomena are well-captured by a description in terms of nonlinear but deterministic partial differential equations. However, noise effects are not understood so far. We focused on the effects of thermal fluctuations, since they are significant on a molecular scale.
We simulated the nonequilibrium dynamics of a Potts model with three cyclic states (s = 0, 1, and 2) [1]. The neighboring sites of the same states have an attraction to induce a phase separation between different states, and they have a cyclic state-energy-difference in the rock–paper–scissors manner. Both forward and backward flips are considered by the Monte Carlo method. It is a model system for chemical reactions on a catalytic surface or molecular transport through a membrane. For the reaction case, the three states are reactant, product, and unoccupied sites. For the transport case, the molecule can bind to both surfaces and flip between these two states. For one cycle, a reaction proceeds in bulk in the former case, and a molecule is transported across the membrane in the latter case. In this study, we consider the cyclically symmetric condition, i.e., the self-energy difference of the successive states is constant as ε0 - ε1 = ε1 - ε2 = ε2 - ε0’ = h. This model can be tuned from thermal-equilibrium to far-from-equilibrium conditions and corresponds to the standard Potts model at h = 0.
We found two dynamic modes: homogeneous cycling mode and spiral wave mode. At a low cycling energy h between two states, the homogeneous dominant states cyclically change as s = 0 → 1 → 2 → 0 via nucleation and growth, as shown in Fig. 1(a). In contrast, spiral waves are formed at high energy h, as shown in Fig. 1(b). The waves are generated from the contacts of three states and rotate around them. The homogeneous cycling mode is newly found in this study, whereas the spiral waves have been reported in continuum models and other lattice models.
For large systems, a discontinuous transition occurs from these cyclic homogeneous phases to spiral waves, while the opposite transition is absent. Conversely, these two modes can temporally coexist for small systems, and the ratio of the two modes continuously changes with increasing h. The transition from the homogeneous cycling to spiral wave modes occurs by the nucleation of the third state during the domain growth, i.e., it is determined by the competition of the nucleation and growth. The opposite transition from the spiral wave to homogeneous cycling modes occurs through the stochastic disappearance of three-state contacts. With increasing system size, the former transition rate increases, but the latter rate exponentially decreases. The transition character is changed by these size dependencies.
References
- [1] H. Noguchi, F. van Wijland, and J.-B. Fournier, J. Chem. Phys. 161, 025101 (2024).