Numerical investigation of the Brownian $q=2$ Potts Model

In active matter, such as the Vicsek Model of flocking, particles possesses an internal degree of freedom, such as their director, which is subject to interaction with other particles, provided they are within a certain range. In an effort to understand better the interplay between spatial and internal degrees of freedom, we study numerically a variation of the $q=2$ Potts Model on and off the lattice, where particles are additionally subject to Brownian motion. The lack of a feedback of the internal degrees of freedom to the spatial degrees of freedom renders this model generically non-equilibrium. We confirm previous work that showed that the static exponents of the phase transition are unaffected by the diffusion. In contrast to previous work, we show that the formation of ordered clusters is not undermined by diffusion, but should rather be thought of as an effective form of interaction. We demonstrate how our numerical findings can be understood on the basis of the well-established Model A, B and C: Off lattice, the Brownian $q=2$ Potts Model is Model C. On the lattice, it is Model A with an additional (irrelevant, conserved) Model B noise.