We investigate the phase transitions of the Brownian spin models in two dimensions. In these models, particles carrying the Potts or clock spins diffuse freely and interact with others within a fixed distance. The diffusion leads to the competition between spin fluctuation dynamics and particle density fluctuation dynamics. As a result of the competition, we find that Brownian $q$-state Potts model undergoes a continuous phase transition even for $q>4$, which is contrasted with a discontinuos phase transition of the equilibrium system in two-dimensional lattice. On the other hand, the Brownian $p$-state clock model for $q>4$ undergoes the double BKT transitions between paramagnetic, quasi-long range ordered phase, and the ordered phase. We present our numerical simulation results and analytic argument for the unversal nature of the phase transitions. We also discuss a possible extension of our works with passive Brwonian particles to those with self-propelled active particles.
[Reference] Chul-Ung Woo, Heiko Rieger, and Jae Dong Noh, "Suppression of discontinuos phase transitions by particle diffusion", Phys. Rev. E {\bf 105}, 054144 (2022).