Diffusion in Coulomb environment and a phase transition
I talk homogenization of diffusion in the two-dimensional Euclid space in a periodic Coulomb environment. That is, I consider a periodic point process in the plane. The diffusion has the repulsive interaction with the two-dimensional Coulomb potential with inverse temperature $\beta$ to each particle in the periodic point process. We first prove that the diffusion is diffusive with non-degenerated effective diffusion constant $\gamma$. We then remove one particle from the environment and consider the diffusive scaling limit of the diffusion. Then its new effective constant depends on inverse temperature $\beta$. It has a phase transition whose critical point is given explicitly in terms of the first effective diffusion constant $\gamma$ of the periodic homogenization problem. Using this result, we present explicit bounds for the critical point of the self-diffusion matrices of the strict two-dimensional Coulomb interacting Brownian motions for inverse temperature $\beta$. Moreover, we discuss the case d-dimensional case with d \ge 3.
