Many-particle systems: stationary solutions, phase transitions, and distinguished limits

We consider a class of weakly interacting particles/diffusion processes living either on the torus or in Euclidean space. We first introduce the corresponding mean-field PDE which is an aggregation-diffusion equation. We then proceed to study the existence of nontrivial stationary solutions and of phase transitions for both the case with linear diffusion and the case with porous medium-type degenerate diffusion. We provide necessary and sufficient conditions on the interaction term for the existence of phase transitions for the associated free energy and characterise interactions for which the phase transitions are continuous or discontinuous. We also provide a sufficient set of conditions for the existence of saddle points of the free energy. Finally, we discuss the behaviour of the underlying particle system under the diffusive rescaling in the large particle limit and the effect the presence of phase transitions has on this limit.