  In this paper, we study the set of stationary solutions of the
Vlasov-Fokker-Planck (VFP) equation. This equation describes the time evolution
of the probability distribution of a particle moving under the influence of a
double-well potential, an interaction potential, a friction force and a
stochastic force. We prove, under suitable assumptions, that the VFP equation
does not have a unique stationary solution and that there exists a phase
transition. Our study relies on the recent results by Tugaut and coauthors
regarding the McKean-Vlasov equation.
