  We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {\bf 19}1-24
(1978)]. In particular, we study stationary states of the mean field limit for
a system of weakly interacting diffusions moving in a multi-well potential
energy landscape, coupled via a Curie-Weiss type (quadratic) interaction
potential. The location and depth of the local minima of the potential are
either deterministic or random. We characterize the structure and nature of
bifurcations and phase transitions for this system, by means of extensive
numerical simulations and of analytical calculations for an explicitly solvable
model. Our numerical experiments are based on Monte Carlo simulations, the
numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov
equation), the minimization of the free energy functional and a continuation
algorithm for the stationary solutions.
