Abstract: We consider a nonlinear parabolic equation that describes the meanfield limit for a system of N diffusions in a bounded domain interacting through a finiterange potential V. We show that for all potentials that are not of positive type (characterized by the positivity of the Fourier transform of V) the system has a firstorder phase transition at the critical temperature, which is manifested by the nonuniqueness/instability of the steady solutions.
An interesting feature of the model is that the "trivial" steady state characterized by the uniform density remains locally stable in the subcritical region while for sufficiently large perturbations the dynamics prefers other, spatially nonuniform stady states.
