Kinetic Description of Multiscale Phenomena

The Annual Kinetic FRG Meeting
September 21-25, 2009

CSIC Building (#406), Seminar Room 4122.

Phase transitions for the Vlasov-McKean model

Dr. Vladislav Panferov

California State University, Northridge

Abstract:  We consider a nonlinear parabolic equation that describes the mean-field limit for a system of N diffusions in a bounded domain interacting through a finite-range 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 first-order 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.