SEMINAIRE D'ANALYSE NUMERIQUE
Année universitaire 2012-2013
Jeudi 22 novembre 2012 :
Jian-Guo LIU
(Duke University)
Phase transition for Smoluchowski equation.
Smoluchowski equation on sphere is a mean field equation
for large interacting oriented/rod-like particles in some physical
and biologic system such as vicsek flocking dynamics,
ferromagnetism and polymers. There is a threshold in noise level or
particle density which gives the phase transition from isotropic
to anisotropic states. In this talk, I will focus on the dipolar
interaction kernel in Smoluchowski equation and show that the
anisotropic states are given by Fisher-von Mises distribution
and dynamical solutions converges exponentially to the isotropic
and anisotropic states for sub-critical and super-critical cases,
respectively. I will also show algebraic convergence for the
critical case. For the space inhomogeneous dynamics, we derived
the hydrodynamic systems and nonlinear diffusion systems for ordered
and disordered regimes, respectively. This is a joint work
with Amic Frouvelle of Dauphine and Pierre Degond of Toulouse.
