A note on phase transitions for the Smoluchowski equation with dipolar potential

Pierre Degond (1) , Amic Frouvelle (2) , Jian-Guo Liu (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Department of Physics and Department of Mathematics

In this note, we study the phase transitions arising in a modified Smoluchowski equation on the sphere with dipolar potential. This equation models the competition between alignment and diffusion, and the modification consists in taking the strength of alignment and the intensity of the diffusion as functions of the order parameter. We characterize the stable and unstable equilibrium states. For stable equilibria, we provide the exponential rate of convergence. We detail special cases, giving rise to second order and first order phase transitions, respectively. We study the hysteresis diagram, and provide numerical illustrations of this phenomena.

Domaines: Physique mathématique [math-ph]

Pierre Degond, Amic Frouvelle, Jian-Guo Liu. A note on phase transitions for the Smoluchowski equation with dipolar potential. Hyperbolic Problems: Theory, Numerics, Applications, Jun 2012, Padova, Italy. pp.179-192. ⟨hal-00765704⟩