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

Liu, Jian-Guo; Frouvelle, Amic; Degond, Pierre

Liu, Jian-Guo; Frouvelle, Amic; Degond, Pierre (2012), A note on phase transitions for the Smoluchowski equation with dipolar potential, Hyperbolic Problems: Theory, Numerics, Applications, 2012-06, Padova, ITALY

Type

Communication / Conférence

External document link

https://www.aimsciences.org/book/AM/volume/Volume%208

Date

2012

Conference title

Hyperbolic Problems: Theory, Numerics, Applications

Conference date

2012-06

Conference city

Padova

Conference country

ITALY

Metadata

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Author(s)

Liu, Jian-Guo
Frouvelle, Amic

Degond, Pierre

Abstract (EN)

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.

Subjects / Keywords

Smoluchowski equation; equilibrium states