Macroscopic limits and phase transition in a system of self-propelled particles

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

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Duke Physics

We investigate systems of self-propelled particles with alignment interaction. Compared to previous work, the force acting on the particles is not normalized and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space inhomogeneous extension of a previous work by Frouvelle & Liu in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a non-linear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a hydrodynamic model for self-alignment interactions previously derived in Degond & Motsch. However, the modified normalization of the force gives rise to different convection speeds and the resulting model may lose its hyperbolicity in some regions of the state space.

Domaines: Physique mathématique [math-ph]

Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Macroscopic limits and phase transition in a system of self-propelled particles. Journal of Nonlinear Science, 2013, 23 (3), pp.427-456. ⟨10.1007/s00332-012-9157-y⟩. ⟨hal-00621766⟩