Macroscopic Limits and Phase Transition in a System of Self-propelled Particles

We investigate systems of self-propelled particles with alignment interaction. Compared to previous work (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012), 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 (Frouvelle and Liu, Dynamics in a kinetic model of oriented particles with phase transition, 2012), 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 nonlinear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, Communicated by Eva Kanso. P. Degond ( ) · A. Frouvelle Institut de Mathématiques de Toulouse, Université de Toulouse; UPS, INSA, UT1, UTM; 31062 Toulouse, France e-mail: [email protected] A. Frouvelle e-mail: [email protected] P. Degond · A. Frouvelle Institut de Mathématiques de Toulouse UMR 5219; CNRS; 31062 Toulouse, France J.-G. Liu Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA e-mail: [email protected]