Phase transition and diffusion among socially interacting self-propelled agents

Alethea B T Barbaro , Pierre Degond (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider a hydrodynamic model of swarming behavior derived from the kinetic description of a particle system combining a noisy Cucker-Smale consensus force and self-propulsion. In the large self-propulsion force limit, we provide evidence of a phase transition from disordered to ordered motion which manifests itself as a change of type of the limit model (from hyperbolic to diffusive) at the crossing of a critical noise intensity. In the hyperbolic regime, the resulting model, referred to as the 'Self-Organized Hydrodynamics (SOH)', consists of a system of compressible Euler equations with a speed constraint. We show that the range of SOH models obtained by this limit is restricted. To waive this restriction, we compute the Navier-Stokes diffusive corrections to the hydrodynamic model. Adding these diffusive corrections, the limit of a large propulsion force yields unrestricted SOH models and offers an alternative to the derivation of the SOH using kinetic models with speed constraints. Acknowledgements: This work has been supported by the french 'Agence Nationale pour la Recherche (ANR)' in the frame of the contracts 'MOTIMO' (ANR-11-MONU-009-01) and 'CBDif-Fr' (ANR-08-BLAN-0333-01). 1

Mots clés: Swarm; Cucker-Smale model; Vicsek model; self-propulsion; hydrodynamic model; diffusion; Chapman-Enskog expansion AMS subject classification: 35L60; 35K55; 35Q80; 82C05; 82C22; 82C70; 92D50

Domaines: Mathématiques [math]

Alethea B T Barbaro, Pierre Degond. Phase transition and diffusion among socially interacting self-propelled agents. Discrete and Continuous Dynamical Systems - Series B, 2014. ⟨hal-02498833⟩