# American Institute of Mathematical Sciences

July  2014, 19(5): 1249-1278. doi: 10.3934/dcdsb.2014.19.1249

## Phase transition and diffusion among socially interacting self-propelled agents

 1 Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7058, United States 2 Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

Received  July 2012 Revised  October 2012 Published  April 2014

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-propulsive 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 propulsive force yields unrestricted SOH models and offers an alternative to the derivation of the SOH using kinetic models with speed constraints.
Citation: Alethea B. T. Barbaro, Pierre Degond. Phase transition and diffusion among socially interacting self-propelled agents. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1249-1278. doi: 10.3934/dcdsb.2014.19.1249
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##### References:
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