Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming

Abstract

We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker-Smale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure.

References in corpus (9)

• Novel type of phase transition in a system of self-driven particles
• Self-organization in systems of self-propelled particles
• A new model for self-organized dynamics and its flocking behavior
• A well-posedness theory in measures for some kinetic models of collective motion
• State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System
• Spontaneously ordered motion of self-propelled particles
• Self-organised complex aerial displays of thousands of starlings: a model
• Mean-field limit for the stochastic Vicsek model
• Macroscopic limits and phase transition in a system of self-propelled particles