Local stability of perfect alignment for a spatially homogeneous kinetic model
arXiv:1403.5233 · doi:10.1007/s10955-014-1062-3
Abstract
We prove the nonlinear local stability of Dirac masses for a kinetic model of alignment of particles on the unit sphere, each point of the unit sphere representing a direction. A population concentrated in a Dirac mass then corresponds to the global alignment of all individuals. The main difficulty of this model is the lack of conserved quantities and the absence of an energy that would decrease for any initial condition. We overcome this difficulty thanks to a functional which is decreasing in time in a neighborhood of any Dirac mass (in the sense of the Wasserstein distance). The results are then extended to the case where the unit sphere is replaced by a general Riemannian manifold.
References in corpus (8)
- Novel type of phase transition in a system of self-driven particles
- Interaction Ruling Animal Collective Behaviour Depends on Topological rather than Metric Distance: Evidence from a Field Study
- Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis
- Macroscopic limits and phase transition in a system of self-propelled particles
- Uniform convergence to equilibrium for granular media
- Kinetic limits for pair-interaction driven master equations and biological swarm models
- Kinetic hierarchy and propagation of chaos in biological swarm models
- Complete characterization of convergence to equilibrium for an inelastic Kac model