Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics

Abstract

We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function.

Archive for Rational Mechanics and Analysis (2014) accepted

References in corpus (8)

• Novel type of phase transition in a system of self-driven particles
• Collective motion of self-propelled particles interacting without cohesion
• Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis
• State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System
• Macroscopic limits and phase transition in a system of self-propelled particles
• Hydrodynamic models of self-organized dynamics: derivation and existence theory
• Phase transitions in swarming systems: A recent debate
• Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators

Cited by in corpus (20)

• Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus
• Propagation of chaos: a review of models, methods and applications. II. Applications
• Global regularity for the fractional Euler alignment system
• Phase Transitions in a kinetic flocking model of Cucker-Smale type
• Consensus-Based Optimization on Hypersurfaces: Well-Posedness and Mean-Field Limit
• Traveling Bands, Clouds, and Vortices of Chiral Active Matter
• Global weak solutions for Kolmogorov-Vicsek type equations with orientational interactions
• Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions
• Structure preserving schemes for the continuum Kuramoto model: phase transitions
• Phase transitions and macroscopic limits in a BGK model of body-attitude coordination
• Kinetic models for topological nearest-neighbor interactions
• Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles
• Coupled Self-Organized Hydrodynamics and Navier-Stokes models: local well-posedness and the limit from the Self-Organized Kinetic-fluid models
• A Finite Volume Method for Continuum Limit Equations of Nonlocally Interacting Active Chiral Particles
• Uniform in time weak propagation of chaos on the torus
• Propagation of chaos: a review of models, methods and applications. I. Models and methods
• Bulk topological states in a new collective dynamics model
• How environment affects active particle swarms: a case study
• From Kinetic Flocking Model of Cucker-Smale Type to Self-Organized Hydrodynamic model
• Macroscopic limit of a Fokker-Planck model of swarming rigid bodies