A Boltzmann model for rod alignment and schooling fish

Abstract

We consider a Boltzmann model introduced by Bertin, Droz and Greegoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the equilibria of this Boltzmann model and we rigorously show the existence of a pitchfork bifurcation when a parameter measuring the inverse of the noise intensity crosses a critical threshold. The analysis is carried over rigorously when there are only finitely many non-zero Fourier modes of the noise distribution. In this case, we can show that the critical exponent of the bifurcation is exactly 1/2. In the case of an infinite number of non-zero Fourier modes, a similar behavior can be formally obtained thanks to a method relying on integer partitions first proposed by Ben-Naim and Krapivsky.

22 pages, 3 figures

References in corpus (25)

• Novel type of phase transition in a system of self-driven particles
• Collective motion
• Onset of collective and cohesive motion
• Collective motion of self-propelled particles interacting without cohesion
• Non-equilibrium clustering of self-propelled rods
• A new model for self-organized dynamics and its flocking behavior
• Boltzmann and hydrodynamic description for self-propelled particles
• 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
• Continuum limit of self-driven particles with orientation interaction
• Hydrodynamics of self-propelled hard rods
• Phase transitions in systems of self-propelled agents and related network models
• On a kinetic model for a simple market economy
• Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models
• Mesoscopic theory for fluctuating active nematics
• Macroscopic limits and phase transition in a system of self-propelled particles
• Hydrodynamic models of self-organized dynamics: derivation and existence theory
• Nonequilibrium Statistical Mechanics of Self-propelled Hard Rods
• Dynamics in a kinetic model of oriented particles with phase transition
• Kinetic limits for pair-interaction driven master equations and biological swarm models
• A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters
• Phase transition and diffusion among socially interacting self-propelled agents
• Kinetic hierarchy and propagation of chaos in biological swarm models
• Alignment of Rods and Partition of Integers
• Collective Behavior of Self Propelling Particles with Kinematic Constraints; The relation between the discrete and the continuous description

Cited by in corpus (6)

• Propagation of chaos: a review of models, methods and applications. II. Applications
• Global regularity for the fractional Euler alignment system
• A rigorous derivation from the kinetic Cucker-Smale model to the pressureless Euler system with nonlocal alignment
• Nematic alignment of self-propelled particles in the macroscopic regime
• Continuum dynamics of the intention field under weakly cohesive social interactions
• An age-structured continuum model for myxobacteria