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.

Symmetry-breaking phase transitions in highly concentrated semen

“…In [36], the dependences of the macroscopic parameters c 1 and c 2 upon n are related to how the microscopic alignment and noise intensities depend on the local particle alignment. In particular, for adequate choices of the model parameters, phase transitions as the density crosses a threshold n c from disordered states (corresponding to c 1 ( n ) = c 2 ( n ) = 0 for n < n c ) to aligned states (corresponding to c 1 ( n ), c 2 ( n ) > 0 for n > n c ) can be obtained [36].…”

“…In particular, for adequate choices of the model parameters, phase transitions as the density crosses a threshold n c from disordered states (corresponding to c 1 ( n ) = c 2 ( n ) = 0 for n < n c ) to aligned states (corresponding to c 1 ( n ), c 2 ( n ) > 0 for n > n c ) can be obtained [36]. However, assumptions on the microscopic parameters translate into properties of the macroscopic parameters in a non-obvious way.…”

“…In [48], a derivation of the TT model is proposed. The approach differs from [35,36] in that the Vicsek model is replaced by a Boltzmann model describing binary interactions. However, the use of binary collisions for highly concentrated suspensions is subject to caution.…”

Hydrodynamic limits for kinetic flocking models of Cucker-Smale type

“…The alignment between particles is imposed via localized versions of the Cucker-Smale or Motsch-Tadmor reorientation procedure [26,36,34,18,19,41] leading to relaxation terms to the mean velocity modulated or not by the density of particles. By scaling the relaxation time towards the asymptotic cruise speed, or equivalently, penalizing the balance between friction and self-propulsion, this alignment interaction leads asymptotically to variations of the classical kinetic Vicsek-Fokker-Planck equation with velocities on the sphere, see [45,29,33,27,28,11,12]. It was shown in [2] that particular versions of the localized kinetic Cucker-Smale model can lead to phase transitions driven by noise.…”

Dimensional Reduction of a Multiscale Model Based on Long Time Asymptotics

“…While the kinetic formulation of gas dynamics is studied theoretically in [28], many applications in various fields are described in the literature, along with specific numerical methods, modeling for instance fluid dynamics [25], strain and stress in mechanical or biomechanical models [5], production models [21], crowd [13,33] models, predator-prey [14] and other biological systems [29].…”