Symmetry-breaking phase transitions

Mean-field kinetic models of alignment dynamics

In mean-field kinetic models of self-organized dynamics inspired from the Vicsek model, the distribution of particle orientations sometimes exhibits symmetry-breaking phase transitions due to multiple 'Local Thermodynamical' equilibria (LTE). These equilbria can be:

- Isotropic LTE corresponding to random motion (top picture). Such equilibra are rotationally symmetric.
- Anisotropic LTE corresponding to directed motion and described by the Von-Mises distribution (bottom picture). Such equilibria are not rotationally symmetric: a rotation of the velocity-space leads to another LTE still given by a Von-Mises distribution but pointing in a different direction.

isotropic LTE: histogram of orientations (left) ; plot of the particle directions (right)

anisotropic LTE corresponding to directed motion: histogram of orientations (left) ; plot of the particle directions (right)

Anisotropic LTE's appear only when the local density reaches a threshold and simultaneously, isotropic LTE becomes unstable.

The associated macroscopic dynamics exhibit two distinct phases

- A disordered phase at low densities associated to the isotropic LTE.At large spatio-temporal scales, this phase is described by a nonlinear diffusion equation for the density
- An ordered phase at large densities associated to the Von-Mises LTE. At large scales, this phase is described by "Self-Organized Hydrodynamics" (SOH)

This phase transition is second-order (see picture below, left). Depending on the interaction parameters, first-order phase transitions leading to hysteresis phenomena may appear (see picture below, right). The stability and instability of the various equilibria has been established in full mathematical rigor.

This picture shows the order parameter (a measure of the alignment of the particles) as a function of the density, for a second order phase transition (left) and for a first order one (right). The various curves correspond to various values of the physical parameters. Second order phase transitions exihibit a continuous transition from zero order (i.e. disorder) to finite order, while first order phase transitions show a discontinuous behavior of the order parameter, as well as a range of densities where both disordered and ordered equilibria coexist and are stable.

Such symmetry-breaking phase transitions also appear in the Cucker-Smale model with self-propulsion.

Boltzmann models of alignment dynamics

Phase transitions have also been found in a Boltzmann model for rod alignment, the 'BDG model' (after Bertin, Droz and Gregoire, J. Phys A: Math. Theor. 42, 445001 (2009) ; see section 'propagation of chaos' for a description of the BDG model). For the noiseless BDG model, dirac deltas (corresponding to all particles pointing in the same direction) have been proved to be stable and explicit convergence rates (in the Wasserstein metric) are given.

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