Single to Double Mill Small Noise Transition via Semi-Lagrangian Finite Volume Methods
This work provides numerical evidence for a phase transition in kinetic swarming models, relevant for understanding collective behavior in biological and physical systems.
The authors demonstrate that double mills are more stable than single mills under stochastic perturbations in swarming dynamics, and numerically observe a phase transition from single to double mill as noise increases, transitioning to disordered states at higher noise levels.
We show that double mills are more stable than single mills under stochastic perturbations in swarming dynamic models with basic attraction-repulsion mechanisms. In order to analyse accurately this fact, we will present a numerical technique for solving kinetic mean field equations for swarming dynamics. Numerical solutions of these equations for different sets of parameters will be presented and compared to microscopic and macroscopic results. As a consequence, we numerically observe a phase transition diagram in term of the stochastic noise going from single to double mill for small stochasticity fading gradually to disordered states when the noise strength gets larger. This bifurcation diagram at the inhomogeneous kinetic level is shown by carefully computing the distribution function in velocity space.