Communications in Mathematical Sciences

Volume 14 (2016)

Number 4

Single to double mill small noise transition via semi-Lagrangian finite volume methods

Pages: 1111 – 1136



José A. Carrillo (Department of Mathematics, Imperial College London, South Kensington Campus, London, United Kingdom)

Axel Klar (Department of Mathematics, Technische Universität Kaiserslautern, Germany; and Fraunhofer ITWM, Kaiserslautern, Germany)

Andreas Roth (Department of Mathematics, Technische Universität Kaiserslautern, Germany)


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 this fact accurately, 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 terms 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.


self-propelled interacting particles, mean-field equations, semi-Lagrangian method, finite volume method, milling solution, phase transition

2010 Mathematics Subject Classification

35B40, 74S10, 82C22, 92C15

Published 5 June 2023