Communications in Mathematical Sciences

Volume 13 (2015)

Number 6

Macroscopic models of collective motion with repulsion

Pages: 1615 – 1638



Pierre Degond (Department of Mathematics, Imperial College London, United Kingdom)

Giacomo Dimarco (Department of Mathematics, University of Ferrara, Italy)

Thi Bich Ngoc Mac (Institut de Mathématiques de Toulouse, France)

Nan Wang (Department of Mathematics, National University of Singapore)


We study a system of self-propelled particles which interact with their neighbors via alignment and repulsion. The particle velocities result from self-propulsion and repulsion by close neighbors. The direction of self-propulsion is continuously aligned to that of the neighbors, up to some noise. A continuum model is derived starting from a mean-field kinetic description of the particle system. It leads to a set of non conservative hydrodynamic equations. We provide a numerical validation of the continuum model by comparison with the particle model. We also provide comparisons with other self-propelled particle models with alignment and repulsion.


Fokker–Planck equation, macroscopic limit, Von Mises–Fisher distribution, generalized collision invariants, non-conservative equations, self-organized hydrodynamics, self-propelled particles, alignment, repulsion

2010 Mathematics Subject Classification

35L60, 35Qxx, 82C22, 82C70, 92D50

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