Young Researchers Workshop: Kinetic descriptions in theory and applications

Kinetic equations and self-organized band formations

Sébastien Motsch

Arizona State University

Self-organization is an ubiquitous phenomenon in nature which can be observed in a variety of different contexts and scales, with examples ranging from fish schools, swarms of birds or locusts, to flocks of bacteria. The observation of such global patterns can often be reproduced in models based on simple interactions between neighboring particles. In this talk, we focus on a particular interaction dynamics introduced by Degond, Frouvelle and Liu closely related to the one described in the seminal paper of Vicsek and collaborators. We present a numerical scheme for this kinetic equation which preserves many physical properties of the model (e.g. entropy inequality, positivity) allowing to study the long-time behavior of the dynamics. We observe the emergence of 'band formations', i.e. the density concentrates along a curve. The dynamics of these bands are various (e.g. static, oscillatory) and remain to be understood analytically.