2019/10/07 by Pierre Degond, Degond, Pierre, Sara Merino-Aceituno +1
Computer Science · Mathematics · Physics and Astronomy · #35K99 #35L60 #5Q80 #82C22 #82C31 #82C44 #82C70 #92D50 #Distributed Control Multi-Agent Systems #FOS: Physical sciences #Mathematical Physics (math-ph) #Micro and Nano Robotics #Nonlinear Dynamics and Pattern Formation #math-ph #math.MP #msc:35K99 #msc:35L60 #msc:5Q80 #msc:82C22 #msc:82C31 #msc:82C44 #msc:82C70 #msc:92D50
paper · pdf · doi:10.48550/arxiv.1910.02865
arxiv created 2019/10/07 · openalex publication_date 2019/10/07 · arxiv updated 2019/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics. After diffusive rescaling of the kinetic equation, we formally show that the distribution function converges to an equilibrium distribution in particle direction, whose local density and mean direction satisfies a cross-diffusion system. We show that the system is consistent with symmetries typical of a nematic material. The derivation is carried over by means of a Hilbert expansion. It requires the inversion of the linearized collision operator for which we show that the generalized collision invariants, a concept introduced to overcome the lack of momentum conservation of the system, plays a central role. This cross diffusion system poses many new challenging questions.