# American Institute of Mathematical Sciences

June  2017, 22(4): 1295-1327. doi: 10.3934/dcdsb.2017063

## A continuum model for nematic alignment of self-propelled particles

 1 Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom 2 Faculty of Mathematics, University of Vienna, Austria, Current affiliation: Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA 3 Université de Toulouse; UPS, INSA, UT1, UTM, CNRS; Institut de Mathématiques de Toulouse, UMR 5219, France, and Department of Mathematics, Imperial College London, United Kingdom, Current affiliation: Inst. für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, 52056, Germany

* Corresponding author: Pierre Degond

Received  May 2016 Revised  October 2016 Published  February 2017

A continuum model for a population of self-propelled particles interacting through nematic alignment is derived from an individual-based model. The methodology consists of introducing a hydrodynamic scaling of the corresponding mean field kinetic equation. The resulting perturbation problem is solved thanks to the concept of generalized collision invariants. It yields a hyperbolic but non-conservative system of equations for the nematic mean direction of the flow and the densities of particles flowing parallel or anti-parallel to this mean direction. Diffusive terms are introduced under a weakly non-local interaction assumption and the diffusion coefficient is proven to be positive. An application to the modeling of myxobacteria is outlined.

Citation: Pierre Degond, Angelika Manhart, Hui Yu. A continuum model for nematic alignment of self-propelled particles. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1295-1327. doi: 10.3934/dcdsb.2017063
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$M_0(\theta)$ for $\kappa=0.5, 2,10$ (red-dotted, black-solid, blue-dashed).
$g(\theta)$ for $\kappa=0.5, 2,10$ (red-dotted, black-solid, blue-dashed).
Local dynamics for $\lambda(\rho)$ given by 82. The arrows mark the flow field in the $(\rho_+,\rho_-)$ plane. The red-dotted and green-dashed lines show the values for which $\lambda(\rho_+)\rho_--\lambda(\rho_-)\rho_+=0$. The blue-solid line shows the threshold values $\rho_++\rho_-=2\sqrt{\frac{\lambda_0}{\lambda_1}}$.
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