https://doi.org/10.1140/epjst/e2014-02192-0
Regular Article
Towards a quantitative kinetic theory of polar active matter
1 Department of Physics, North Dakota State University, Fargo, ND 58108-6050, USA
2 Max-Planck-Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
a e-mail: thomas.ihle@ndsu.edu
Received: 31 January 2014
Revised: 14 April 2014
Published online: 12 June 2014
A recent kinetic approach for Vicsek-like models of active particles is reviewed. The theory is based on an exact Chapman- Kolmogorov equation in phase space. It can handle discrete time dynamics and “exotic” multi-particle interactions. A nonlocal mean-field theory for the one-particle distribution function is obtained by assuming molecular chaos. The Boltzmann approach of Bertin, et al., Phys. Rev. E 74, 022101 (2006) and J. Phys. A 42, 445001 (2009), is critically assessed and compared to the current approach. In Boltzmann theory, a collision starts when two particles enter each others action spheres and is finished when their distance exceeds the interaction radius. The average duration of such a collision, τ0, is measured for the Vicsek model with continuous time-evolution. If the noise is chosen to be close to the flocking threshold, the average time between collisions is found to be roughly equal to τ0 at low densities. Thus, the continuous-time Vicsek-model near the flocking threshold cannot be accurately described by a Boltzmann equation, even at very small density because collisions take so long that typically other particles join in, rendering Boltzmann's binary collision assumption invalid. Hydrodynamic equations for the phase space approach are derived by means of a Chapman-Enskog expansion. The equations are compared to the Toner-Tu theory of polar active matter. New terms, absent in the Toner-Tu theory, are highlighted. Convergence problems of Chapman-Enskog and similar gradient expansions are discussed.
© EDP Sciences, Springer-Verlag, 2014
