Eur. Phys. J. Special Topics 157, 111-122 (2008)
https://doi.org/10.1140/epjst/e2008-00634-x

A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms

¹ Max-Planck-Institute for Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany
² Center for Information Services and High Performance Computing, Technische Universität Dresden, Zellescher Weg 12, 01069 Dresden, Germany
³ Physikalisch-Technische Bundesanstalt, Abbestr. 2-12, 10587 Berlin, Germany

Abstract

A mean-field approach (MFA) is proposed for the analysis of orientational order in a two-dimensional system of stochastic self-propelled particles interacting by local velocity alignment mechanism.
The treatment is applied to the cases of ferromagnetic (F) and liquid-crystal (LC) alignment. In both cases, MFA yields a second order phase transition for a critical noise strength and a scaling exponent of 1/2 for the respective order parameters. 
We find that the critical noise amplitude η_c at which orientational order emerges in the LC case is smaller than in the F-alignment case, i.e. η^LC_C<η^F_C.
A comparison with simulations of individual-based models with F- resp. LC-alignment shows that the predictions about the critical behavior and the qualitative relation between the respective critical noise amplitudes are correct.