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

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 ${\eta }_c$ at which orientational order emerges in the LC case is smaller than in the F-alignment case, i.e. ${\eta }_C^{LC}<{\eta }_C^F$. 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.

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