Critical assessment of von Mises distribution and an infinite series ansatz for self-propelled particles

Abstract

We consider a Vicsek model of self-propelled particles with bounded confidence, where each particle interacts only with neighbors that have a similar direction. Depending on parameters, the system exhibits a continuous or discontinuous polar phase transition from the isotropic phase to a phase with a preferred direction. In a recent paper [1] the von Mises distribution was proposed as an ansatz for polar ordering. In the present system the time evolution of the angular distribution can be solved in Fourier space. We compare the results of the Fourier analysis with the ones obtained by using the von Mises distribution ansatz. In the latter case the qualitative behavior of the system is recovered correctly. However, quantitatively there are serious deviations. We introduce an extended von Mises distribution ansatz such that a second term takes care of the next two Fourier modes. With the extended ansatz we find much better quantitative agreement. As an alternative approach we also use a Gaussian and a geometric series ansatz in Fourier space. The geometric series ansatz is analytically handable but fails for very weak noise, the Gaussian ansatz yields better results but it is not analytically treatable.