Swarming Transition in Super-Diffusive Self-Propelled Particles

Abstract

A super-diffusive Vicsek model is introduced in this paper that incorporates Levy flights with exponent (Formula presented.). The inclusion of this feature leads to an increase in the fluctuations of the order parameter, ultimately resulting in the disorder phase becoming more dominant as (Formula presented.) increases. The study finds that for (Formula presented.) values close to two, the order–disorder transition is of the first order, while for small enough values of (Formula presented.), it shows degrees of similarities with the second-order phase transitions. The article formulates a mean field theory based on the growth of the swarmed clusters that accounts for the decrease in the transition point as (Formula presented.) increases. The simulation results show that the order parameter exponent (Formula presented.), correlation length exponent (Formula presented.), and susceptibility exponent (Formula presented.) remain constant when (Formula presented.) is altered, satisfying a hyperscaling relation. The same happens for the mass fractal dimension, information dimension, and correlation dimension when (Formula presented.) is far from two. The study reveals that the fractal dimension of the external perimeter of connected self-similar clusters conforms to the fractal dimension of Fortuin–Kasteleyn clusters of the two-dimensional (Formula presented.) Potts (Ising) model. The critical exponents linked to the distribution function of global observables vary when (Formula presented.) changes.