Swarming Transition in Super-Diffusive Self-Propelled Particles

A super-diffusive Vicsek model is introduced in this paper that incorporates Levy flights with exponent [Formula: see text]. 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: see text] increases. The study finds that for [Formula: see text] values close to two, the order–disorder transition is of the first order, while for small enough values of [Formula: see text] , 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: see text] increases. The simulation results show that the order parameter exponent [Formula: see text] , correlation length exponent [Formula: see text] , and susceptibility exponent [Formula: see text] remain constant when [Formula: see text] is altered, satisfying a hyperscaling relation. The same happens for the mass fractal dimension, information dimension, and correlation dimension when [Formula: see text] 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: see text] Potts (Ising) model. The critical exponents linked to the distribution function of global observables vary when [Formula: see text] changes.