Swarming Transition in Super-Diffusive Self-Propelled Particles

A super-diffusive Vicsek model is introduced in this paper that incorporates Levy flights with exponent α. 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 α increases. The study finds that for α values close to two, the order-disorder transition is of the first order, while for small enough values of α, 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 α increases. The simulation results show that the order parameter exponent β, correlation length exponent ν, and susceptibility exponent γ remain constant when α is altered, satisfying a hyperscaling relation. The same happens for the mass fractal dimension, information dimension, and correlation dimension when α 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 Q=2 Potts (Ising) model. The critical exponents linked to the distribution function of global observables vary when α changes.