On the relation between Vicsek and Kuramoto models of spontaneous synchronization

Abstract: The Vicsek model for the self-propelled particles is investigated with the respect to the introduction of the stochastic perturbation of the dynamics. It is shown that such a dependence can be thought in terms of the isomorphism of the Vicsek model with the Kuramoto model of spontaneous synchronization. They are isomorphic at least within the mean-field approach. The isomorphism between two models allows to state the dependence of the type of the transition in Vicsek model on the noise perturbation. Two types of noise the scalar and the vector ones lead to qualitatively different behavior with continuous and the discontinuous transition to ordered state correspondingly. New type of the stochastic perturbation - mixed noise is proposed. It is the weighted superposition of the scalar and vector noises. The corresponding phase diagram noise amplitude vs. interaction strength is obtained and the tricritical behavior for Vicsek model is demonstrated.