On the relation between Vicsek and Kuramoto models of spontaneous synchronization

TL;DR

This paper explores the connection between the Vicsek and Kuramoto models of synchronization, showing their isomorphism under mean-field approximation and analyzing how different noise types affect phase transitions.

Contribution

It establishes an isomorphism between Vicsek and Kuramoto models and introduces a mixed noise type, revealing tricritical behavior in the phase diagram.

Findings

Scalar and vector noises lead to continuous and discontinuous transitions.

The isomorphism allows transfer of insights between models.

A new mixed noise type affects the nature of phase transitions.

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.