Noise-driven Synchronization of Vicsek Model in Mean

TL;DR

This paper provides a rigorous mathematical proof that in the Vicsek model, maintaining noise within a certain bound guarantees synchronization in mean, advancing the theoretical understanding of phase transitions in collective behavior.

Contribution

It offers the first comprehensive theoretical proof showing noise bounds lead to synchronization in the Vicsek model, filling a gap in the understanding of its phase transition.

Findings

Existence of a noise bound for synchronization

Mathematical proof of phase transition behavior

Enhanced understanding of noise's role in self-organization

Abstract

The Vicsek model has long stood as a pivotal framework in exploring collective behavior and self-organization, captivating the scientific community with its compelling dynamics. However, understanding how noise influences synchronization within this model and its associated phase transition characteristics has presented significant challenges. While numerous studies have focused on simulations due to the model's mathematical complexity, comprehensive theoretical analyses remain sparse. In this paper, we deliver a rigorous mathematical proof demonstrating that for any initial configuration of the Vicsek model, there exists a bound on noise amplitude such that if the noise amplitude is maintained within this bound, the system will achieve synchronization in mean. This finding not only lays a solid mathematical groundwork for the Vicsek model's phase transition theory but also underscores the critical role of noise in collective dynamics, enhancing our understanding of self-organizing systems in stochastic environments.