2025/02/19 by Wei Su, Su, Wei, Yongguang Yu +3
Computer Science · #Adaptation and Self-Organizing Systems (nlin.AO) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Neural Networks and Applications #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2502.13993
openalex publication_date 2025/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.