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Noise-driven Synchronization of Vicsek Model in Mean

Wei Su, Yongguang Yu, Ge Chen

TL;DR

This work addresses how noise influences synchronization and phase transitions in the Vicsek model by shifting focus from almost-sure synchronization to synchronization in mean. It develops a rigorous framework demonstrating the existence of a noise bound $\bar{\delta}$, dependent on system parameters, such that for $\delta\in(0,\bar{\delta}]$ the system achieves $\tau$-synchronization in mean for any initial configuration, i.e., $\limsup_{t\to\infty}\mathbb{E}\,d_{\theta}(t)\le\tau$. The key contributions include a constructive argument leveraging a noise-control sequence to drive cohesion in finite time and probabilistic bounds that guarantee mean synchronization, complemented by simulations validating the theory. This provides a mathematical foundation for Vicsek model phase-transition theory and highlights the constructive role of bounded noise in promoting order in stochastic, self-organizing systems.

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.

Noise-driven Synchronization of Vicsek Model in Mean

TL;DR

This work addresses how noise influences synchronization and phase transitions in the Vicsek model by shifting focus from almost-sure synchronization to synchronization in mean. It develops a rigorous framework demonstrating the existence of a noise bound , dependent on system parameters, such that for the system achieves -synchronization in mean for any initial configuration, i.e., . The key contributions include a constructive argument leveraging a noise-control sequence to drive cohesion in finite time and probabilistic bounds that guarantee mean synchronization, complemented by simulations validating the theory. This provides a mathematical foundation for Vicsek model phase-transition theory and highlights the constructive role of bounded noise in promoting order in stochastic, self-organizing systems.

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.

Paper Structure

This paper contains 5 sections, 5 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Main Results
  4. Simulations
  5. Conclusions

Key Result

Theorem 3.1

Given any configuration $x_i(0)\in [0,B]^2$, $\theta_i(0)\in[-\pi,\pi)$, $i\in\mathcal{V}$ and $v>0, r\in(0,B)$ of the systems (Model:Vicsekmodel) - (Model:neighbor), suppose the noises $\{\xi_i(t)\}_{i\in\mathcal{V},t\geq 1}$ are zero-mean random variables with independent and identical distributio

Figures (2)

  • Figure 1: $\phi(t)$
  • Figure 2: The average value of $d_\theta(t)$ across 50 simulation runs

Theorems & Definitions (10)

  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof