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Absence of $O (2)$ symmetry in the Vicsek model

Yushin Takahashi, Kota Mitsui, Tsuyoshi Mizohata, Hideyuki Miyahara

Abstract

The phase transition in the Vicsek model is widely believed to be associated with spontaneous symmetry breaking of the two-dimensional rotational symmetry $O (2)$. In this paper, we revisit the original Vicsek model introduced in [Phys. Rev. Lett. 75, 1226] and demonstrate that the model lacks $O (2)$ symmetry. As a consequence, we numerically demonstrate that the phase transition reported in the original paper vanishes when the global phase is adaptively chosen.

Absence of $O (2)$ symmetry in the Vicsek model

Abstract

The phase transition in the Vicsek model is widely believed to be associated with spontaneous symmetry breaking of the two-dimensional rotational symmetry . In this paper, we revisit the original Vicsek model introduced in [Phys. Rev. Lett. 75, 1226] and demonstrate that the model lacks symmetry. As a consequence, we numerically demonstrate that the phase transition reported in the original paper vanishes when the global phase is adaptively chosen.

Paper Structure

This paper contains 7 sections, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vicsek model
  3. Translational transformation and $O (2)$ symmetry
  4. Numerical simulations
  5. Continuous-time limit of the arithmetic-mean Vicsek model
  6. Remark
  7. Conclusions

Figures (1)

  • Figure 1: Time evolution of the order parameter, Eq. \ref{['main_eq_def_order-parameter_Vicsek-model_001_001']}, of (upper) the $\arctan$ Vicsek model, Eq. \ref{['main_eq_definition_original-Vicsek-model_001_001']}, and (lower) the arithmetic-mean Vicsek model, Eq. \ref{['main_eq_mean-angle_angle-Vicsek-model_001_001']}. We set $N = 1600$, $L = 8.0$, $v_\mathrm{abs} = 0.01$, $r_\mathrm{V} = 0.1$, $\eta = 0.75$, and $\Delta t = 1.0$. We run simulations 30 times to compute standard errors. In the inset, we plot the data for $t \in [0, 200]$ to see the relaxation process.