Noisy multistate voter model for flocking in finite dimensions

TL;DR

This paper investigates a noisy multistate voter model for flocking, showing how particle motion and noise influence collective order, with phase transitions characterized in static and moving particle scenarios.

Contribution

It introduces a model combining voter dynamics with particle motion, revealing how movement sustains order and characterizing phase transitions in finite dimensions.

Findings

Static particles reach disorder at any noise level in the thermodynamic limit.

Moving particles exhibit an order-disorder transition at a noise level proportional to their speed.

Finite-size transition noise scales inversely with system size, depending on dimensionality.

Abstract

We study a model for the collective behavior of self-propelled particles subject to pairwise copying interactions and noise. Particles move at a constant speed $v$ on a two--dimensional space and, in a single step of the dynamics, each particle adopts the direction of motion of a randomly chosen neighboring particle, with the addition of a perturbation of amplitude $\eta$ (noise). We investigate how the global level of particles' alignment (order) is affected by their motion and the noise amplitude $\eta$. In the static case scenario $v=0$ where particles are fixed at the sites of a square lattice and interact with their first neighbors, we find that for any noise $\eta_c>0$ the system reaches a steady state of complete disorder in the thermodynamic limit, while for $\eta=0$ full order is eventually achieved for a system with any number of particles $N$. Therefore, the model displays a transition at zero noise when particles are static, and thus there are no ordered steady states for a finite noise ($\eta>0$). We show that the finite-size transition noise vanishes with $N$ as $\eta_c^{1D} \sim N^{-1}$ and $\eta_c^{2D} \sim \left(N \ln N \right)^{-1/2}$ in one and two--dimensional lattices, respectively, which is linked to known results on the behavior of a type of noisy voter model for catalytic reactions. When particles are allowed to move in the space at a finite speed $v>0$, an ordered phase emerges, characterized by a fraction of particles moving in a similar direction. The system exhibits an order-disorder phase transition at a noise amplitude $\eta_c>0$ that is proportional to $v$, and that scales approximately as $\eta_c \sim v \, (-\ln v)^{-1/2}$ for $v \ll 1$. These results show that the motion of particles is able to sustain a state of global order in a system with voter-like interactions.