Finite-size scaling analysis and dynamic study of the critical behavior of a model for the collective displacement of self-driven individuals

Abstract

The Vicsek model (VM) [T. Vicsek, Phys. Rev. Lett. 75, 1226 (1995)], for the description of the collective behavior of self-driven individuals, assumes that each of them adopts the average direction of movement of its neighbors, perturbed by an external noise. A second-order transition between a state of ordered collective displacement (low-noise limit) and a disordered regime (high-noise limit) was found early on. However, this scenario has recently been challenged by Grégory and Chaté [G. Grégory and H. Chaté, Phys. Rev. Lett. 92, 025702 (2004)] who claim that the transition of the VM may be of first order. By performing extensive simulations of the VM, which are analyzed by means of both finite-size scaling methods and a dynamic scaling approach, we unambiguously demonstrate the critical nature of the transition. Furthermore, the complete set of critical exponents of the VM, in d=2 dimensions, is determined. By means of independent methods--i.e., stationary and dynamic measurements--we provide two tests showing that the standard hyperscaling relationship dnu-2beta=gamma holds, where beta, nu, and gamma are the order parameter, correlation length, and "susceptibility" critical exponents, respectively. Furthermore, we established that at criticality, the correlation length grows according to xi-t1z, with z approximately = 1.27(3) , independently of the degree of order of the initial configuration, in marked contrast with the behavior of the XY model.

Equilibrium and dynamical behavior in the Vicsek model for self-propelled particles under shear

“…2 shows the time evolution of χ for the cases with noise equal to η = 0 070, 0 093 and 0 100, with other parameters given by ρ = 1/8 (N = 1058) and 0 = 0 1, in the absence of shear. The expected power-law behavior can be clearly distinguished at η = 0 093, in good agreement with previous results [30]. This is an indication that the parameters are tuned so that the system is at the critical point.…”

First-order phase transition in a model of self-propelled particles with variable angular range of interaction

“…A complete treatment of the numerical simulation of a phase transition would require a full-fledged finite size scaling (FSS) analysis [32,33]. This theory of finite size scaling was originally developed for the equilibrium phase transition, but now it is known that much of this analysis is applicable to far-from-equilibrium phase transitions also, such as the ones observed in the Vicsek model [14,[34][35][36]. Here we do not wish to carry out a full finite size scaling analysis for the phase transitions we have observed, because that would require very large system sizes and times, and is quite prohibitive for us …”

“…But in fact there does appear to be a second order phase transition which is obscured by the finite size effects. This can be seen from the behavior of the variance of the order parameter σ 2 (A, L) as a function of the neighborhood area A, defined by [14,36] …”

Scale invariance in natural and artificial collective systems: a review

“…These systems are often described using a minimal flocking model, first described by Vicsek et al [176], in which a set of self-propelled point particles advance at a fixed speed and tend to align their heading direction with other particles within an interaction range. The Vicsek model has been extended in multiple ways to describe a variety of collective motion systems [177][178][179][180][181]. Animal groups are expected to have scale-invariant behaviour, because they must display coherent collective dynamics regardless of the group size.…”