A noisy multi-state voter model for flocking dynamics

LOSCAR, ERNESTO S.; BAGLIETTO, GABRIEL; FEDERICO VAZQUEZ

La Plata

Workshop; Workshop on Statistical Mechanics of Swarming Behaviour; 2019

We introduce and study a simple model for flocking with voter-like interactions. At every step of the dynamics, a random particle adopts (copy) the direction of motion of a randomly chosen neighboring particle. We study the dynamics of the model in mean-field (all-to-all interactions), which is equivalent to that of a multi-state voter model with the addition of imperfection or noise in the copying process. We found that the global alignment of particles measured by an order parameter $psi$ in $[0, 1]$, depend on the amplitude $eta$ of the noise and the population size $N$. In the case of perfect copying $eta=0$ the system reaches an absorbing configuration with complete order ($psi=1$) for all values of N. However, for any degree of imperfection $eta>0$, we show that the average value of $psi$ at the stationary state decreases with N as $langle psi rangle simeq 6/(pi^2 eta^2 N)$ for $eta ll 1$ and $eta^2 N >1$, and thus the system becomes totally disordered in the thermodynamic limit $N to infty$. We also show that $langle psi rangle simeq 1-1.64 eta^2 N$ in the vanishing small error limit $eta to 0$, which implies that complete order is never achieved for $eta > 0$. These results are supported by Monte Carlo simulations of the model, which allow to study other scenarios as well.