Flocking dynamics with voter-like interactions

Abstract

We study the collective motion of a large set of self-propelled particles subject to voter-like interactions. Each particle moves on a two-dimensional space at a constant speed in a direction that is randomly assigned initially. Then, at every step of the dynamics, each particle adopts the direction of motion of a randomly chosen neighboring particle. We investigate the time evolution of the global alignment of particles measured by the order parameter ϕ, until complete order ϕ = 1.0 is reached (polar consensus). We find that ϕ increases as t 1/2 for short times and approaches exponentially fast to 1.0 for long times. Also, the mean time to consensus τ varies nonmonotonically with the density of particles ρ, reaching a minimum at some intermediate density ρ min . At ρ min , the mean consensus time scales with the system size N as τ min ∼ N 0.765 , and thus the consensus is faster than in the case of all-to-all interactions (large ρ) where τ = 2N . We show that the fast consensus, also observed at intermediate and high densities, is a consequence of the segregation of the system into clusters of equally-oriented particles which breaks the balance of transitions between directional states in well mixed systems.

Multistate voter model with imperfect copying

“…Here |•| is the absolute value, while Θ m (t) is the state of particle m (m = 1, .., N) at time t. The parameter ψ (0 ≤ ψ ≤ 1) is similar to that introduced in the context of flocking dynamics to quantify the degree of global alignment in a system of moving particles [9,28], while the parameter σ θ is a measure of the width of the distribution of angular states.…”

“…where we have assumed an initial value s τ (τ ) = S = N ≫ 1 and introduced the prefactor 2 that appears in the parallel update (see [9]). Note also that s τ is 1 after consensus is reached in a time of order N. Then, if at time t there are s τ (t) ≤ N surviving states that we callδ τ k (t) [k = 1, .., s τ (t)], we can express the states' variance as σ 2…”

“…The copying mechanism is also used in the voter model for opinion formation [7,8], where each individual adopts the opinion of one of its neighbors in the population. More recently, this type of social imitation rule was introduced to study the flocking dynamics of a large group of animals [9], for instance birds, where each bird aligns its flying direction with that of a nearby random bird. In the case of all-to-all interactions, this flocking voter model is equivalent to the well known multi-state voter model (MSVM) [10,11] for opinion dynamics, where the moving direction of a bird is associated to its opinion or decision.…”

Species exclusion and coexistence in a noisy voter model with a competition-colonization tradeoff

“…Other early works investigated the dynamics of catalytic reactions [43][44][45], and, more recently, the dynamics of the NVM has also been studied in complex networks [46][47][48]. A multistate NVM has been proposed to explain the emergence of flocking out from pairwise stochastic interactions [49][50][51]. Experimental evidence of these noisy voter-like interactions was found recently in groups of fish, where schooling is induced by the intrinsic noise that arises from the finite number of interacting individuals [52].…”

Species exclusion and coexistence in a noisy voter model with a competition-colonization tradeoff

“…Other early works investigated the dynamics of catalytic reactions [43][44][45] and, more recently, the dynamics of the NVM has also been studied in complex networks [46][47][48]. A multi-state NVM has been proposed to explain the emergence of flocking out from pairwise stochastic interactions [49,50]. Experimental evidence of these noisy voter-like interactions was found recently in groups of fish, where schooling is induced by the intrinsic noise that arises from the finite number of interacting individuals [51].…”