Ordering kinetics with long-range interactions: interpolating between voter and Ising models

Abstract

Abstract We study the ordering kinetics of a generalization of the voter model with long-range interactions, the p-voter model, in one dimension. It is defined in terms of Boolean variables S i , agents or spins, located on sites i of a lattice, each of which takes in an elementary move the state of the majority of p other agents at distances r chosen with probability P ( r ) ∝ r − α . For p = 2 the model can be exactly mapped onto the case with p = 1, which amounts to the voter model with long-range interactions decaying algebraically. For 3 ⩽ p < ∞ , instead, the dynamics falls into the universality class of the one-dimensional Ising model with long-ranged coupling constant J ( r ) = P ( r ) quenched to small finite temperatures. In the limit p → ∞ , a crossover to the (different) behavior of the long-range Ising model quenched to zero temperature is observed. Since for p > 3 a closed set of differential equations cannot be found, we employed numerical simulations to address this case.