Coarsening and metastability of the long-range voter model in three dimensions
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Cited by 2 publications
References 52 publications
Ordering kinetics with long-range interactions: interpolating between voter and Ising models
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.
Ordering kinetics with long-range interactions: interpolating between voter and Ising models
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.
General properties of the response function in a class of solvable non-equilibrium models
We study the non-equilibrium response function R i j ( t , t ′ ) , namely the variation of the local magnetization ⟨ S i ( t ) ⟩ on site i at time t as an effect of a perturbation applied at the earlier time t′ on site j, in a class of solvable spin models characterized by the vanishing of the so-called asymmetry. This class encompasses both systems brought out of equilibrium by the variation of a thermodynamic control parameter, as after a temperature quench, or intrinsically out of equilibrium models with violation of detailed balance. The one-dimensional Ising model and the voter model (on an arbitrary graph) are prototypical examples of these two situations which are used here as guiding examples. Defining the fluctuation-dissipation ratio X i j ( t , t ′ ) = β R i j / ( ∂ G i j / ∂ t ′ ) , where G i j ( t , t ′ ) = ⟨ S i ( t ) S j ( t ′ ) ⟩ is the spin–spin correlation function and β is a parameter regulating the strength of the perturbation (corresponding to the inverse temperature when detailed balance holds), we show that, in the quite general case of a kinetics obeying dynamical scaling, on equal sites this quantity has a universal form X i i ( t , t ′ ) = ( t + t ′ ) / ( 2 t ) , whereas lim t → ∞ X i j ( t , t ′ ) = 1 / 2 for any ij couple. The specific case of voter models with long-range interactions is thoroughly discussed.
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