Domain statistics in the relaxation of the one-dimensional Ising model with strong long-range interactions

Corberi, Federico; Kumar, M. Pawan; Lippiello, Eugenio; Politi, Paolo

doi:10.1016/j.chaos.2023.113681

Cited by 5 publications

(2 citation statements)

References 14 publications

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“…When interactions are sufficiently long-ranged, due to the one-to-all character of the coupling, there can be a size dependence at any time [26]. In the following we show that, in the present model, this happens for α ⩽ 2, and we determine such N dependence for the quantity L(t) in the coarsening stage.…”

Section: Size Dependence In the Coarsening Stagementioning

confidence: 52%

Kinetics of the one-dimensional voter model with long-range interactions

Corberi,

Castellano

2024

J. Phys. Complex.

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The voter model is an extremely simple yet nontrivial prototypical model of ordering dynamics, which has been studied in great detail. Recently, a great deal of activity has focused on long-range statistical physics models, where interactions take place among faraway sites, with a probability slowly decaying with distance. In this paper, we study analytically the one-dimensional long-range voter model, where an agent takes the opinion of another at distance r with probability ∝ r−α. The model displays rich and diverse features as α is changed. For α > 3 the behavior is similar to the one of the nearest-neighbor version, with the formation of ordered domains whose typical size grows as R(t) ∝ t1/2 until consensus (a fully ordered configuration) is reached. The correlation function C(r,t) between two agents at distance r obeys dynamical scaling with sizeable corrections at large distances r > r∗(t), slowly fading away in time. For 2 < α ≤ 3 violations of scaling appear, due to the simultaneous presence of two lengh-scales, the size of domains growing as t(α−2)/(α−1), and the distance L(t) ∝ t1/(α−1) over which correlations extend. For α ≤ 2 the system reaches a partially ordered stationary state, characterised by an algebraic correlator, whose lifetime diverges in the thermodynamic limit of infinitely many agents, so that consensus is not reached. For a finite system escape towards the fully ordered configuration is finally promoted by development of large distance correlations. In a system of N sites, global consensus is achieved after a time T ∝ N2 for α>3,T ∝Nα−1 for2<α≤3,andT ∝N forα≤2.

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Section: Size Dependence In the Coarsening Stagementioning

confidence: 52%

Kinetics of the one-dimensional voter model with long-range interactions

Corberi,

Castellano

2024

J. Phys. Complex.

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“…In recent years, there has been a growing interest in the investigation of long-range statistical models far from equilibrium, particularly concerning phase-ordering kinetics [11,[18][19][20][21][22][23][24][25][26], as well as other related aspects [27][28][29]. When long-range interactions are present, much less is known of the aging properties [11].…”

Section: J Stat Mech (2024) 053204mentioning

confidence: 99%

Aging properties of the voter model with long-range interactions

Corberi,

Smaldone

2024

J. Stat. Mech.

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We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, S i = ± 1 , positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P ( r ) ∝ r − α . Employing both analytical and numerical methods, we compute the two-time correlation function G ( r ; t , s ) ( t ⩾ s ) between the state of a variable Si at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A ( t , s ) = G ( r = 0 ; t , s ) , decays algebraically for α > 1 as [ L ( t ) / L ( s ) ] − λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ = 1 / ( α − 1 ) for 1 < α ⩽ 2 . For α ⩽ 1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α. The space-dependent correlation G ( r ; t , s ) obeys a scaling symmetry G ( r ; t , s ) = g [ r / L ( s ) ; L ( t ) / L ( s ) ] for α > 2. Similarly, for 1 < α ⩽ 2 , one has G ( r ; t , s ) = g [ r / L ( t ) ; L ( t ) / L ( s ) ] , where the length L regulating two-time correlations now differs from the coherence length as L ∝ L δ , with δ = 1 + 2 ( 2 − α ) .

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

Corberi,

dello Russo,

Smaldone

2024

J. Stat. Mech.

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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.

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Domain statistics in the relaxation of the one-dimensional Ising model with strong long-range interactions

Cited by 5 publications

References 14 publications

Kinetics of the one-dimensional voter model with long-range interactions

Kinetics of the one-dimensional voter model with long-range interactions

Aging properties of the voter model with long-range interactions

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

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