Nonequilibrium spin models with Ising universal behaviour

Oliveira, Mário J. de; Mendes, J. F. F.; Santos, M.

doi:10.1088/0305-4470/26/10/006

Cited by 135 publications

(210 citation statements)

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“…Again, the VM, given by r h = 1 2 − h 8 , is an exception. Its marginal situation in this context has already been noted [10,11] in the particular case where the transition probabilities only depend on h, a condition expressed here by r −s,h = 1 − r s,h . One is then left with only two free parameters, p b ≡ r 4 , p i ≡ r 2 , which can be respectively interpreted [11] as a measure of the strength of 'bulk' and 'interfacial' noise.…”

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confidence: 68%

“…To our knowledge, there has been only one attempt to determining the scaling exponents governing the approach to the usual VM. Within the reduced (p i , p b ) parameter plane defined above, the authors of [10] noted first that increasing p i along the zero-bulk-noise line p b = 0, up to the VM point (p i = 1 4 ), the (static) magnetization m s jumps from 0 to ±1. Thus, the order parameter exponent β = 0, and, for such a first-order d = 2 nonequilibrium critical point, the susceptibility and the correlation length exponents should satisfy γ = 2ν.…”

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confidence: 99%

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Critical Coarsening without Surface Tension: The Universality Class of the Voter Model

et al. 2001

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We show that the two-dimensional voter model, usually considered to only be a marginal coarsening system, represents a broad class of models for which phase-ordering takes place without surface tension. We argue that voter-like growth is generically observed at order-disorder nonequilibrium transitions solely driven by interfacial noise between dynamically symmetric absorbing states.PACS numbers: 02.50. Ey, 05.70.Fh, 64.60.C, 64.60.Ht Coarsening phenomena occur in a large variety of situations in and out of physics, ranging from the demixion of alloys [1] to population dynamics [2]. At a fundamental level, phase-ordering challenges our capacity to deal with nonequilibrium systems and our understanding of the mechanisms determining different universality classes. In many cases, phase competition is driven by surface tension, leading to 'curvature-driven' growth. Coarsening patterns are then characterized by a single length scale L(t) ∼ t 1/z , where the exponent z only depends on general symmetry and conservation properties of the system [1]. For instance, z = 2 for the common case of a non-conserved scalar order parameter (NCOP), a large class including the Ising model. In this context, the twodimensional voter model (VM) [3], a caricatural process in which sites on a square lattice adopt the opinion of a randomly-chosen neighbor, stands out as an exception. Its coarsening process, which gives rise to patterns with clusters of all sizes between 1 and √ t ( Fig. 1) [4,5], is characterized by a slow, logarithmic decay of the density of interfaces ρ ∼ 1/ ln t (as opposed to the algebraic decay ρ ∼ 1/L(t) ∼ t −1/z of curvature-driven growth). The marginality of the VM is usually attributed to the exceptional character of its analytic properties [4][5][6].In this Letter, we show that, in fact, large classes of models exhibit the same type of domain growth as the simple VM, without being endowed with any of its peculiar symmetry and integrability properties. We argue that voter-like coarsening is best defined by the absence of surface tension and that it is generically observed at the transitions between disordered and fully-ordered phases in the absence of bulk fluctuations, when these nonequilibrium transitions are driven by interfacial noise only. Finally, we discuss the universality of the scaling properties associated with voter-like critical points.We first review the properties of the usual two-state VM, emphasizing those of importance for defining generalized models. A 'voter' (or spin) residing on site x of a hypercubic lattice can have two different opinions s x = ±1. In any space dimension d, an elementary move consists in randomly choosing one site and assigning to it the opinion of one of its randomly chosen nearest neighbors (n.n.). This ensures that the two homogeneous configurations (where all spins are either + or −) are absorbing states, and that the model is Z 2 -symmetric under global inversion (s x → −s x ). Recasting the dynamic rule in terms of n.n. pair updating, a pair of opposite spins +− ...

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Critical Coarsening without Surface Tension: The Universality Class of the Voter Model

et al. 2001

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“…In two dimensions general Z 2 symmetric update rules were investigated by (Achahbar et al, 1996;Dornic et al, 2001;Drouffe and Godréche, 1999;Oliveira et al, 1993). The essence of this -following (Drouffe and Godréche, 1999) -is described below.…”

Section: Competing Dynamics Added To Spin-flipmentioning

confidence: 99%

“…if h = 4, the local spin aligns with its neighbors with a probability p 2 < 1. Simulations (Oliveira et al, 1993) revealed that the transition line between the low and high temperature regions is Ising type except for the endpoint (p 1 = 1,p 2 = β γ ηa ν ⊥ 0.33(2) 1.16(6) 0.13(4) 0.62(3) TABLE V Critical exponents of the d = 2 randomly driven lattice gas 0.75), that is first-order and corresponds to the voter model class. The local persistence exponent was also found to be constant along the line θ l ∼ 0.22 (Drouffe and Godréche, 1999) in agreement with that of the A-model (see Table IV) except for the VM point.…”

Section: Competing Dynamics Added To Spin-flipmentioning

confidence: 99%

Universality classes in nonequilibrium lattice systems

2004

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This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.

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“…This irregularity prevents the observation of expected spirals to be formed by the rotating vortex arms for smooth interfaces [9,14]. In the absence of cyclic invasion (and related vortex rotation) some features of the rough interfaces are already studied by several authors considering the twostate voter model [15][16][17][18][19][20]. In the presence of cyclic invasion, however, the topological and geometrical features of domain structure are not yet investigated rigorously even for smooth interfaces.…”

Section: Introductionmentioning

confidence: 99%

Three-state cyclic voter model extended with Potts energy

Szabó

Szolnoki

2002

Phys. Rev. E

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The cyclically dominated voter model on a square is extended by taking into consideration the variation of Potts energy during the nearest neighbor invasions. We have investigated the effect of surface tension on the self-organizing patterns maintained by the cyclic invasions. A geometrical analysis is also developed to study the three-color patterns. These investigations indicate clearly that in the "voter model" limit the pattern evolution is governed by the loop creation due to the overhanging during the interfacial roughening. Conversely, in the presence of surface tension the evolution is governed by spiral formation whose geometrical parameters depend on the strength of cyclic dominance.

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Nonequilibrium spin models with Ising universal behaviour

Cited by 135 publications

References 24 publications

Critical Coarsening without Surface Tension: The Universality Class of the Voter Model

Critical Coarsening without Surface Tension: The Universality Class of the Voter Model

Universality classes in nonequilibrium lattice systems

Three-state cyclic voter model extended with Potts energy

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