Critical finite-range scaling in scalar-field theories and Ising models

Rikvold, Per Arne; Gorman, B. M.; Novotny, M. A.

doi:10.1103/physreve.47.1474

Cited by 11 publications

(18 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the critical region, this mapping to the continuum field theory agrees with the formulation in Ref. [34]. The resulting Euler-Lagrange equation for stationary solutions is the continuous analogue to Eq.…”

Section: The Q1di Modelsupporting

confidence: 83%

“…In Ref. [34] we showed, both analytically and numerically, that the critical equilibrium N-scaling relations for the Q1DI model are those of an N×∞ cylindrical Ising model above its upper critical dimension, which can be obtained from a standard one-dimensional φ 4 field theory by a single rescaling of the correlation length. Here we extend our study of the scaling behavior to metastable states, where β>β c and H =0, by studying the scaling behavior of the analytic continuation of the free energy across the first-order phase transition.…”

Section: The Q1di Modelmentioning

confidence: 99%

“…Our main focus here will be to relate the finite-size (or equivalently for this model, finiterange) scaling behavior of the constrained free energy to the scaling behavior of the analytic continuation of the free energy for an equivalent one-dimensional long-range field-theoretical model. This work represents an extension to metastable states of our recent study [34] of critical equilibrium finite-range scaling in the Q1DI model.…”

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

1994

Self Cite

View full text Add to dashboard Cite

We apply both a scalar field theory and a recently developed transfer-matrix method to study the stationary properties of metastability in a two-state model with weak, long-range interactions: the N ×∞ quasi-one-dimensional Ising model. Using the field theory, we find the analytic continuationf of the free energy across the first-order transition, assuming that the system escapes the metastable state by nucleation of noninteracting droplets. We find that corrections to the field-dependence are substantial, and by solving the EulerLagrange equation for the model numerically, we have verified the form of the free-energy cost of nucleation, including the first correction. In the transfermatrix method we associate with subdominant eigenvectors of the transfer matrix a complex-valued "constrained" free-energy density f α computed directly from the matrix. For the eigenvector with an associated magnetization most strongly opposed to the applied magnetic field, f α exhibits finite-range scaling behavior in agreement withf over a wide range of temperatures and fields, extending nearly to the classical spinodal. Some implications of these results for numerical studies of metastability are discussed.

show abstract

Section: The Q1di Modelsupporting

confidence: 83%

Section: The Q1di Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

1994

Self Cite

View full text Add to dashboard Cite

show abstract

“…Much clearer results were obtained in a study by Liu, Pavlenko, and Evans (LPE) [26], who considered a lattice-gas reaction-diffusion model similar to the ZGB model with CO desorption, in which adsorbed CO molecules were allowed to diffuse to adjacent empty sites at a finite rate, h. This leads to an effective diffusion length ∼ h 1/2 . In analogy with earlier studies of equilibrium Ising systems of linear size L with equal interaction constants of range ≤ R, which derived a crossover parameter L/R 2 for two-dimensional systems [48][49][50][51][52][53], LPE obtained 'effective critical exponents' from finite-size scaling analysis of Monte Carlo simulations. When plotted vs L/h, their results showed good data collapse and a monotonic trend over about two decades of the crossover parameter from Ising exponents for L/h ≫ 1 to mean-field for L/h ≪ 1.…”

Section: Introductionmentioning

confidence: 95%

Monte Carlo simulations of the critical properties of a Ziff-Gulari-Barshad model of catalytic CO oxidation with long-range reactivity

Chan

Rikvold

2015

Phys. Rev. E

View full text Add to dashboard Cite

The Ziff-Gulari-Barshad (ZGB) model, a simplified description of the oxidation of carbon monoxide (CO) on a catalyst surface, is widely used to study properties of nonequilibrium phase transitions. In particular, it exhibits a nonequilibrium, discontinuous transition between a reactive and a CO poisoned phase. If one allows a nonzero rate of CO desorption (k), the line of phase transitions terminates at a critical point (k(c)). In this work, instead of restricting the CO and atomic oxygen (O) to react to form carbon dioxide (CO(2)) only when they are adsorbed in close proximity, we consider a modified model that includes an adjustable probability for adsorbed CO and O atoms located far apart on the lattice to react. We employ large-scale Monte Carlo simulations for system sizes up to 240×240 lattice sites, using the crossing of fourth-order cumulants to study the critical properties of this system. We find that the nonequilibrium critical point changes from the two-dimensional Ising universality class to the mean-field universality class upon introducing even a weak long-range reactivity mechanism. This conclusion is supported by measurements of cumulant fixed-point values, cluster percolation probabilities, correlation-length finite-size scaling properties, and the critical exponent ratio β/ν. The observed behavior is consistent with that of the equilibrium Ising ferromagnet with additional weak long-range interactions [T. Nakada, P. A. Rikvold, T. Mori, M. Nishino, and S. Miyashita, Phys. Rev. B 84, 054433 (2011)]. The large system sizes and the use of fourth-order cumulants also enable determination with improved accuracy of the critical point of the original ZGB model with CO desorption.

show abstract

“…We define interactions in an Ld`1 xoo cylindrical geometry can be mapped to a one-dimensional In Refs. [24] and [43] it was shown that a two-state model with long-range local effective field, Hd; = J (m;.,.1 + m;-1), and replace m with mg elsewhere.…”

mentioning

confidence: 99%