Effect of anisotropy on finite-size scaling in percolation theory

Masihi, Mohsen; King, Peter R.; Nurafza, Peyman

doi:10.1103/physreve.74.042102

Cited by 26 publications

(14 citation statements)

References 11 publications

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“…There have already been numerous studies of anisotropic continuum percolation, some using analytic calculations [e. g., [15][16][17] others are based on Monte Carlo simulations [e. g., 13,18] or experiments [e. g., 19]. There are also detailed studies of anisotropic lattice percolation [20][21][22].…”

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confidence: 99%

Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals

Klatt¹,

Schröder‐Turk²,

Mecke³

2017

J. Stat. Mech.

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Abstract. We examine the interplay between anisotropy and percolation, i. e., the spontaneous formation of a system spanning cluster in an anisotropic model. We simulate an extension of a benchmark model of continuum percolation, the Boolean model, which is formed by overlapping grains. Here we introduce an orientation bias of the grains that controls the degree of anisotropy of the generated patterns. We analyze in the Euclidean plane the percolation thresholds above which percolating clusters in x-and in y-direction emerge. Only in finite systems, distinct differences between effective percolation thresholds for different directions appear. If extrapolated to infinite system sizes, these differences vanish independent of the details of the model. In the infinite system, the uniqueness of the percolating cluster guarantees a unique percolation threshold. While percolation is isotropic even for anisotropic processes, the value of the percolation threshold depends on the model parameters, which we explore by simulating a score of models with varying degree of anisotropy. To which precision can we predict the percolation threshold without simulations? We discuss analytic formulas for approximations (based on the excluded area or the Euler characteristic) and compare them to our simulation results. Empirical parameters from similar systems allow for accurate predictions of the percolation thresholds (with deviations of < 5% in our examples), but even without any empirical parameters, the explicit approximations from integral geometry provide, at least for the systems studied here, lower bounds that capture well the qualitative dependence of the percolation threshold on the system parameters (with deviations of 5%-30%). As an outlook, we suggest further candidates for explicit and geometric approximations based on second moments of the so-called Minkowski functionals.

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confidence: 99%

Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals

Klatt¹,

Schröder‐Turk²,

Mecke³

2017

J. Stat. Mech.

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“…Among hundreds of subjects we can mention the study of finite size effects at first order transitions by gaussian approximation [7][8], the Gibbs ensemble [9], five dimensional Ising model [10][11], percolation models [12][13], stochastic sandpiles [14], six-dimensional Ising system [15], Baxter-Wu model [16], two dimensional anisotropic Heisenberg model [17]...…”

Section: Introductionmentioning

confidence: 99%

Theoretical investigation of finite size effects at DNA melting

Buyukdagli

Joyeux

2007

Phys. Rev. E

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We investigated how the finiteness of the length of the sequence affects the phase transition that takes place at DNA melting temperature. For this purpose, we modified the Transfer Integral method to adapt it to the calculation of both extensive (partition function, entropy, specific heat, etc) and non-extensive (order parameter and correlation length) thermodynamic quantities of finite sequences with open boundary conditions, and applied the modified procedure to two different dynamical models. We showed that rounding of the transition clearly takes place when the length of the sequence is decreased. We also performed a finite-size scaling analysis of the two models and showed that the singular part of the free energy can indeed be expressed in terms of an homogeneous function. However, both the correlation length ξ and the average separation between paired bases y diverge at the melting transition, so that it is no longer clear to which of these two quantities the length L of the system should be compared. Moreover, Josephson's identity is satisfied for none of the investigated models, so that the derivation of the characteristic exponents which appear, for example, in the expression of the specific heat, requires some care.(♯)

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“…The difference between this value and the infinite threshold value in site percolation (i.e., p c = 0.593) as emphasized by Masihi et al (2006) is due to the anisotropy of the system caused by different effective size of the system in the x and y directions which results in a shift in the threshold value appearing in the longer direction of the system (i.e., the y-direction). Now for each permeability distribution function, consider the threshold permeability value along the y-axis (the representative permeability threshold over all realizations) to be the threshold permeability value that gives p = 0.62 (i.e., the 62nd percentile).…”

Section: Connected Cluster Of High Permeability Valuesmentioning

confidence: 99%

“…Moreover, in reality the spatial distributions of permeability maps are neither isotropic nor uncorrelated. The effect of anisotropy or the existence of spatial correlation in the permeability distribution results in different percolation threshold along the x or y axis and consequently a shift in the percolation quantities such as the connected sand fraction and the effective permeability master curves (see Masihi et al 2006;Sadeghnejad et al 2010 for more details on the impact of anisotropy on percolation quantities). Estimation of the effective permeability in such practical cases is a challenge.…”

Section: Percolation Theory Approachmentioning

confidence: 99%

Estimation of the Effective Permeability of Heterogeneous Porous Media by Using Percolation Concepts

Masihi

Gago²,

King³

2016

Transp Porous Med

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In this paper we present new methods to estimate the effective permeability (k eff ) of heterogeneous porous media with a wide distribution of permeabilities and various underlying structures, using percolation concepts. We first set a threshold permeability (k th ) on the permeability density function and use standard algorithms from percolation theory to check whether the high permeable grid blocks (i.e., those with permeability higher than k th ) with occupied fraction of "p" first forms a cluster connecting two opposite sides of the system in the direction of the flow (high permeability flow pathway). Then we estimate the effective permeability of the heterogeneous porous media in different ways: a power law (k eff = k th p m ), a weighted power average (with k g the geometric average of the permeability distribution) and a characteristic shape factor multiplied by the permeability threshold value. We found that the characteristic parameters (i.e., the exponent "m") can be inferred either from the statistics and properties of percolation subnetworks at the threshold point (i.e., high and low permeable regions corresponding to those permeabilities above and below the threshold permeability value) or by comparing the system properties with an uncorrelated random field having the same permeability distribution. These physically based approaches do not need fitting to the experimental data of effective permeability measurements to estimate the model parameter (i.e., exponent m) as is usually necessary in empirical methods. We examine the order of accuracy of these methods on different layers of 10th SPE model and found very good estimates as compared to the values determined from the commercial flow simulators.

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Effect of anisotropy on finite-size scaling in percolation theory

Cited by 26 publications

References 11 publications

Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals

Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals

Theoretical investigation of finite size effects at DNA melting

Estimation of the Effective Permeability of Heterogeneous Porous Media by Using Percolation Concepts

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