Precise determination of the critical percolation threshold for the three-dimensional “Swiss cheese” model using a growth algorithm

Lorenz, Christian D.; Ziff, Robert M.

doi:10.1063/1.1338506

Cited by 197 publications

(165 citation statements)

References 45 publications

Supporting

Mentioning

148

Contrasting

Order By: Relevance

“…These include the pore-size functions (the distribution of the distance from a randomly chosen location in the void phase to the closest phase boundary), 46 the quantizer error (a moment of the poresize function, which is related to the principal relaxation time), 32,46 the order metric τ (a measure of the translational order of point configurations), 18 and the percolation threshold or the critical radius (the radius of the spheres at which a specific phase becomes connected) of each phase. [47][48][49][50] We compare the aforementioned physical and geometrical properties of our two-phase system derived from decorated stealthy ground states, as a function of the tuning parameter χ, with those of two other two-phase media: (1) equilibrium disordered (fluid) hard-sphere systems and (2) decorated Poisson point processes (idealgas configurations). The former has short-range order that is tunable by its volume fraction but no long-range order.…”

Section: -45mentioning

confidence: 99%

See 1 more Smart Citation

Transport, geometrical, and topological properties of stealthy disordered hyperuniform two-phase systems

Zhang

Stillinger

Torquato

2016

The Journal of Chemical Physics

122

View full text Add to dashboard Cite

Disordered hyperuniform many-particle systems have attracted considerable recent attention, since they behave like crystals in the manner in which they suppress large-scale density fluctuations, and yet also resemble statistically isotropic liquids and glasses with no Bragg peaks. One important class of such systems is the classical ground states of "stealthy potentials." The degree of order of such ground states depends on a tuning parameter χ. Previous studies have shown that these ground-state point configurations can be counterintuitively disordered, infinitely degenerate, and endowed with novel physical properties (e.g., negative thermal expansion behavior). In this paper, we focus on the disordered regime (0 < χ < 1/2) in which there is no long-range order, and control the degree of short-range order. We map these stealthy disordered hyperuniform point configurations to two-phase media by circumscribing each point with a possibly overlapping sphere of a common radius a: the "particle" and "void" phases are taken to be the space interior and exterior to the spheres, respectively. The hyperuniformity of such two-phase media depend on the sphere sizes: While it was previously analytically proven that the resulting two-phase media maintain hyperuniformity if spheres do not overlap, here we show numerically that they lose hyperuniformity whenever the spheres overlap. We study certain transport properties of these systems, including the effective diffusion coefficient of point particles diffusing in the void phase as well as static and time-dependent characteristics associated with diffusion-controlled reactions. Besides these effective transport properties, we also investigate several related structural properties, including pore-size functions, quantizer error, an order metric, and percolation thresholds. We show that these transport, geometrical and topological properties of our two-phase media derived from decorated stealthy ground states are distinctly different from those of equilibrium hard-sphere systems and spatially uncorrelated overlapping spheres. As the extent of short-range order increases, stealthy disordered two-phase media can attain nearly maximal effective diffusion coefficients over a broad range of volume fractions while also maintaining isotropy, and therefore may have practical applications in situations where ease of transport is desirable. We also show that the percolation threshold and the order metric are positively correlated with each other, while both of them are negatively correlated with the quantizer error. In the highly disordered regime (χ → 0), stealthy point-particle configurations are weakly-perturbed ideal gases. Nevertheless, reactants of diffusion-controlled reactions decay much faster in our two-phase media than in equilibrium hard-sphere systems of similar degrees of order, and hence indicate that the formation of large holes is strongly suppressed in the former systems.

show abstract

Section: -45mentioning

confidence: 99%

“…For example, to accurately determine the percolation threshold of 3D fully penetrable spheres, Ref. 48 employed systems of up to N = 7 × 10 8 particles. The whole system is divided into smaller cubes and the content particles in each cube is generated only when such cube is being probed.…”

Section: E Calculating Percolation Thresholdsmentioning

confidence: 99%

Transport, geometrical, and topological properties of stealthy disordered hyperuniform two-phase systems

Zhang

Stillinger

Torquato

2016

The Journal of Chemical Physics

122

View full text Add to dashboard Cite

show abstract

“…H o w e v e r , w e n e e d t o t h e p e r c o l a t i o n threshold as it depends on the details of the system. Examples of applying percolation theory to uncorrelated (or even correlated) continuum systems that check the universality and determination of the percolation threshold of different models can be found elsewhere [Gawlinski and Stanley, 1981;Lee and Torquato., 1990;King, 1990;Berkowitz, 1995;Lorenz and Ziff, 2001;Baker et al, 2002].…”

Section: Percolation Theorymentioning

confidence: 99%

Percolation Approach in Underground Reservoir Modeling

Masihi¹,

King²

2012

Water Resources Management and Modeling

View full text Add to dashboard Cite

“…The value η c for which the transition occurs is known from numerical experiments to be η c = 0.34189(2) [40] and η c = 0.341888(3) [41] which is to date the most precise estimate of the critical filling.…”

Section: B Continuum Modelmentioning

confidence: 99%

Conformal invariance in three dimensional percolation

Gori¹,

Trombettoni²

2015

J. Stat. Mech.

View full text Add to dashboard Cite

The aim of the paper is to present numerical results supporting the presence of conformal invariance in three dimensional statistical mechanics models at criticality and to elucidate the geometric aspects of universality. As a case study we study three dimensional percolation at criticality in bounded domains. Both on discrete and continuous models of critical percolation, we test by numerical experiments the invariance of quantities in finite domains under conformal transformations focusing on crossing probabilities. Our results show clear evidence of the onset of conformal invariance in finite realizations especially for the continuum percolation models. Finally we propose a simple analytical function approximating the crossing probability among two spherical caps on the surface of a sphere and confront it with the numerical results.

show abstract

Precise determination of the critical percolation threshold for the three-dimensional “Swiss cheese” model using a growth algorithm

Cited by 197 publications

References 45 publications

Transport, geometrical, and topological properties of stealthy disordered hyperuniform two-phase systems

Transport, geometrical, and topological properties of stealthy disordered hyperuniform two-phase systems

Percolation Approach in Underground Reservoir Modeling

Conformal invariance in three dimensional percolation

Contact Info

Product

Resources

About