Nonuniversality and continuity of the critical covered volume fraction in continuum percolation

Meester, Ronald; Roy, Rahul; Sarkar, Anish

doi:10.1007/bf02186282

Cited by 41 publications

(38 citation statements)

References 8 publications

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“…Assuming that clusters have a tree-like structure, we find that the percolation threshold η c depends on the shape of the log-normal distribution and, interestingly, that η c for d → ∞ can be smaller that the threshold for monodisperse spheres, in contrast to what is expected at finite dimensions [24].…”

Section: Summary and Discussionmentioning

confidence: 74%

“…In the hypothesis that closed loops are irrelevant also for spheres of unbounded size, we show that η c for d → ∞ is not universal, as it depends on the parameters of the distribution, and that it can be smaller than the critical threshold of monodisperse spheres, in contrast to what is expected for finite dimensions [24].…”

Section: Introductionmentioning

confidence: 60%

“…Interestingly, from Eq. (59) we also see that η c can be smaller than the critical threshold of monodisperse spheres (η c = 2 −d ), contrary to what is expected in finite dimensions [24]. We note that a critical threshold smaller than the monodisperse sphere limit in large dimensions has been found also for the case of radii distributions with d-dependent weights [25,26].…”

Section: The Case Of Unbounded Distribution Of the Radiimentioning

confidence: 67%

“…This last point has been formally confirmed in Ref. [24], although it may not hold true in the limit of infinite dimensions d [25,26].…”

Section: Introductionmentioning

confidence: 72%

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Continuum percolation of polydisperse hyperspheres in infinite dimensions

Grimaldi

2015

Phys. Rev. E

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We analyze the critical connectivity of systems of penetrable d-dimensional spheres having size distributions in terms of weighed random geometrical graphs, in which vertex coordinates correspond to random positions of the sphere centers and edges are formed between any two overlapping spheres. Edge weights naturally arise from the different radii of two overlapping spheres. For the case in which the spheres have bounded size distributions, we show that clusters of connected spheres are tree-like for d → ∞ and they contain no closed loops. In this case we find that the mean cluster size diverges at the percolation threshold density ηc → 2 −d , independently of the particular size distribution. We also show that the mean number of overlaps for a particle at criticality zc is smaller than unity, while zc → 1 only for spheres with fixed radii. We explain these features by showing that in the large dimensionality limit the critical connectivity is dominated by the spheres with the largest size. Assuming that closed loops can be neglected also for unbounded radii distributions, we find that the asymptotic critical threshold for systems of spheres with radii following a lognormal distribution is no longer universal, and that it can be smaller than 2 −d for d → ∞.

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Section: Summary and Discussionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 60%

Section: The Case Of Unbounded Distribution Of the Radiimentioning

confidence: 67%

“…This last point has been formally confirmed in Ref. [24], although it may not hold true in the limit of infinite dimensions d [25,26].…”

Section: Introductionmentioning

confidence: 72%

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Continuum percolation of polydisperse hyperspheres in infinite dimensions

Grimaldi

2015

Phys. Rev. E

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“…For randomly packed spheres, when the packing density p d is greater than a threshold density p c , clusters become connected to each other, and the size of the largest cluster approaches the size of the whole system [17, 19]. At this percolation threshold p c , the volume V of a cluster of random spheres scale with the length R of the cluster as V ∝ R D , with a characteristic exponent D = 2.5 in three-dimensional space [17, 18].…”

Section: Scaling Behaviormentioning

confidence: 99%

Geometry of protein shape and its evolutionary pattern for function prediction and characterization

Liang

2009

2009 Annual International Conference of the IEEE Engineering in Medicine and Biology Society

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Proteins contain thousands or more atoms and have complex shapes. We discuss here the computation of protein packing defects, in the form of voids and pockets, from experimentally resolved protein structures, and the nature of their distribution and scaling behavior, as well as their origin. We further discuss how evolutionary selection pressure due to biological function unaltered by selection pressure due to constraints from folding and stability can be isolated and estimated, and how such information can be used to predict protein function and characterize binding properties of enzymes.

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Flow, Transport, and Reaction in Porous Media: Percolation Scaling, Critical‐Path Analysis, and Effective Medium Approximation

Hunt

Sahimi

2017

Reviews of Geophysics

125

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We describe the most important developments in the application of three theoretical tools to modeling of the morphology of porous media and flow and transport processes in them. One tool is percolation theory. Although it was over 40 years ago that the possibility of using percolation theory to describe flow and transport processes in porous media was first raised, new models and concepts, as well as new variants of the original percolation model are still being developed for various applications to flow phenomena in porous media. The other two approaches, closely related to percolation theory, are the critical‐path analysis, which is applicable when porous media are highly heterogeneous, and the effective medium approximation—poor man's percolation—that provide a simple and, under certain conditions, quantitatively correct description of transport in porous media in which percolation‐type disorder is relevant. Applications to topics in geosciences include predictions of the hydraulic conductivity and air permeability, solute and gas diffusion that are particularly important in ecohydrological applications and land‐surface interactions, and multiphase flow in porous media, as well as non‐Gaussian solute transport, and flow morphologies associated with imbibition into unsaturated fractures. We describe new applications of percolation theory of solute transport to chemical weathering and soil formation, geomorphology, and elemental cycling through the terrestrial Earth surface. Wherever quantitatively accurate predictions of such quantities are relevant, so are the techniques presented here. Whenever possible, the theoretical predictions are compared with the relevant experimental data. In practically all the cases, the agreement between the theoretical predictions and the data is excellent. Also discussed are possible future directions in the application of such concepts to many other phenomena in geosciences.

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Nonuniversality and continuity of the critical covered volume fraction in continuum percolation

Cited by 41 publications

References 8 publications

Continuum percolation of polydisperse hyperspheres in infinite dimensions

Continuum percolation of polydisperse hyperspheres in infinite dimensions

Geometry of protein shape and its evolutionary pattern for function prediction and characterization

Flow, Transport, and Reaction in Porous Media: Percolation Scaling, Critical‐Path Analysis, and Effective Medium Approximation

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