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Kundu S, Mandal D. Breaking universality in random sequential adsorption on a square lattice with long-range correlated defects. Phys Rev E 2021; 103:042134. [PMID: 34006012 DOI: 10.1103/physreve.103.042134] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2021] [Accepted: 04/07/2021] [Indexed: 11/07/2022]
Abstract
Jamming and percolation transitions in the standard random sequential adsorption of particles on regular lattices are characterized by a universal set of critical exponents. The universality class is preserved even in the presence of randomly distributed defective sites that are forbidden for particle deposition. However, using large-scale Monte Carlo simulations by depositing dimers on the square lattice and employing finite-size scaling, we provide evidence that the system does not exhibit such well-known universal features when the defects have spatial long-range (power-law) correlations. The critical exponents ν_{j} and ν associated with the jamming and percolation transitions, respectively, are found to be nonuniversal for strong spatial correlations and approach systematically their own universal values as the correlation strength is decreased. More crucially, we have found a difference in the values of the percolation correlation length exponent ν for a small but finite density of defects with strong spatial correlations. Furthermore, for a fixed defect density, it is found that the percolation threshold of the system, at which the largest cluster of absorbed dimers first establishes the global connectivity, gets reduced with increasing the strength of the spatial correlation.
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Affiliation(s)
- Sumanta Kundu
- Department of Earth and Space Science, Osaka University, 560-0043 Osaka, Japan
| | - Dipanjan Mandal
- Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
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3
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Oliveira EA, Pires RS, Oliveira RS, Furtado V, Herrmann HJ, Andrade JS. A universal approach for drainage basins. Sci Rep 2019; 9:9845. [PMID: 31285496 PMCID: PMC6614398 DOI: 10.1038/s41598-019-46165-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2018] [Accepted: 06/17/2019] [Indexed: 12/02/2022] Open
Abstract
Drainage basins are essential to Geohydrology and Biodiversity. Defining those regions in a simple, robust and efficient way is a constant challenge in Earth Science. Here, we introduce a model to delineate multiple drainage basins through an extension of the Invasion Percolation-Based Algorithm (IPBA). In order to prove the potential of our approach, we apply it to real and artificial datasets. We observe that the perimeter and area distributions of basins and anti-basins display long tails extending over several orders of magnitude and following approximately power-law behaviors. Moreover, the exponents of these power laws depend on spatial correlations and are invariant under the landscape orientation, not only for terrestrial, but lunar and martian landscapes. The terrestrial and martian results are statistically identical, which suggests that a hypothetical martian river would present similarity to the terrestrial rivers. Finally, we propose a theoretical value for the Hack’s exponent based on the fractal dimension of watersheds, γ = D/2. We measure γ = 0.54 ± 0.01 for Earth, which is close to our estimation of γ ≈ 0.55. Our study suggests that Hack’s law can have its origin purely in the maximum and minimum lines of the landscapes.
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Affiliation(s)
- Erneson A Oliveira
- Programa de Pós Graduacção em Informática Aplicada, Universidade de Fortaleza, 60811-905, Fortaleza, Ceará, Brazil. .,Mestrado Profissional em Ciências da Cidade, Universidade de Fortaleza, 60811-905, Fortaleza, Ceará, Brazil. .,Departamento de Física, Universidade Federal do Ceará, Campus do Pici, 60451-970, Fortaleza, Ceará, Brazil.
| | - Rilder S Pires
- Departamento de Física, Universidade Federal do Ceará, Campus do Pici, 60451-970, Fortaleza, Ceará, Brazil
| | - Rubens S Oliveira
- Departamento de Física, Universidade Federal do Ceará, Campus do Pici, 60451-970, Fortaleza, Ceará, Brazil
| | - Vasco Furtado
- Programa de Pós Graduacção em Informática Aplicada, Universidade de Fortaleza, 60811-905, Fortaleza, Ceará, Brazil
| | - Hans J Herrmann
- Departamento de Física, Universidade Federal do Ceará, Campus do Pici, 60451-970, Fortaleza, Ceará, Brazil.,PMMH, ESPCI, 7 quai St Bernard, 75005, Paris, France.,ETH Zürich, Computational Physics for Engineering Materials, Institute for Building Materials, Wolfgang-Pauli-Strasse 27, Hit, CH-8093, Zürich, Switzerland
| | - José S Andrade
- Departamento de Física, Universidade Federal do Ceará, Campus do Pici, 60451-970, Fortaleza, Ceará, Brazil
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Carvalho L, Patricio P, Ponte S, Heisenberg CP, Almeida L, Nunes AS, Araújo NAM, Jacinto A. Occluding junctions as novel regulators of tissue mechanics during wound repair. J Cell Biol 2018; 217:4267-4283. [PMID: 30228162 PMCID: PMC6279375 DOI: 10.1083/jcb.201804048] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2018] [Revised: 08/15/2018] [Accepted: 09/05/2018] [Indexed: 01/02/2023] Open
Abstract
Simple epithelial repair is mediated by the contraction of an actomyosin cable and cellular rearrangements at the wound edge. Carvalho et al. show that occluding junctions are required for epithelial repair by regulating these cellular rearrangements and tissue mechanical properties. In epithelial tissues, cells tightly connect to each other through cell–cell junctions, but they also present the remarkable capacity of reorganizing themselves without compromising tissue integrity. Upon injury, simple epithelia efficiently resolve small lesions through the action of actin cytoskeleton contractile structures at the wound edge and cellular rearrangements. However, the underlying mechanisms and how they cooperate are still poorly understood. In this study, we combine live imaging and theoretical modeling to reveal a novel and indispensable role for occluding junctions (OJs) in this process. We demonstrate that OJ loss of function leads to defects in wound-closure dynamics: instead of contracting, wounds dramatically increase their area. OJ mutants exhibit phenotypes in cell shape, cellular rearrangements, and mechanical properties as well as in actin cytoskeleton dynamics at the wound edge. We propose that OJs are essential for wound closure by impacting on epithelial mechanics at the tissue level, which in turn is crucial for correct regulation of the cellular events occurring at the wound edge.
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Affiliation(s)
- Lara Carvalho
- Chronic Diseases Research Center, NOVA Medical School/Faculdade de Ciências Médicas, Universidade NOVA de Lisboa, Lisbon, Portugal
| | - Pedro Patricio
- Chronic Diseases Research Center, NOVA Medical School/Faculdade de Ciências Médicas, Universidade NOVA de Lisboa, Lisbon, Portugal.,Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de Lisboa, Lisbon, Portugal.,Centro de Física Teórica e Computacional, Universidade de Lisboa, Campo Grande, Lisbon, Portugal
| | - Susana Ponte
- Chronic Diseases Research Center, NOVA Medical School/Faculdade de Ciências Médicas, Universidade NOVA de Lisboa, Lisbon, Portugal
| | | | - Luis Almeida
- Centre National de la Recherche Scientifique/Sorbonne Université/Team Mamba, French Institute for Research in Computer Science and Automation Paris, Laboratoire Jacques-Louis Lions, BC187, Paris, France
| | - André S Nunes
- Centro de Física Teórica e Computacional, Universidade de Lisboa, Campo Grande, Lisbon, Portugal.,Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Lisbon, Portugal
| | - Nuno A M Araújo
- Centro de Física Teórica e Computacional, Universidade de Lisboa, Campo Grande, Lisbon, Portugal.,Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Lisbon, Portugal
| | - Antonio Jacinto
- Chronic Diseases Research Center, NOVA Medical School/Faculdade de Ciências Médicas, Universidade NOVA de Lisboa, Lisbon, Portugal .,The Discoveries Centre for Regenerative and Precision Medicine, Universidade NOVA de Lisboa, Lisbon, Portugal
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5
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Burger B, Andrade JS, Herrmann HJ. A comparison of hydrological and topological watersheds. Sci Rep 2018; 8:10586. [PMID: 30002379 PMCID: PMC6043487 DOI: 10.1038/s41598-018-28470-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2017] [Accepted: 06/15/2018] [Indexed: 11/09/2022] Open
Abstract
We introduce the hydrological watershed, a watershed where water can penetrate the soil, and compare it with the topological watershed for a two-dimensional landscape. For this purpose, we measure the fractal dimension of the hydrological watershed for different penetration depths and different grid sizes. Through finite size scaling, we find that the fractal dimension is 1.31 ± 0.02 which is significantly higher than the fractal dimension of the topological watershed. This indicates that the hydrological watershed belongs to a new universality class. We also find that, as opposed to the topological watershed, the hydrodynamic watershed can exhibit disconnected islands.
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Affiliation(s)
- B Burger
- IfB, HIT G23.1, ETH Zürich, Zürich, 8093, Switzerland.
| | - J S Andrade
- IfB, HIT G23.1, ETH Zürich, Zürich, 8093, Switzerland.,Departamento de Física, Universidade Federal do Ceará, Fortaleza, 60451-970, Ceará, Brazil
| | - H J Herrmann
- IfB, HIT G23.1, ETH Zürich, Zürich, 8093, Switzerland.,Departamento de Física, Universidade Federal do Ceará, Fortaleza, 60451-970, Ceará, Brazil
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6
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de Castro CP, Luković M, Pompanin G, Andrade RFS, Herrmann HJ. Schramm-Loewner evolution and perimeter of percolation clusters of correlated random landscapes. Sci Rep 2018; 8:5286. [PMID: 29588474 PMCID: PMC5869597 DOI: 10.1038/s41598-018-23489-x] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/12/2017] [Accepted: 03/05/2018] [Indexed: 11/30/2022] Open
Abstract
Motivated by the fact that many physical landscapes are characterized by long-range height-height correlations that are quantified by the Hurst exponent H, we investigate the statistical properties of the iso-height lines of correlated surfaces in the framework of Schramm-Loewner evolution (SLE). We show numerically that in the continuum limit the external perimeter of a percolating cluster of correlated surfaces with H ∈ [-1, 0] is statistically equivalent to SLE curves. Our results suggest that the external perimeter also retains the Markovian properties, confirmed by the absence of time correlations in the driving function and the fact that the latter is Gaussian distributed for any specific time. We also confirm that for all H the variance of the winding angle grows logarithmically with size.
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Affiliation(s)
- C P de Castro
- Instituto de Física, Universidade Federal da Bahia, Campus Universitário da Federacção, Salvador, BA, 40170-115, Brazil.
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland.
| | - M Luković
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland
| | - G Pompanin
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland
| | - R F S Andrade
- Instituto de Física, Universidade Federal da Bahia, Campus Universitário da Federacção, Salvador, BA, 40170-115, Brazil
| | - H J Herrmann
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland
- Departamento de Física, Universidade Federal do Ceará, Fortaleza, Ceará, 60451-970, Brazil
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Abnormal grain growth mediated by fractal boundary migration at the nanoscale. Sci Rep 2018; 8:1592. [PMID: 29371608 PMCID: PMC5785526 DOI: 10.1038/s41598-018-19588-4] [Citation(s) in RCA: 20] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2017] [Accepted: 01/04/2018] [Indexed: 11/16/2022] Open
Abstract
Modern engineered materials are composed of space-filling grains or domains separated by a network of interfaces or boundaries. Such polycrystalline microstructures have the capacity to coarsen through boundary migration. Grain growth theories account for the topology of grains and the connectivity of the boundary network in terms of the familiar Euclidian dimension and Euler’s polyhedral formula, both of which are based on integer numbers. However, we recently discovered an unusual growth mode in a nanocrystalline Pd-Au alloy, in which grains develop complex, highly convoluted surface morphologies that are best described by a fractional dimension of ∼1.2 (extracted from the perimeters of grain cross sections). This fractal value is characteristic of a variety of domain growth scenarios—including explosive percolation, watersheds of random landscapes, and the migration of domain walls in a random field of pinning centers—which suggests that fractal grain boundary migration could be a manifestation of the same universal behavior.
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8
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de Castro CP, Luković M, Andrade RFS, Herrmann HJ. The influence of statistical properties of Fourier coefficients on random Gaussian surfaces. Sci Rep 2017; 7:1961. [PMID: 28512343 PMCID: PMC5434017 DOI: 10.1038/s41598-017-02135-y] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2016] [Accepted: 03/15/2017] [Indexed: 12/04/2022] Open
Abstract
Many examples of natural systems can be described by random Gaussian surfaces. Much can be learned by analyzing the Fourier expansion of the surfaces, from which it is possible to determine the corresponding Hurst exponent and consequently establish the presence of scale invariance. We show that this symmetry is not affected by the distribution of the modulus of the Fourier coefficients. Furthermore, we investigate the role of the Fourier phases of random surfaces. In particular, we show how the surface is affected by a non-uniform distribution of phases.
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Affiliation(s)
- C P de Castro
- Instituto de Física, Universidade Federal da Bahia, Campus Universitário da Federação, Salvador, BA, 40170-115, Brazil. .,Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland.
| | - M Luković
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland
| | - R F S Andrade
- Instituto de Física, Universidade Federal da Bahia, Campus Universitário da Federação, Salvador, BA, 40170-115, Brazil
| | - H J Herrmann
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland.,Departamento de Física, Universidade Federal do Ceará, Fortaleza, Ceará, 60451-970, Brazil
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9
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Schrenk KJ, Hilário MR, Sidoravicius V, Araújo NAM, Herrmann HJ, Thielmann M, Teixeira A. Critical Fragmentation Properties of Random Drilling: How Many Holes Need to Be Drilled to Collapse a Wooden Cube? PHYSICAL REVIEW LETTERS 2016; 116:055701. [PMID: 26894717 DOI: 10.1103/physrevlett.116.055701] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2015] [Indexed: 06/05/2023]
Abstract
A solid wooden cube fragments into pieces as we sequentially drill holes through it randomly. This seemingly straightforward observation encompasses deep and nontrivial geometrical and probabilistic behavior that is discussed here. Combining numerical simulations and rigorous results, we find off-critical scale-free behavior and a continuous transition at a critical density of holes that significantly differs from classical percolation.
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Affiliation(s)
- K J Schrenk
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - M R Hilário
- Departamento de Matemática, Universidade Federal de Minas Gerais, Avenida Antonio Carlos, 6627-P.O. Box 702-30161-970, Belo Horizonte, Minas Gerais, Brazil
- Section de Mathématiques, Université de Genève, 2-4 Rue du Lièvre, 1211 Genève, Switzerland
| | - V Sidoravicius
- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA
- New York University-Shanghai, 1555 Century Avenue, Pudong New Area, Shanghai 200122, China
- CEMADEN, Avenida Doutor Altino Bondensan, 500, São José dos Campos, São Paulo 12247-016, Brazil
| | - N A M Araújo
- Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal and Centro de Física Teórica e Computacional, Universidade de Lisboa, 1749-016 Lisboa, Portugal
| | - H J Herrmann
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
- Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil
| | - M Thielmann
- Bayerisches Geoinstitut, University of Bayreuth, Universitätsstraße 30, 95440 Bayreuth, Germany
| | - A Teixeira
- Instituto Nacional de Matemática Pura e Aplicada, Est. Dona Castorina, 110, 22460-320 Rio de Janeiro, Rio de Janeiro, Brazil
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11
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Cho YS, Kahng B. Two Types of Discontinuous Percolation Transitions in Cluster Merging Processes. Sci Rep 2015; 5:11905. [PMID: 26148542 PMCID: PMC5387406 DOI: 10.1038/srep11905] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2015] [Accepted: 05/29/2015] [Indexed: 11/23/2022] Open
Abstract
Percolation is a paradigmatic model in disordered systems and has been applied to various natural phenomena. The percolation transition is known as one of the most robust continuous transitions. However, recent extensive studies have revealed that a few models exhibit a discontinuous percolation transition (DPT) in cluster merging processes. Unlike the case of continuous transitions, understanding the nature of discontinuous phase transitions requires a detailed study of the system at hand, which has not been undertaken yet for DPTs. Here we examine the cluster size distribution immediately before an abrupt increase in the order parameter of DPT models and find that DPTs induced by cluster merging kinetics can be classified into two types. Moreover, the type of DPT can be determined by the key characteristic of whether the cluster kinetic rule is homogeneous with respect to the cluster sizes. We also establish the necessary conditions for each type of DPT, which can be used effectively when the discontinuity of the order parameter is ambiguous, as in the explosive percolation model.
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Affiliation(s)
- Y S Cho
- Center for Complex Systems Studies and CTP, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
| | - B Kahng
- Center for Complex Systems Studies and CTP, Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
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12
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Daryaei E. Loop-erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution. Phys Rev E 2014; 90:022129. [PMID: 25215710 DOI: 10.1103/physreve.90.022129] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2014] [Indexed: 11/07/2022]
Abstract
We study the scaling limit of a planar loop-erased random walk (LERW) on the percolation cluster, with occupation probability p≥p(c). We numerically demonstrate that the scaling limit of planar LERW(p) curves, for all p>p(c), can be described by Schramm-Loewner evolution (SLE) with a single parameter κ that is close to the normal LERW in a Euclidean lattice. However, our results reveal that the LERW on critical incipient percolation clusters is compatible with SLE, but with another diffusivity coefficient κ. Several geometrical tests are applied to ascertain this. All calculations are consistent with SLE(κ), where κ=1.732±0.016. This value of the diffusivity coefficient is outside the well-known duality range 2≤κ≤8. We also investigate how the winding angle of the LERW(p) crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to p(c). For finite systems, two crossover exponents and a scaling relation can be derived. This finding should, to some degree, help us understand and predict the existence of conformal invariance in disordered and fractal landscapes.
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Affiliation(s)
- E Daryaei
- Department of Physics, Faculty of Basic Sciences, University of Neyshabur, P.O. Box 91136-899, Neyshabur, Iran
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13
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Daryaei E, Rouhani S. Loop-erased random walk on a percolation cluster: crossover from Euclidean to fractal geometry. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062101. [PMID: 25019719 DOI: 10.1103/physreve.89.062101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/14/2013] [Indexed: 06/03/2023]
Abstract
We study loop-erased random walk (LERW) on the percolation cluster, with occupation probability p ≥ p_{c}, in two and three dimensions. We find that the fractal dimensions of LERW_{p} are close to normal LERW in a Euclidean lattice, for all p>p_{c}. However, our results reveal that LERW on critical incipient percolation clusters is fractal with d_{f}=1.217 ± 0.002 for d=2 and 1.43 ± 0.02 for d=3, independent of the coordination number of the lattice. These values are consistent with the known values for optimal path exponents in strongly disordered media. We investigate how the behavior of the LERW_{p} crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to p_{c}. For finite systems, two crossover exponents and a scaling relation can be derived. This work opens up a theoretical window regarding the diffusion process on fractal and random landscapes.
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Affiliation(s)
- E Daryaei
- Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran and Faculty of Basic Sciences, University of Neyshabur, P.O. Box 91136-899, Neyshabur, Iran
| | - S Rouhani
- Department of Physics, Sharif University of Technology, P.O. Box 11155-9161, Tehran, Iran
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14
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Schrenk KJ, Araújo NAM, Ziff RM, Herrmann HJ. Retention capacity of correlated surfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062141. [PMID: 25019758 DOI: 10.1103/physreve.89.062141] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2014] [Indexed: 06/03/2023]
Abstract
We extend the water retention model [C. L. Knecht et al., Phys. Rev. Lett. 108, 045703 (2012)] to correlated random surfaces. We find that the retention capacity of discrete random landscapes is strongly affected by spatial correlations among the heights. This phenomenon is related to the emergence of power-law scaling in the lake volume distribution. We also solve the uncorrelated case exactly for a small lattice and present bounds on the retention of uncorrelated landscapes.
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Affiliation(s)
- K J Schrenk
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland
| | - N A M Araújo
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland
| | - R M Ziff
- Center for the Study of Complex Systems and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA
| | - H J Herrmann
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093, Zurich, Switzerland and Departamento de Física, Universidade Federal do Ceará, 60451-970, Fortaleza, Ceará, Brazil
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15
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Kim S, Cho YS, Araújo NAM, Kahng B. Crossover behavior of conductivity in a discontinuous percolation model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032113. [PMID: 24730796 DOI: 10.1103/physreve.89.032113] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/16/2014] [Indexed: 06/03/2023]
Abstract
When conducting bonds are occupied randomly in a two-dimensional square lattice, the conductivity of the system increases continuously as the density of those conducting bonds exceeds the percolation threshold. Such a behavior is well known in percolation theory; however, the conductivity behavior has not been studied yet when the percolation transition is discontinuous. Here we investigate the conductivity behavior through a discontinuous percolation model evolving under a suppressive external bias. Using effective medium theory, we analytically calculate the conductivity behavior as a function of the density of conducting bonds. The conductivity function exhibits a crossover behavior from a drastically to a smoothly increasing function beyond the percolation threshold in the thermodynamic limit. The analytic expression fits well our simulation data.
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Affiliation(s)
- Seongmin Kim
- Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
| | - Y S Cho
- Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
| | - N A M Araújo
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - B Kahng
- Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
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16
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Schrenk KJ, Posé N, Kranz JJ, van Kessenich LVM, Araújo NAM, Herrmann HJ. Percolation with long-range correlated disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052102. [PMID: 24329209 DOI: 10.1103/physreve.88.052102] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2013] [Indexed: 06/03/2023]
Abstract
Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as functions of the Hurst exponent H, which characterizes the degree of spatial correlation among the occupation of sites. In particular, we study the fractal dimension of the largest cluster and the scaling behavior of the second moment of the cluster size distribution, as well as the complete and accessible perimeters of the largest cluster. Concerning the inner structure and transport properties of the largest cluster, we analyze its shortest path, backbone, red sites, and conductivity. Finally, bridge site growth is also considered. We propose expressions for the functional dependence of the critical exponents on H.
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Affiliation(s)
- K J Schrenk
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - N Posé
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - J J Kranz
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - L V M van Kessenich
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - N A M Araújo
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - H J Herrmann
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland and Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil
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Andrade RFS, Herrmann HJ. Percolation model with continuously varying exponents. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:042122. [PMID: 24229131 DOI: 10.1103/physreve.88.042122] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2013] [Indexed: 06/02/2023]
Abstract
This work analyzes a percolation model on the diamond hierarchical lattice (DHL), where the percolation transition is retarded by the inclusion of a probability of erasing specific connected structures. It has been inspired by the recent interest on the existence of other universality classes of percolation models. The exact scale invariance and renormalization properties of DHL leads to recurrence maps, from which analytical expressions for the critical exponents and precise numerical results in the limit of very large lattices can be derived. The critical exponents ν and β of the investigated model vary continuously as the erasing probability changes. An adequate choice of the erasing probability leads to the result ν=∞, like in some phase transitions involving vortex formation. The percolation transition is continuous, with β>0, but β can be as small as desired. The modified percolation model turns out to be equivalent to the Q→1 limit of a Potts model with specific long range interactions on the same lattice.
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Affiliation(s)
- R F S Andrade
- Instituto de Física, Universidade Federal da Bahia, 40210-210 Salvador, Brazil
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Andrade RFS, Herrmann HJ. Exact evaluation of the cutting path length in a percolation model on a hierarchical network. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:042113. [PMID: 23679379 DOI: 10.1103/physreve.87.042113] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2013] [Indexed: 06/02/2023]
Abstract
This work presents an approach to evaluate the exact value of the fractal dimension of the cutting path d(f)(CP) on hierarchical structures with finite order of ramification. Our approach is based on a renormalization group treatment of the universality class of watersheds. By making use of the self-similar property, we show that d(f)(CP) depends only on the average cutting path (CP) of the first generation of the structure. For the simplest Wheastone hierarchical lattice (WHL), we present a mathematical proof. For a larger WHL structure, the exact value of d(f)(CP) is derived based on a computer algorithm that identifies the length of all possible CP's of the first generation.
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Affiliation(s)
- R F S Andrade
- Instituto de Física, Universidade Federal da Bahia, 40210-210, Salvador, Brazil
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Abstract
When dynamics in a system proceeds under suppressive external bias, the system can undergo an abrupt phase transition, as can happen when an epidemic spreads. Recently, an explosive percolation (EP) model was introduced to understand such phenomena. The order of the EP transition has not been clarified in a unified framework covering low-dimensional systems and the mean-field limit. We introduce a stochastic model in which a rule for dynamics is designed to avoid the formation of a spanning cluster through competitive selection in Euclidean space. We use heuristic arguments to show that in the thermodynamic limit and depending on a control parameter, the EP transition can be either continuous or discontinuous if d < d(c) and is always continuous if d ≥ d(c), where d(c) is the spatial dimension and d is the upper critical dimension.
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Affiliation(s)
- Y S Cho
- Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
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Moreira AA, Oliveira CLN, Hansen A, Araújo NAM, Herrmann HJ, Andrade JS. Fracturing highly disordered materials. PHYSICAL REVIEW LETTERS 2012; 109:255701. [PMID: 23368480 DOI: 10.1103/physrevlett.109.255701] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/06/2012] [Indexed: 06/01/2023]
Abstract
We investigate the role of disorder on the fracturing process of heterogeneous materials by means of a two-dimensional fuse network model. Our results in the extreme disorder limit reveal that the backbone of the fracture at collapse, namely, the subset of the largest fracture that effectively halts the global current, has a fractal dimension of 1.22 ± 0.01. This exponent value is compatible with the universality class of several other physical models, including optimal paths under strong disorder, disordered polymers, watersheds and optimal path cracks on uncorrelated substrates, hulls of explosive percolation clusters, and strands of invasion percolation fronts. Moreover, we find that the fractal dimension of the largest fracture under extreme disorder, d(f) = 1.86 ± 0.01, is outside the statistical error bar of standard percolation. This discrepancy is due to the appearance of trapped regions or cavities of all sizes that remain intact till the entire collapse of the fuse network, but are always accessible in the case of standard percolation. Finally, we quantify the role of disorder on the structure of the largest cluster, as well as on the backbone of the fracture, in terms of a distinctive transition from weak to strong disorder characterized by a new crossover exponent.
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Affiliation(s)
- A A Moreira
- Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil
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Daryaei E, Araújo NAM, Schrenk KJ, Rouhani S, Herrmann HJ. Watersheds are Schramm-Loewner evolution curves. PHYSICAL REVIEW LETTERS 2012; 109:218701. [PMID: 23215620 DOI: 10.1103/physrevlett.109.218701] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2012] [Indexed: 06/01/2023]
Abstract
We show that in the continuum limit watersheds dividing drainage basins are Schramm-Loewner evolution (SLE) curves, being described by one single parameter κ. Several numerical evaluations are applied to ascertain this. All calculations are consistent with SLE(κ), with κ = 1.734 ± 0.005, being the only known physical example of an SLE with κ<2. This lies outside the well-known duality conjecture, bringing up new questions regarding the existence and reversibility of dual models. Furthermore, it constitutes a strong indication for conformal invariance in random landscapes and suggests that watersheds likely correspond to a logarithmic conformal field theory with a central charge c ≈ -7/2.
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Affiliation(s)
- E Daryaei
- Computational Physics for Engineering Materials, Institut f. Baustoffe, ETH Zurich, Wolfgang-Pauli-Street 27, 8093 Zurich, Switzerland.
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Abstract
Many resources, such as oil, gas, or water, are extracted from porous soils and their exploration is often shared among different companies or nations. We show that the effective shares can be obtained by invading the porous medium simultaneously with various fluids. Partitioning a volume in two parts requires one division surface while the simultaneous boundary between three parts consists of lines. We identify and characterize these lines, showing that they form a fractal set consisting of a single thread spanning the medium and a surrounding cloud of loops. While the spanning thread has fractal dimension 1.55 ± 0.03, the set of all lines has dimension 1.69 ± 0.02. The size distribution of the loops follows a power law and the evolution of the set of lines exhibits a tricritical point described by a crossover with a negative dimension at criticality.
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Fehr E, Schrenk KJ, Araújo NAM, Kadau D, Grassberger P, Andrade JS, Herrmann HJ. Corrections to scaling for watersheds, optimal path cracks, and bridge lines. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:011117. [PMID: 23005378 DOI: 10.1103/physreve.86.011117] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2012] [Indexed: 06/01/2023]
Abstract
We study the corrections to scaling for the mass of the watershed, the bridge line, and the optimal path crack in two and three dimensions (2D and 3D). We disclose that these models have numerically equivalent fractal dimensions and leading correction-to-scaling exponents. We conjecture all three models to possess the same fractal dimension, namely, d(f) =1.2168 ± 0.0005 in 2D and d(f) = 2.487 ± 0.003 in 3D, and the same exponent of the leading correction, Ω = 0.9 ± 0.1 and Ω=1.0 ± 0.1, respectively. The close relations between watersheds, optimal path cracks in the strong disorder limit, and bridge lines are further supported by either heuristic or exact arguments.
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Affiliation(s)
- E Fehr
- IfB, ETH Zürich, CH-8093 Zürich, Switzerland
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