Introduction to Percolation Theory

Stauffer, Dietrich; Aharony, Amnon

doi:10.4324/9780203211595

Cited by 4,736 publications

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“…Above a critical obstacle density n * c the infinite cluster ceases to exist and only the hierarchy of disconnected finite clusters prevails. The transition is of purely geometric origin and is accompanied by a series of power laws reminiscent of a continuous phase transition [19]. Directly at the critical density n * c the infinite cluster becomes a self-similar structure with fractal dimension ‡ d f = 2.53 implying that its total weight is subextensive.…”

Section: Lorentz Model and Continuum Percolationmentioning

confidence: 99%

Anomalous transport of a tracer on percolating clusters

Spanner¹,

Höfling²,

Schröder‐Turk³

et al. 2011

J. Phys.: Condens. Matter

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Abstract. We investigate the dynamics of a single tracer exploring a course of fixed obstacles in the vicinity of the percolation transition for particles confined to the infinite cluster. The mean-square displacement displays anomalous transport, which extends to infinite times precisely at the critical obstacle density. The slowing down of the diffusion coefficient exhibits power-law behavior for densities close to the critical point and we show that the mean-square displacement fulfills a scaling hypothesis. Furthermore, we calculate the dynamic conductivity as response to an alternating electric field. Last, we discuss the non-gaussian parameter as an indicator for heterogeneous dynamics.

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Section: Lorentz Model and Continuum Percolationmentioning

confidence: 99%

Anomalous transport of a tracer on percolating clusters

Spanner¹,

Höfling²,

Schröder‐Turk³

et al. 2011

J. Phys.: Condens. Matter

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“…The key parameter a in such a model is assumed to depend on two types of mechanisms: firstly, a random distribution of sources within the cell walls and/or, secondly a distribution of locks holding up dislocations within the cell walls. Their analysis shows that percolating slip clusters conform to the universality class of percolation theory (Stauffer and Aharony 1992). Furthermore, it can be effectively used to describe slip transmission in stage III, and that a well-defined percolation threshold can be identified.…”

Section: N M Ghoniem Et Almentioning

confidence: 99%

Multiscale modelling of nanomechanics and micromechanics: an overview

Ghoniem

Busso

Kioussis

et al. 2003

Philosophical Magazine

136

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Recent advances in analytical and computational modelling frameworks to describe the mechanics of materials on scales ranging from the atomistic, through the microstructure or transitional, and up to the continuum are reviewed. It is shown that multiscale modelling of materials approaches relies on a systematic reduction in the degrees of freedom on the natural length scales that can be identified in the material. Connections between such scales are currently achieved either by a parametrization or by a 'zoom-out' or 'coarse-graining' procedure. Issues related to the links between the atomistic scale, nanoscale, microscale and macroscale are discussed, and the parameters required for the information to be transferred between one scale and an upper scale are identified. It is also shown that seamless coupling between length scales has not yet been achieved as a result of two main challenges: firstly, the computational complexity of seamlessly coupled simulations via the coarse-graining approach and, secondly, the inherent difficulty in dealing with system evolution stemming from time scaling, which does not permit coarse graining over temporal events. Starting from the Born-Oppenheimer adiabatic approximation, the problem of solving quantum mechanics equations of motion is first reviewed, with successful applications in the mechanics of nanosystems. Atomic simulation methods (e.g. molecular dynamics, Langevin dynamics and the kinetic Monte Carlo method) and their applications at the nanoscale are then discussed. The role played by dislocation dynamics and statistical mechanics methods in understanding microstructure self-organization, heterogeneous plastic deformation, material instabilities and failure phenomena is also discussed.

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“…If one views p = sin 2 θ as connection probability of two neighboring loops of opposite chirality, our two-channel model without intra-band tunnellings at nodes is analogous to a bond-percolation problem. It is well-known that a percolation cluster exists at p ≥ p c = 1/2 or J ≥ J c = 1 for a square lattice 35 . Therefore, an extended state is formed by strong mixing.…”

Section: B Discussion Of Finite-size Effectmentioning

confidence: 99%

“…The equipotential contour of value V 0 is the boundary of land and water. According to the percolation theory 35 , the percolation threshold of a twodimensional (2D) continuum model is p c = 1/2, where p c is the occupation probability of the medium (the land or the water). For simplicity, we suppose that the distribution of the random potential is symmetric around zero.…”

Section: The Semiclassical Model Including Inter-landau-band Mixingmentioning

confidence: 99%

Possible existence of a band of extended states induced by inter-Landau-band mixing in a quantum Hall system

Xiong

Wang

Niu

et al. 2006

J. Phys.: Condens. Matter

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The mixing of states with opposite chirality in quantum Hall system is shown to have delocalization effect. It is possible that extended states may form bands because of this mixing, as is shown through a numerical calculation on a two-channel network model. Based on this result, a new phase diagram with a narrow metallic phase separating two adjacent QH phases and/or separating a QH phase from the insulating phase is proposed. The data from recent non-scaling experiments is reanalyzed and shown that they seem to be consistent with the new phase diagram. However, due to finite-size effects, further study on large system size is still needed to conclude whether there are extended state bands in thermodynamic limit.

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Introduction to Percolation Theory

Cited by 4,736 publications

References 0 publications

Anomalous transport of a tracer on percolating clusters

Anomalous transport of a tracer on percolating clusters

Multiscale modelling of nanomechanics and micromechanics: an overview

Possible existence of a band of extended states induced by inter-Landau-band mixing in a quantum Hall system

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