Exact result for the effective conductivity of a continuum percolation model

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Abstract

A random two-dimensional checkerboard of squares of conductivities 1 and δ in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys σ*= δ at p=1/2. It is rigorously found here that to leading order as δ→0, this exact result holds for all p in the interval (1-pc,pc), where pc0.59 is the site percolation probability, not just at p=1/2. In particular, σ*(p,δ)= δ +O(δ), as δ→0, which is argued to hold for complex δ as well. The analysis is based on the identification of a ''symmetric'' backbone, which is statistically invariant under interchange of the components for any p(1-pc,pc), like the entire checkerboard at p=1/2. This backbone is defined in terms of ''choke points'' for the current, which have been observed in an experiment.

Original languageEnglish (US)
Pages (from-to)2114-2117
Number of pages4
JournalPhysical Review B
Volume50
Issue number4
DOIs
StatePublished - Jan 1 1994

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chokes
continuums
conductivity
proportion
intervals

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics

Cite this

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title = "Exact result for the effective conductivity of a continuum percolation model",
abstract = "A random two-dimensional checkerboard of squares of conductivities 1 and δ in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys σ*= δ at p=1/2. It is rigorously found here that to leading order as δ→0, this exact result holds for all p in the interval (1-pc,pc), where pc0.59 is the site percolation probability, not just at p=1/2. In particular, σ*(p,δ)= δ +O(δ), as δ→0, which is argued to hold for complex δ as well. The analysis is based on the identification of a ''symmetric'' backbone, which is statistically invariant under interchange of the components for any p(1-pc,pc), like the entire checkerboard at p=1/2. This backbone is defined in terms of ''choke points'' for the current, which have been observed in an experiment.",
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Exact result for the effective conductivity of a continuum percolation model. / Berlyand, Leonid V.; Golden, K.

In: Physical Review B, Vol. 50, No. 4, 01.01.1994, p. 2114-2117.

Research output: Contribution to journalArticle

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N2 - A random two-dimensional checkerboard of squares of conductivities 1 and δ in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys σ*= δ at p=1/2. It is rigorously found here that to leading order as δ→0, this exact result holds for all p in the interval (1-pc,pc), where pc0.59 is the site percolation probability, not just at p=1/2. In particular, σ*(p,δ)= δ +O(δ), as δ→0, which is argued to hold for complex δ as well. The analysis is based on the identification of a ''symmetric'' backbone, which is statistically invariant under interchange of the components for any p(1-pc,pc), like the entire checkerboard at p=1/2. This backbone is defined in terms of ''choke points'' for the current, which have been observed in an experiment.

AB - A random two-dimensional checkerboard of squares of conductivities 1 and δ in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys σ*= δ at p=1/2. It is rigorously found here that to leading order as δ→0, this exact result holds for all p in the interval (1-pc,pc), where pc0.59 is the site percolation probability, not just at p=1/2. In particular, σ*(p,δ)= δ +O(δ), as δ→0, which is argued to hold for complex δ as well. The analysis is based on the identification of a ''symmetric'' backbone, which is statistically invariant under interchange of the components for any p(1-pc,pc), like the entire checkerboard at p=1/2. This backbone is defined in terms of ''choke points'' for the current, which have been observed in an experiment.

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