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Conduction and Diffusion in Percolating Systems

Encyclopedia of Complexity and Systems Science

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Definition of the Subject and Its Importance

The problem of determining the macroscopic structural and transport properties of microscopically nonuniform materials has a long history and is of such central importance that it finds applications in an astonishingly wide range of areas of science and technology, from developmental biology to xerography (Milton 2002; Sahimi 2003a, b). Consider systems that consist of two phases, each of which is homogeneous in its properties, mixed in some way to create an inhomogeneous material. Let one phase occupy a fraction ϕ of the volume. If ϕ ≪ 1, so that the inhomogeneous structure is in some sense dilute, the possibility arises of determining the effective properties of the material as an expansion in powers of ϕ. Simple examples of this, when the dilute phase consists of identical spheres distributed in some reasonable manner, are the prediction variously associated with the names of Maxwell, Clausius, Mossotti, Lorenz, Lorentz, and others that...

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Abbreviations

Backbone:

In a lattice percolation problem above the percolation threshold, that fraction of the infinite cluster with two disjoint connections to infinity.

Correlation length (ξ):

Length scale in a randomly structured system over which the system cannot be regarded as homogeneous.

Critical exponent:

Exponent characterizing the dominant behavior of an observable quantity near a percolation threshold, e.g., ξ ~ constant × (p c − p)−ν as p ↑ p c.

Effective medium approximation:

Approximate description of an inhomogeneous system obtained by matching averaged local fluctuations in properties in a self-consistent manner.

Flux:

Vector-valued function (in a continuum) or signed scalar (in a discrete system) quantifying transport or conduction rates.

Lattice:

Discrete structure (network/graph) of sites (nodes/vertices) connected by bonds (links/edges), including periodic lattices, random lattices, and treelike or self-similar pseudolattices.

Percolation theory:

Idealized model of a random medium. In the classical discrete case, the bonds of a lattice are independently open with probability p (Bernoulli bond percolation) or the sites of a lattice are independently occupied with probability p (Bernoulli site percolation). There are various continuum analogues.

Percolation threshold (p c):

Dividing point in parameter space separating cases where long-range connectivity is precluded (infinite connected sets exist with probability 0) from those where long-range connectivity occurs (infinite connected sets exist with positive probability).

Percolative system:

Random two-phase continuous or discrete system in which one phase is deemed void or nonconducting; usually a percolation threshold exists in such systems.

Potential (V):

Function distributed over space or over the sites of a network from which steady-state transport or conduction may be determined.

Pseudolattice:

A nonperiodic discrete structure of sites (nodes/vertices) and bonds (links/edges), most commonly either topologically treelike or geometrically self-similar.

Random walk:

Model of random motion, especially on lattices, consisting of a sequence of steps separated by constant or random time intervals.

Recurrent random walk:

Random walk process on a lattice for which the walker returns to the starting site with probability 1.

Renormalization:

Converting a system, either exactly or approximately, to a related system with a different characteristic length scale.

Transient random walk:

Random walk process on a lattice for which the walker has probability less than 1 of returning to the starting site.

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  1. School of Mathematics and Statistics, University of Melbourne, Melbourne, VIC, 3010, Australia

    Barry D. Hughes

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Hughes, B.D. (2014). Conduction and Diffusion in Percolating Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_93-2

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  1. Latest

    Conduction and Diffusion in Percolating Systems
    Published:
    04 December 2020

    DOI: https://doi.org/10.1007/978-3-642-27737-5_93-3

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    Conduction and Diffusion in Percolating Systems
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    26 December 2014

    DOI: https://doi.org/10.1007/978-3-642-27737-5_93-2

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