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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Consider an arbitrary network of nodes connected by links, each of which isa resistor with a specified electrical resistance. Suppose that this network isconnected to the leads of a battery. Two natural scenarios are:(a) the “bus-bar geometry” (Fig. 1), in which the network is connected to two parallel lines (intwo dimensions), plates (in three dimensions), etc., and the battery is connected across thetwo plates, and (b) the “two-point geometry”, in which a battery isconnected to two distinct nodes, so that a current I injected at a one node and the same current withdrawn from theother node. In both cases, a basic question is: what is the nature of the current flowthrough the network?

Figure 1
figure 1_220

Resistor networks in a the bus-bar geometry, and b the two-point geometry

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Abbreviations

Conductance (G):

The relation between the current I in an electrical network and the applied voltage V: \( { I=GV } \).

Conductance exponent (t):

The relation between the conductance G and the resistor (or conductor) concentration p near the percolation threshold: \( G\sim (p-p_c)^t \).

Effective medium theory (EMT):

A theory to calculate the conductance of a heterogeneous system that is based on a homogenization procedure.

Fractal:

A geometrical object that is invariant at any scale of magnification or reduction.

Multifractal:

A generalization of a fractal in which different subsets of an object have different scaling behaviors.

Percolation :

Connectivity of a random porous network.

Percolation threshold p c :

The transition between a connected and disconnected network as the density of links is varied.

Random resistor network:

A percolation network in which the connections consist of electrical resistors that are present with probability p and absent with probability \( { 1-p } \).

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  1. Center for Polymer Studies and Department of Physics, Boston University, Boston, USA

    Sidney Redner

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  1. Sidney Redner
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Redner, S. (2009). Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_220

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