Effective Conductivity of Percolation Media
The analogy between percolation theory and theory of second-order phase transition is introduced. Effective conductivity is considered as the order parameter of phase transitions. Calculations of critical indexes are provided. Different models of percolat
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Effective Conductivity of Percolation Media
5.1
Analogy with the Phenomenological Theory of Second-Order Phase Transitions. Scaling and Critical Exponents
Once Broadbent and Hammersly [4] had introduced the percolation threshold and discovered that various geometrical and physical characteristics of percolation systems depend on the proximity to percolation threshold s and this is a power dependence and critical indices describing it are universal (see Chap. 4), it could not but suggest the idea that percolation systems behave similarly to systems with second-order phase transitions. This idea was first put forward and realized in [15, 54]. To describe the results of these works and their subsequent development, it is necessary to give some brief information, the basic terms and relationships of the phenomenological theory of second-order phase transitions detailed, for instance, in [25, 30, 39]. The trend in the theory of phase transitions is to use the so-called proximity to phase transition point (temperature)t ¼ T Tc or t ¼ ðT Tc Þ=Tc . In the latest definition t is dimensionless; the order parameter η—the value characterizing the properties of system as a whole, changes considerably with a phase transition, i.e. in passing through Tc . In the case of ferromagnetics, the order parameter is ferromagnetic magnetization, and then Tc is the Curie temperature. Correlation radius (length) rc , which is also frequently designated as n, shows the order of magnitude of distances where correlation between the order parameter fluctuations decreases considerably. This magnitude appears in the correlation function [compare to (2.2.5)]: GðrÞ en ; r
© Springer Science+Business Media, LLC 2016 A.A. Snarskii et al., Transport Processes in Macroscopically Disordered Media, DOI 10.1007/978-1-4419-8291-9_5
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5 Effective Conductivity of Percolation Media
Fig. 5.1 Dependence of the order parameter η on the proximity to phase transition point Tc
and shows a power dependence on the proximity to the transition point n jtjm ;
ð5:1:1Þ
where m is critical index of correlation length.Above and below the transition point, the order parameter, just like the correlation length, has power-mode behavior (Fig. 5.1): g hjtjc ; g jt jb ;
t [ 0ðT [ Tc Þ; t \ 0ðT \ Tc Þ;
c [ 0; b [ 0;
ð5:1:2Þ ð5:1:3Þ
where c and b are certain critical indices, h is an external field (dimensionless), its nature is different for various order parameters g, in the case of ferromagnetism it is a magnetic field. In the area of t \ 0; T \ Tc (Fig. 5.1a) g tb there is a spontaneous magnetization separating some direction (hence the name—nonsymmetric phase). In the area of t [ 0; T [ Tc (Fig. 5.1b) g h jtjc the spontaneous magnetization occurs in the presence of an external magnetic field. Note that at T [ Tc the order parameter is different from zero only in the presence of an external field h. The external field (its dimensionless value is designated as h) “smears the phase transition,” “the discrete point of the phase transition disappear
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