Conductivity of continuum percolating systems

Abstract

We study the conductivity of a class of disordered continuum systems represented by the Swiss-cheese model, where the conducting medium is the space between randomly placed spherical holes, near the percolation threshold. This model can be mapped onto a bond percolation model where the conductance sigma of randomly occupied bonds is drawn from a probability distribution of the form sigma(-a). Employing the methods of renormalized field theory we show to arbitrary order in epsilon expansion that the critical conductivity exponent of the Swiss-cheese model is given by t(SC)(a) = (d-2)nu + max[phi,(1-a)(-1)], where d is the spatial dimension and nu and phi denote the critical exponents for the percolation correlation length and resistance, respectively. Our result confirms a conjecture that is based on the "nodes, links, and blobs" picture of percolation clusters.

Effect of Polydispersity on Diffusion in Random Obstacle Matrices

“…Recent theoretical and simulation studies for a two-dimensional Lorentz model suggested that even though the non-universality of the transport exponent µ − β might originate from a sufficiently strong power-law singularity of the transition rate distribution between pores in three dimensions, narrow gaps responsible for the singularity would not be relevant in two dimensions and the transport exponent for lattices should be recovered. 18,19,21,23 We cannot conclusively determine whether the exponent µ-β is universal, but our analysis suggests this is the case if the system is ergodic. In order to investigate whether the transport exponent µ − β is universal, we estimate the transition rate (W) between two pores and its distribution (ρ(W )).…”

Piezoresistivity and tunneling-percolation transport in apparently nonuniversal systems

“…2 For example, for two-dimensional networks, the exponent t is equal to 1/͑1−␣ ϱ ͒ for ␣ ϱ տ 0.23, while for lower values of ␣ ϱ the dc transport becomes universal with t Ӎ 1.3. [13][14][15][16] The fact that for finite L / values the distribution function h L ͑g͒ displays the behavior plotted in Fig. 2 suggests that the network conductance G follows Eq.…”

Multifractal properties of resistor diode percolation

“…ψ 0 is related to the fractal dimension D B of DP clusters via D B = 1 + ψ 0 /ν ⊥ − z (cf. [29,31]). Equation (3.50) is in agreement with the ε-expansions of ν ⊥ , z [3,47], and D B (see Refs.…”