Voltage distribution in a two-component random system

Abstract

A disordered medium composed of randomly arranged metal and insulator, both with finite conductance, is considered. The distribution of voltage drops in such two-component random system has been calculated both analytically and numerically. It is shown that the distribution N(y) of the logarithm of voltage drops, yϭϪln(2), is the sum of several members, N ck (y) and N ik (y), kϭ0,1,2,. .. . Members N ck (y) describe the voltage distribution in the metallic phase. Members N ik (y) describe the voltage distribution in the insulating component. The subsequent members are shifted subsequently on the y axis by an amount of 2k ln(hL 1/()), where is the crossover exponent and is the percolation correlation length exponent. The zero-order member of the N ck family is governed by the multifractal spectrum f (␣), where ␣ϭy/lnL, found originally for the random resistor network. The zero-order member of the N ik family is governed by the multifractal spectrum (␣) found originally for the random resistor superconductor network. The next members are built from two components. The first one is the scaled repetition of N c0 for the N ck family or N i0 for the N ik family. The other one is the distribution of voltage drops in such percolation objects like dangling ends, isolated clusters for the N ck family or clusters perimeter for the N ik family.

Critical Behavior of Nonlinear Properties in Percolating Superconductor/Nonlinear-Normal Conductor Networks

“…This approach may be useful when one is asked to discuss the percolation effect on nonlinear properties qualitatively and can be used as a basic step to describe the critical behavior of arbitrary nonlinear response. For realistic system, the ratio of the poor conductivity and the good conductivity may not be zero,' 11,28,29 ! we can take one step forward to discuss the crossover effect and scaling behavior in such a system, which will be reported elsewhere.…”

Hotspots in two-phase conducting media

“…The distribution of the currents flowing through individual resistors and the corresponding voltage drops has been studied in great detail. 20,29,30 We use these known results to validate our model and therefore represents the first step for the investigation of the dissipated power distribution. To this end, we introduce dimensionless quantities i nm = I nm LR eff /V and v nm = V nm L/V, where V nm = I nm R nm .…”

Critical currents and voltages in weakly nonlinear inhomogeneous media

“…for χ d = χ m = 0). In the linear problem the behaviour of the moments W Dk , W Mk was found both above and below the percolation threshold [3][4][5][6][7][8][9][10]. It is summarized in table 1.…”

“…where ν is the percolation correlation length exponent. Important special cases are t (2) ≡ t and q(2) ≡ q, which characterize the linear conductivity behaviour above and below the percolation threshold, [σ ∼ σ m τ t for p > p c and σ ∼ σ d |τ | −q for p < p c [13,14], and the resistance noise exponents κ = 2t − t (4), and κ = 2q − q(4) [15,16]. The relations in table 1 enable one to find the behaviour of the effective cubic-nonlinear susceptibility in the neighbourhood of the percolation threshold [10,17] χ ∼ χ m τ t (4)…”

“…The condition 'weak' requires σ ( r) χ( r)| K( r)| 2 . Stroud and Hui [1] and Aharony [2] have shown that in the limit of low fields the overall j versus K dependence is also cubic, j = σ K + χ| K| 2 K. The effective conductivity σ is given by σ = σ ( r) K( r) 2 / K( r) 2 , whereas the effective nonlinear susceptibility, χ, is related to the fourth moment of the local field distribution, χ = χ( r) K( r) 4 / K( r) 4 .…”