Critical behaviour of non-linear susceptibility in random non-linear resistor networks

Abstract

The critical behaviour of non-linear susceptibility of a two-component composite is studied in this paper. The first component of fraction p is non-linear and obeys a current-field (J -E) characteristic of the form J = g 1 v + χ 1 v β while the second component of fraction q is linear with J = g 2 v. Near the percolation threshold p c or q c , we examine the conductorinsulator (C-I) limit (g 2 = 0) and superconductor-conductor (S-C) limit (g 2 = +∞). For the C-I limit and p > p c , the effective linear and non-linear response functions behave as g e ≈ (p − p c ) t and χ e (β) ≈ (p − p c ) t 2 (β) , respectively. For the S-C limit and q < q c , g e and χ e (β) are found to diverge as g e ≈ (q c − q) −s and χ e (β) ≈ (q c − q) s 2 (β) . Within the effectivemedium approximation, the exponents are found to be s = t = 1 and s 2 (β) = t 2 (β) = (β +1)/2, p c = 1/d and q c = (d − 1)/d. By using a connection between the non-linear response of the random non-linear composite problem and the resistance or conductance fluctuations of the corresponding random linear composite problem, the exponents t 2 (β) and s 2 (β) are found to berespectively, where t (s) is the conductivity exponent in a C-I(S-C) composite, d is the dimension of the composite and ν is the correlation-length exponent in d dimensions, κ((β + 1)/2) and κ ((β + 1)/2) are given by ψ R ((β + 1)/2) + [(β + 1)/2](dν − ζ R ) = κ((β + 1)/2) + dν, ψ G ((β + 1)/2) + [(β + 1)/2](dν − ζ G ) = κ ((β + 1)/2) + dν, where ψ R ((β + 1)/2)(ψ G ((β + 1)/2)) characterizes the scaling of the [(β + 1)/2]th cumulant of the global resistance (conductance) distribution due to local resistance (conductance) fluctuations in the corresponding linear C-I(S-C) composites, ζ R = t − (d − 2)ν and ζ G = s + (d − 2)ν.We prove that t 2 (β) is a monotonically increasing function of β while s 2 (β) is a monotonically decreasing function of β, which have the following special values:The critical behaviour of the non-linear susceptibility in a C-I composite is very different from that of the non-linear susceptibility in a S-C composite and some unexpected results of the critical behaviour of non-linear susceptibility in a C-I network are reported in this paper.

Critical currents and voltages in weakly nonlinear inhomogeneous media

“…These agree with [29] and [30] and disagree with [27] and [28]. Again, the critical field K c can be defined as a field at which the linear and nonlinear terms become comparable, K c = (εσ/χ) 1/(2k−2) .…”

“…which only in part agrees with that of Blumenfeld and Bergman [19]. They incorrectly proposed χ ∼ L d δσ k c , which made the effective parameter χ size dependent and has led to incorrect analyses performed in some papers on related subjects [27][28][29].…”

Critical properties of nonlinear susceptibilities for weakly nonlinear composites

“…where δσ 2 e is the mean square fluctuation of the effective linear conductivity σ e . The above equation can be easily generalized to the effective nonlinear susceptibility χ e (β) for arbitrary β [13,14]: (10) where δσ (β+2)/2 e is defined as the higher-order cumulant. Consequently, the quantity δσ…”

“…, by replacing ξ with L (not L with ξ ). In short, L cannot be replaced by ξ in any case, and L and ξ must not be confused [13].…”

“…For the case where β = 2, these exponents have been studied in references [11,12] on the basis of the effective-medium approximation (EMA), and in references [2,12] on the basis of the relation to the noise exponents k(2) (or k ( 2)) which characterize the divergence of the relative resistance (or conductance) fluctuations in linear random composites. The qualitative results obtained using the EMA for arbitrary β have been reported [7,13]. It is our aim here to determine the dependence of the exponents u(β) and v(β) on the nonlinear exponent β by relating the nonlinear response of the random composite problem to the resistance (or conductance) fluctuations of the corresponding problem.…”