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.
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