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 while the second component of fraction q is linear with . Near the percolation threshold or , we examine the conductor - insulator (C - I) limit and superconductor - conductor (S - C) limit . For the C - I limit and , the effective linear and non-linear response functions behave as and , respectively. For the S - C limit and , and are found to diverge as and . Within the effective-medium approximation, the exponents are found to be s = t = 1 and , and . 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 and are found to be , , respectively, where t(s) is the conductivity exponent in a C - I(S - C) composite, d is the dimension of the composite and is the correlation-length exponent in d dimensions, and are given by , , where characterizes the scaling of the th cumulant of the global resistance (conductance) distribution due to local resistance (conductance) fluctuations in the corresponding linear C - I(S - C) composites, and . We prove that is a monotonically increasing function of while is a monotonically decreasing function of , which have the following special values: and ; and ; and . 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.