RECIPROCITY RELATIONS FOR THE EFFECTIVE ELECTRICAL-CONDUCTIVITY OF RANDOMLY INHOMOGENEOUS-MEDIA IN THE FRACTAL REGIME

An exact relation for the realization-averaged effective conductivitie s in the fractal region is found for two-dimensional randomly inhomoge neous media. It has the form {sigma(e)(tau,L)} X{1/sigma(e)(-tau,L)}(- 1) = sigma(e)(2)(tau= 0,L much greater than xi), where xi is the corre lation length (the self-averaging scale), L is the size of the system, tau= (p -p(c))/p(c), and p(c) is the percolation threshold. For L muc h greater than xi, the system is self-averaged, and the relation trans forms into the Dykhne reciprocity relation, A. M. Dykhne, Zh. Eksp. Te or. Fit. 59, 110 (1970) [Sov. Phys. JETP 32, 63 (1971)] sigma(e)(tau)s igma(e)(-tau)= sigma(e)(2)(tau=0)= sigma(1) sigma(2). A similar relati on is obtained for media with an exponentially broad distribution of l ocal conductivities, as well as for individual realizations of some de terministic structures. (C) 1998 American Institute of Physics.