Studies on the scaling exponents of conductivity for Sierpinski carpets

Abstract

The size scaling exponents alpha of the DC conductivity of Sierpinski carpets made of conductive paper are measured, and the possible relations between the exponent alpha and the fractal dimension d_f, the symmetry and the topology of carpets are studied. For some sequences of the carpets, the authors find that alpha can be expressed as the convenient power law alpha ^{varies as} (2-d_f)^eta and the exponent eta is not sensitive to the geometry of the carpets and is almost constant (about 0.96) in some region beta (from 0.6 to 1.6), where beta is a control parameter used to describe other properties then fractal dimension, such as the symmetry of the sequences of carpets and is defined by the ratio of the void sections along the x direction to those along the y direction in the carpet. The relation between the spectral dimension d_s and the geometry of the carpet is also discussed.