The Landau-Lifshitz/Looyenga dielectric mixture expression and its   self-similar fractal nature
  In this paper, dielectric permittivity of dielectric mixtures is discussed in view of the spectral density representation method. A distinct representation is derived for predicting the dielectric properties [Tuncer E 2005 J. Phys. Condens. Matter 17 L125]. In order to illustrate the strength of the representation and confirm the proposed hypothesis, Landau-Lifshitz/Looyenga expression is selected, and the structural information of the mixture is extracted. Both a recently developed numerical method to solve inverse integral transforms and the proposed empirical scaled permittivity expression are employed to estimate the spectral density function of the Landau-Lifshitz/Looyenga expression. In the simulations the concentration q of the inclusions phase are varied. The estimated spectral functions for the mixtures with different inclusion concentration compositions show similar spectral density functions, composed of couple of bell-shaped distributions, with coinciding peak locations. We think therefore that the coincidence is an absolute illustration of a self-similar fractal nature of mixture topology (structure) for the considered Landau-Lifshitz/Looyenga expression. Consequently, the spectra are not altered significantly with increased filler concentration level--exhibit a self-similar spectral density functions for different concentration levels. We conclude that the Landau-Lifshitz/Looyenga expression is therefore suitable for complex composite systems that have hierarchical order in their structure, which confirms the finding in the literature. 
Enis Tuncer