A Diagram Technique for Perturbation Theory Calculations of the Effective Conductivity of TwoDimensional Systems
Abstract
The perturbation theory for calculation of the effective conductivity of the plane consisting of pieces of different conductivities is constructed and the convenient diagram technique for this perturbation theory is elaborated. It is shown that for the chessboard perturbative calculations give results which are in agreement with the wellknown formula $\sigma_{eff} = \sqrt{\sigma_1\sigma_2}$. The components of the tensor of effective conductivity for the anisotropic threecolor chessboard are calculated. It is shown that the isotropic (symmetric) part of effective conductivity calculated up to the sixth order of perturbation theory satisfies the Bruggeman effective medium equation for symmetric threecolor structures with equally partitioned components. We also consider isotropic threecolor chessboard with nonequal weights of colors. In this case the perturbation theory already in fourth order contradicts the results following from the Bruggeman equation for nonequal weights.
 Publication:

Soviet Journal of Experimental and Theoretical Physics
 Pub Date:
 December 2000
 DOI:
 10.1134/1.1342894
 arXiv:
 arXiv:condmat/0008012
 Bibcode:
 2000JETP...91.1261K
 Keywords:

 Condensed Matter  Materials Science;
 Mathematical Physics
 EPrint:
 19 pages