  In this paper, we introduce the macroscopic quantity, namely the dispersion
tensor or the \textit{Burnett coefficient}s in the class of generalized
Hashin-Shtrikman micro-structures \cite[page no. 281]{T}. In the case of
two-phase materials associated with the periodic Hashin-Shtrikman structures,
we settle the issue that the dispersion tensor has an unique minimizer, which
is so called Apollonian-Hashin-Shtrikman micro-structure.
