Random composite: stirred or shaken?

Abstract

A James Bond’s (JB) catchphrase “shaken, not stirred" is explored for the problem of effective conductivity of composites. The superconductivity critical index s for the conductivity of random non-overlapping disks turns out to be distinctly different for shaking and stirring protocols. In the case of stirring modeled by random walks the formula s(τ) = 0.5 + 0.8 ³√τ is deduced for evolution of the critical index with the normalized time 0 ≤ τ ≤ 1, which is proportional to the number of random walks and serving as the disorder measure. Strikingly, the coefficient 0.8 is very close to the critical index for shaking protocol and 0.5 is the critical index for regular lattices. The obtained formula for s is based on the analytical solution to the 2D conductivity problem of randomly distributed disks up to O(x¹⁹), where x denotes the concentration of inclusions and its extension to special 3D composites.