Effective conductivity ($⟨a\nabla u⟩=a^{\ast }⟨\nabla u⟩$)

Effective conductivity describes the averaged response of a microscopic conductivity field to a macroscopic temperature or electric-potential gradient. Here the positive conductivity flux is $a\nabla u$; physical heat flux has the opposite sign. For periodic media it is computed by cell correctors, and can be a tensor even for isotropic constituents.