On cross-correlation between thermal gradients and electric fields

An important phenomenon in the field of thermoelectric conversion in certain materials is the Seebeck effect, which is characterized by an electrical field, E being produced by a temperature gradient, E=S∇θ, where S is known as the (isotropic) Seebeck number. The objective of this note is to develop bounds on the effective thermoelectric Seebeck property for heterogeneous mixtures of materials. Specifically, we develop bounds on 〈E〉Ω=S∗〈∇θ〉Ω, where S∗ is the effective Seebeck number for the mixture, where the averaging operator is defined as 〈·〉Ω=def1|Ω|∫Ω(·)dΩ over a statistically representative volume element with domain Ω, using only the pointwise cross-correlation properties of the material and the average thermal fields. .