Fractal aspects of magneto-thermoelectricity: Towards generalized Onsager relations

We present a description and some applications of the basic relationships that may affect the figure of merit in complex multiscale thermoelectric materials. The success of Onsager´s Linear relationships between Fluxes and Forces in explaining the reversible Thermoelectric effects and in deriving Kelvin´s relationships (Onsager 1931) has been extended to include Magneto-Thermo-Electrical effects. Nernst or Nernst-Ettinghausen effects, among others provide further examples of possible applications in thermal technologies. The way in which the material structure is built with a controllable multifractal aspect, alternating at many different scales the grains which, either due to intrinsic crystalline anisotropy or due to a selective doping produce power relationships between the interfacial line lengths and the areas in 2D, or between them area of the surface separating subsets of different material properties and the volumes of the respective grains in 3D. The application of these fractal aspects in order to describe fluxes that may be very different when measured at different scales may also be stated in terms of the relationships between fluxes and forces or between fluxes per unit area and gradients perpendicular to that same area. When basic physical properties that are defined in a very different geometrical way, such as masic properties or surface properties, the need of integrating over all possible scales arises in order to avoid singularities in the theory. The effect of minimum and maximum grain size clusters and their geometrical self similarity is studied in terms of non-linear relationships and of higher order cumulants for several of the Magneto-Thermo-Electric (Devies 1952) and Thermoelectric Effects.