Overview of electrical processes in fractal geometry: from electrodynamic relaxation to superconductivity

The consequences of the parameterization of the fractal set on the electrodynamics of this set are analyzed. The relevance of scaling properties to electrochemical, dielectric, and magnetic relaxations is considered with a special emphasis on the use of noninteger derivative operators in electromagnetism and superconductivity. In electromagnetism, the above analysis includes a brief overview of the main results already obtained, focusing especially on the introduction of dissipative terms in the equation of propagation and on the generalized form of the uncertainty principle in fractal media. The new Laplacian and d´Alembertian operators are evoked as well as the scale relativity on which this new analysis is founded. For superconductivity, the analysis introduces a geometrical interpretation founded on frustration acting not only on topology but on the metric of the space-time in a particular type of fractal geometry. Although this point of view may appear as a breakthrough in the theory of superconductors, the model offers some relations with the theory of fractional statistics and the theory of Anyons