An Entropy Dynamics Approach to Inferring Fractal-Order Complexity in the Electromagnetics of Solids

A fractal-order entropy dynamics model is developed to create a modified form of Maxwell's time-dependent electromagnetic equations. The approach uses an information-theoretic method by combining Shannon's entropy with fractional moment constraints in time and space. Optimization of the cost function leads to a time-dependent Bayesian posterior density that is used to homogenize the electromagnetic fields. Self-consistency between maximizing entropy, inference of Bayesian posterior densities, and a fractal-order version of Maxwell's equations are developed. We first give a set of relationships for fractal derivative definitions and their relationship to divergence, curl, and Laplacian operators. The fractal-order entropy dynamic framework is then introduced to infer the Bayesian posterior and its application to modeling homogenized electromagnetic fields in solids. The results provide a methodology to help understand complexity from limited electromagnetic data using maximum entropy by formulating a fractal form of Maxwell's electromagnetic equations.