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

Area of Science:

• Electromagnetism
• Information Theory
• Fractal Dynamics

Background:

• Maxwell's equations are fundamental to classical electromagnetism.
• Understanding complex electromagnetic fields often requires advanced modeling techniques.
• Information-theoretic approaches offer novel ways to analyze physical systems.

Purpose of the Study:

• To develop a modified form of Maxwell's time-dependent electromagnetic equations using a fractal-order entropy dynamics model.
• To integrate Shannon's entropy and fractional moment constraints for electromagnetic field analysis.
• To establish a self-consistent framework for maximizing entropy and inferring Bayesian posterior densities.

Main Methods:

• Development of a fractal-order entropy dynamics model.
• Application of information-theoretic methods, including Shannon's entropy and fractional moment constraints.
• Optimization of a cost function to derive a time-dependent Bayesian posterior density for field homogenization.
• Formulation of fractal derivative definitions and their relation to differential operators (divergence, curl, Laplacian).

Main Results:

• A novel fractal-order entropy dynamic framework is established.
• The framework enables the inference of Bayesian posterior densities for modeling homogenized electromagnetic fields in solids.
• Demonstration of self-consistency between maximum entropy, Bayesian inference, and fractal Maxwell's equations.
• Establishment of relationships between fractal derivatives and standard differential operators.

Conclusions:

• The developed methodology provides a new approach to understanding complexity in electromagnetic data.
• The fractal-order entropy dynamics model offers a powerful tool for analyzing and homogenizing electromagnetic fields.
• This work bridges concepts from fractal geometry, information theory, and electromagnetism to advance scientific understanding.