Renyi Entropy and Fractional Order Mechanics for Predicting Complex Mechanics of Materials

Recently we employed entropy dynamics, a statistical inference tool that facilitates quantifying posterior probabilities of likely particle positions, to create material models relating fractal polymers networks to their constitutive behaviors.(1) This methodology is applicable to classical mechanics, electromagnetic field theory, and quantum mechanics, thus offering new opportunities to expand our understanding of functional materials. The entropy dynamics approach usually starts by maximizing Shannon entropy of possible particle locations with added constraints to account for particle interactions or motion. Here, we take a broader approach and use the Renyi entropy, a generalization of the Shannon entropy, to build our constitutive models for multi-functional polymers. The Renyi entropy allows us to derive wide-ranging material constitutive models that consolidate other entropy approaches such as max-entropy, min-entropy, and collision entropy. Furthermore, we investigate material properties using fractional moment constraints instead of the widely used integer moment constraints. Finally, we show how our approach provides a way to building models relevant to a broad class of smart materials and structures.