Entropy and information in a fractional order model of anomalous diffusion

Fractional order dynamic models (e.g., systems of ordinary and partial differential equations of non-integer order in time and space) are becoming more popular for characterizing the behavior of complex systems. Justification for such models is typically based on improved fits to experimental data or a reduced mean squared error for models with the same number of fitting parameters. This rationale, however, is relative to the form of the selected fitting function, and is dependent on the order of the derivatives. Nevertheless, there seems to be a recognition that fractional order models work better than integer order models in describing the electrical and mechanical properties of multi-scale, heterogeneous materials. In order to address this issue and to offer a new approach for establishing the utility of fractional order models, we calculate the total Shannon spectral entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation of the continuous time, random walk model of diffusion gives a spectral entropy minimum for normal (i.e., Gaussian) diffusion with surrounding values leading to greater values of spectral entropy.