SPECTRAL ENTROPY AND CUMULATIVE SPECTRAL ENTROPY OF HAUSDORFF DERIVATIVE DIFFUSION MODEL1)

The cumulative spectral entropy in space is derived based on the spectral entropy of the Hausdorff derivative diffusion model for describing the spatial and temporal complexity of the anomalous diffusion process. The individual, the total spectral and the cumulative spectral entropies are investigated by varying the diffusion coefficient and the diffusion time. It is shown that the spectral and the cumulative spectral entropies increase with the decrease of the order of the time Hausdorff derivative α or the space Hausdorff derivative β and are characterized by a heavy tail. With the decrease of the diffusion time or the diffusion coefficient, the normal diffusion ( α = 1,β = 1) sees a faster decay of the individual spectral entropy than the anomalous diffusion, and the corresponding spectral density becomes narrower. Thus, the spectral and the cumulative spectral entropies of the Hausdorff derivative diffusion model can reflect the heterogeneous structure of complex media and the uncertainty of the underlying diffusion process.