Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function

Abstract

Abstract This study analyzes the complexity of ultraslow diffusion process using both the classical Shannon entropy and its general case with inverse Mittag-Leffler function in conjunction with the structural derivative. To further describe the observation process with information loss in ultraslow diffusion, e.g., some defects in the observation process, two definitions of fractional entropy are proposed by using the inverse Mittag-Leffler function, in which the Pade approximation technique is employed to numerically estimate the diffusion entropy. The results reveal that the inverse Mittag-Leffler tail in the propagator of the ultraslow diffusion equation model adds more information to the original distribution with larger entropy. Smaller value of α in the inverse Mittag-Leffler function indicates more complicated of the underlying ultraslow diffusion and corresponds to higher value of entropy. The proposed definitions of fractional entropy can serve as candidates to capture the information loss in ultraslow diffusion.