Computation of the inverse Mittag-Leffler function and its application to modeling ultraslow dynamics

The inverse Mittag-Leffler function has been used to model the logarithmic growth of the mean squared displacement in anomalous diffusion, the restricted mobility of membrane proteins, and the slow viscoelastic creep observed in glasses. These applications are hindered because the inverse Mittag-Leffler function has no explicit form and cannot be approximated by existing methods in the domain x is an element of (0, +infinity). This study proposes a conversion method to compute the inverse Mittag-Leffler function in terms of the Mittag-Leffler function. The new method uses the one- and two-parameter Mittag-Leffler function to compute the inverse Mittag-Leffler function in the target domain. We apply this method to fit data collected in studies of: (i) the ultraslow mobility of beta-barrel proteins in bacterial membranes, (ii) the ultraslow creep observed in high strength self-compacting concrete, and (iii) the ultraslow relaxation seen in various glasses. The results show that the inverse Mittag-Leffler function can capture ultraslow dynamics in all three cases. This method may also be extended to other generalized logarithmic laws.