Ultrafast dynamics modeling via fractional Brownian motion run with Mittag-Leffler clock in porous media

Abstract

Ultrafast diffusion process characterized by unusually large diffusivities is often occurs on porous media and the mean square displacement grows exponentially in time. This paper clarifies the characteristics of ultrafast diffusion and tackles this perplexing problem using the fractional Brownian motion run with a nonlinear clock model. We employ the Mittag-Leffler function as the nonlinear clock, the increments are dependent and obey Gaussian distribution, and the derived corresponding mean square displacement is more widely than the exponential function. A comparison between the power law model and the proposed model with respect to available experimental data verifies that the proposed model is more effective and accurate. Ultraslow diffusion is also studied with the inverse Mittag-Leffler function as the nonlinear clock. The results show that it can capture the ultraslow diffusion process better than the case of the logarithmic model. As the generalization of fractional Brownian motion, fractional Brownian motion run with a nonlinear clock is an alternative model method for extreme anomalous diffusion in complex systems.