The study of fracional anomalous diffusion laws on the hypersphere

In this work, we present an extended mathematical study of the fractional anomalous diffusion of a target particle which moves on a curved surface. To do explicit calculations, we assumed that this particle moves on a hypersphere of radius R. We analyzed the particle dynamics using a generalized fractional Langevin equation approach, based on the Caputo fractional derivative and Laplace transform techniques. We derived three physical quantities:the mean square displacement (MSD), the time diffusion coefficient (TDC), and the velocity autocorrelation function (VACF). We performed exact calculations of their temporal evolution by selecting power-law and Dirac delta memory functions. The results obtained are expressed in terms of Mittag-Leffler-type functions. The introduction of the fractional derivative affected the behavior of balistic regime. We plotted the dynamic quantities for different values fractional derivative order and analyzed their influence on diffusion scaling laws.