Fractional anomalous diffusion laws on spherical surfaces from fractional generalized Langevin equation theory

This work deals with an extended mathematical study of the fractional anomalous diffusion laws of a target particle which moves on a sphere of radius. This movement is provoked by a diffusion of the particle with traps (as molecules and other particles) forming the surface. The fractional anomalous diffusion is a consequence of the complexity of the host medium structure, and it is very slow in comparison with the normal diffusion. Dynamics of the target particle is studied considering three physical quantities, which are the mean square displacement (MSD), the time diffusion coefficient (TDC) and the velocity autocorrelation function (VACF). The study is achieved within the framework of the Fractional Generalized Langevin Equation Theory, with a chosen memory function. The time evolution of MSD and VACF is exactly determined with the help of Laplace transform techniques. An exact calculation of the temporal evolution of the dynamic quantities in question, by choosing a power law memory function. The main result is that, at large time, the movement of the target particle is saturated and MSD goes to a finite value, i.e. 2R2, unlike the anomalous diffusion taking place in spaces of infinite extension.