Hydrodynamic fluctuations in the presence of one parameter Mittag-Leffler friction

The effects of hydrodynamic fluctuations on the subdiffusive motion of a particle subject to one parameter Mittag-Leffler friction are examined by means of the fractional Langevin equation. The particle experiences an overall additive colored noise formed by, on the one hand, the hydrodynamic back flow effects and, on the other hand, an additional contribution predicted by fluctuation dissipation relation. Particle motion may or may not be subject to a restoring force. All possible combinations of forces exerted on the test particle are being studied, and for each of them the generalized response function in terms of multinomial Mittag-Leffler functions is provided. Mean square displacement, normalized velocity and position auto-correlation functions are furnished as special cases of the generalized response function, and their short and long time limits are analytically given. In addition, for the same measures analytical expressions valid for time windows much broader than the usual asymptotic limit are provided, and can be used to fit real life data. We demonstrate that normalized velocity and position auto-correlation functions are the main sources providing information on the effect of hydrodynamic fluctuations on particle motion. Actually, they oppose to friction and to restoring force, and smooth out the anti-persistent character of the motion.