Brownian motion and anomalous diffusion revisited via a fractional\n Langevin equation

In this paper we revisit the Brownian motion on the basis of {the fractional\nLangevin equation which turns out to be a particular case of the generalized\nLangevin equation introduced by Kubo in 1966. The importance of our approach is\nto model the Brownian motion more realistically than the usual one based on the\nclassical Langevin equation, in that it takes into account also the retarding\neffects due to hydrodynamic back-flow, i.e. the added mass and the Basset\nmemory drag. We provide the analytical expressions of the correlation functions\n(both for the random force and the particle velocity) and of the mean squared\nparticle displacement. The random force has been shown to be represented by a\nsuperposition of the usual white noise with a "fractional" noise. The velocity\ncorrelation function is no longer expressed by a simple exponential but\nexhibits a slower decay, proportional to t^{-3/2} for long times, which indeed\nis more realistic. Finally, the mean squared displacement is shown to maintain,\nfor sufficiently long times, the linear behaviour which is typical of normal\ndiffusion, with the same diffusion coefficient of the classical case. However,\nthe Basset history force induces a retarding effect in the establishing of the\nlinear behaviour, which in some cases could appear as a manifestation of\nanomalous diffusion to be correctly interpreted in experimental measurements.\n