Fractal model of the transition from ballistic to diffusive motion of a Brownian particle

A formula is presented for calculation the particle mean square displacement normalized by the square of its mean free path as dependent on the time related to the momentum relaxation time. The obtained equation is a result of the fractal analysis of the particle trajectory being a sequence of linear segments. At very short times the motion is ballistic whereas for long times the particle starts to behave according to Einsteinʹs theory. The slope of a log–log plot of time dependence of mean square displacement changes from two to one. The ballistic to diffusive transition is wider than that described by the Langevin equation and spans more than three decades of time.