From generalized Langevin stochastic dynamics to anomalous diffusion.

Scaling methods are fundamental in all branches of physics. In stochastic process, we usually try to describe the long time behavior of a given time correlation function. In this work we investigate a scaling method for anomalous diffusion in systems with memory that produces good results for long and intermediate times. We will initially present a generalization of the diffusion exponent. Then, we present an asymptotic method to obtain an analytical expression for the diffusion coefficient by introducing a time scale factor λ(t). We found an exact expression for the function λ(t), which allows us to describe the diffusive process. For large times, λ(t) becomes a universal parameter determined by the diffusion exponent. In turn, the analytical results are then compared to the numerical results, with a good matching. Then, we'll show the practical effects of scaling. An important first result is that λ(t) quickly converges to a constant. Another very important point was the classification of new forms of diffusion due to the generalized exponent. In previous works, we verified the existence of ergodic ballistic diffusion with diffusion exponent α=2^{-}. Here, we verify the existence of the nonergodic ballistic diffusion type with the obtainment of the diffusion coefficient α=2. Finally, we show that the scaling works. This method is general and can be applied to various types of stochastic problems.