In the last part, we were interested in the state of the art in spatial neutral theory of Ecology, which problems sound familiar to experts in Statistical Physics. When the derivative of this tempered fractional Brownian motion is used in the fractional Langevin equation, it leads to ballistic diffusion at long times. Instead, at long times it exhibits localization as Ornstein-Uhlenbeck. It is known as tempered fractional Brownian motion and, surprisingly, it does not arrive to the same behaviours than our models. We were also interested in another process defined through a tempering directly done in the fractional Brownion motion definition. When the truncation is done by a weak power-law truncation, we also got a different crossover behaviour from faster to slower superdiffusion and from slower to faster subdiffusion. Deriving analytical expressions of the MSD for the overdamped Langevin equation and the fractional Langevin equation, we find that when the truncation is strongly done, the mentioned crossover appears. We studied the motion of particles driven by tempered fractional Gaussian noise which power-law correlations present a cutoff at some mesoscopic time scale. For example, it happens in viscoelastic systems such as the motion of lipid molecules in lipid bilayer membranes. On the other hand, many experiments show a characteristic crossover from anomalous to normal diffusion. In agreement with our computational results, we were able to obtain analytical expressions for a list of observables too, including temporal and ensemble-average MSD, Ergodicity Breaking parameter, one-point one-time probability density function and p-variation. In addition, we also use an explicit time-dependent factor to model non-stationarity, usually referred to aging in the specialized literature. We find that a family of stochastic processes known as generalised grey Brownian motion can fit correctly that observation. coli cells, where stochastic processes as the continuous-time random walk can explain non-ergodicity but an alternative like the fractional Brownian motion is needed to reproduce p-variation. This is the case of the motion of mRNA molecules inside living E. However, the underlying physics of a plethora of experiments is still not well-understood. On the one hand, anomalous diffusion has been shown to appear extensively in nature and, consecuently, scientists have developed several and different models that can effectively reproduce it. It can be thought as the amount of space the particles have explored in the system. Physically, the MSD is a measure of the deviation of the position of the particles over time. When the relation is faster than linear, it is called superdiffusion and when it is slower, subdiffusion. Anomalous diffusion is a diffusion process which Mean Square Displacement (MSD) is not a linear funtion of time, what is known as normal diffusion.
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