A numerical study of fractional diffusion through a Langevin approach in random media.

The study of Brownian motion has a long history and involves many different formulations. All these formulations show two fundamental common results: the mean square displacement of a diffusing particle scales linearly with time and the probability density function is a Guassian distribution. However standard diffusion is not universal. In literature there are numerous experimental measurements showing non linear diffusion in many fields including physics, biology, chemistry, engineering, astrophysics and others. This behavior can have different physical origins and has been found to occur frequently in spatially disordered systems, in turbulent fluids and plasmas, and in biological media with traps, binding sites or macro-molecular crowding. Langevin approach describes the Brownian motion in terms of a stochastic differential equation. The process of diffusion is driven by two physical parameters, the relaxation or correlation time tau and the velocity diffusivity coefficient Dv. An extension of the classical Langevin approach by means of a population of tau and Dv is here considered to generate a fractional dynamics. This approach supports the idea that fractional diffusion in complex media results from Gaussian processes with random parameters, whose randomness is due to the medium complexity. A statistical characterization of the complex medium in which the diffusion occurs is realized deriving the distributions of these parameters. Specific populations of tau and Dv lead to particular fractional diffusion processes. This approach allows for preserving the classical Brownian motion as basis and it is promising to formulate stochastic processes for biological systems that show complex dynamics characterized by fractional diffusion. A numerical study of this new alternative approach represents the core of the present thesis.