A heterogeneous ensemble of Brownian particles governed by the Langevin equation is considered. The heterogeneity of the ensemble is characterised by a population of friction coefficients. When the frequency histogram of the friction coefficients decreases with a proper power-law, the linear superposition of these Brownian trajectories converges to the fractional Brownian motion. This construction of the fractional Brownian motion provides a dynamical framework that allows for physical perspectives. In this respect, this results is first connected with the so-called generalized gray Brownian motion and later the application for modelling anomalous diffusion in biological systems is discussed.

The first part of the seminar is based on a research in collaboration with Mirko D'Ovidio whose results are still work in progress, and the second part is based on:

M. D'Ovidio, S. Vitali, V. Sposini, O. Sliusarenko, P. Paradisi, G. Castellani, G. Pagnini, Centre-of-mass like superposition of Ornstein-Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion. Fract. Calc. Appl. Anal. 21, 1420-1435 (2018), arxiv:1806.11351

D. Molina-García, T. Minh Pham , P. Paradisi, C. Manzo, G. Pagnini, Fractional kinetics emerging from ergodicity breaking in random media. Phys. Rev. E. 94, 052147 (2016)