A Limit Theorem Clarifying the Physical Origin of Fractional Brownian Motion and Related Gaussian Models of Anomalous Diffusion

Abstract: We consider a dynamical system describing the motion of a test-particle surrounded by $N$ Brownian particles with different masses. Physical principles of conservation of momentum and energy are met. We prove that, in the limit $N\to\infty$, the test-particle diffuses in time according to a quite general (non-Markovian) Gaussian process whose covariance function is determined by the distribution of the masses of the surround-particles. In particular, with proper choices of the distribution of the masses of the surround-particles, we obtain fractional Brownian motion, a mixture of independent fractional Brownian motions with different Hurst parameters and the classical Wiener process. Moreover, we present some distributions of masses of the surround-particles leading to limiting processes which perform transition from ballistic to superdiffusion or from ballistic to classical diffusion. Keywords: fractional Brownian motion, limit theorems for stochastic processes, anomalous diffusion, heterogeneous environment, crowded environment