Anomalous Brownian motion via linear Fokker-Planck equations

Abstract

According to a traditional point of view Boltzmann entropy is intimately related to linear Fokker-Planck equations (Smoluchowski, Klein-Kramers, and Rayleigh equations) that describe a well-known nonequilibrium phenomenon: (normal) Brownian motion of a particle immersed in a thermal bath. Nevertheless, current researches have claimed that non-Boltzmann entropies (Tsallis and Renyi entropies, for instance) may give rise to anomalous Brownian motion through nonlinear Fokker-Planck equations. The novelty of the present article is to show that anomalous diffusion could be investigated within the framework of non-Markovian linear Fokker-Planck equations. So on the ground of this non-Markovian approach to Brownian motion, we find out anomalous diffusion characterized by the mean square displacement of a free particle and a harmonic oscillator in absence of inertial force as well as the mean square momentum of a free particle in presence of inertial force.