Short note on the emergence of fractional kinetics

Abstract

In the present Short Note an idea is proposed to explain the emergence and the observation of processes in complex media that are driven by fractional non-Markovian master equations. Particle trajectories are assumed to be solely Markovian and described by the Continuous Time Random Walk model. But, as a consequence of the complexity of the medium, each trajectory is supposed to scale in time according to a particular random timescale. The link from this framework to microscopic dynamics is discussed and the distribution of timescales is computed. In particular, when a stationary distribution is considered, the timescale distribution is uniquely determined as a function related to the fundamental solution of the space-time fractional diffusion equation. In contrast, when the non-stationary case is considered, the timescale distribution is no longer unique. Two distributions are here computed: one related to the M-Wright/Mainardi function, which is Green's function of the time-fractional diffusion equation, and another related to the Mittag-Leffler function, which is the solution of the fractional-relaxation equation.

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• Fractional kinetics emerging from ergodicity breaking in random media
• Modeling biological systems with an improved fractional Gompertz law
• Anomalous diffusion originated by two Markovian hopping-trap mechanisms
• Finite-energy Lévy-type motion through heterogeneous ensemble of Brownian particles
• Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart
• Emergence of Fractional Kinetics in Spiny Dendrites
• Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers