Modelling intermittent anomalous diffusion with switching fractional Brownian motion

Abstract

The stochastic trajectories of molecules in living cells, as well as the
dynamics in many other complex systems, often exhibit memory in their path
over long periods of time. In addition, these systems can show dynamic
heterogeneities due to which the motion changes along the trajectories. Such
effects manifest themselves as spatiotemporal correlations. Despite the
broad occurrence of heterogeneous complex systems in nature, their analysis is
still quite poorly understood and tools to model them are largely missing. We
contribute to tackling this problem by employing an integral representation
of Mandelbrot's fractional Brownian motion that is compliant with varying
motion parameters while maintaining long memory. Two types of switching
fractional Brownian motion are analysed, with transitions arising from
a Markovian stochastic process and scale-free intermittent processes. We
obtain simple formulas for classical statistics of the processes, namely
the mean squared displacement and the power spectral density. Further, a
method to identify switching fractional Brownian motion based
on the distribution of displacements is described. A validation of the model is given for
experimental measurements of the motion of quantum dots in the cytoplasm of
live mammalian cells that were obtained by single-particle tracking.

Anomalous and ultraslow diffusion of a particle driven by power-law-correlated and distributed-order noises

“…when the second FDT is not satisfied. In both cases, superstatistical and stochastic variations of the diffusion coefficient and anomalous scaling exponent (Hurst exponent) have been analysed recently [47,[82][83][84][85][86]. Studying such concepts in the frameworks developed here will significantly enlarge our current range of stochastic models for disordered systems.…”

Urban Meteorology, Pollutants, Geomorphology, Fractality, and Anomalous Diffusion

“…It is well established that diffusion processes that take place in fractal and/or random media generally present subdiffusive and superdiffusive behaviors induced by the mean geometry and associated to strong correlations in the motion of the particles [93,100,101].…”