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.

Emergence of Fractional Kinetics in Spiny Dendrites

“…The emergence of fractional kinetics in complex media in CTRW was introduced more explicitly as a general concept in [12]. Analogously to the comb-like model presented, in that short note the special case of a survival probability of the Mittag-Leffer type was there derived in terms of a Markovian process with characteristic waiting time properly distributed:…”

The emergence of self-organization in complex systems–Preface

“…The Tsallis entropy and distribution represent an attempt to build a non-standard statistical mechanics for complex systems, and superstatistics proved to be compatible with systems whose complexity originates from non-homogeneity of some physical parameter in the medium supporting the anomalous transport. It is not yet clear if this picture is in agreement with other modeling approaches, such as the fractional one (see [8,9] and the comment to papers [24,25] reported below). Paper [22] illustrates the application of kinetic theory and hydrodynamic equations to a granular binary mixture, which is an interesting prototype of complexity.…”

“…This result is even more interesting as it has been depicted in the framework of a complex biological system. In recent years, the derivation of fractional operators in the context of disordered, non-homogeneous media has attracted some interest (see, e.g., [8,9]), and paper [25] surely gives an important contribution in shedding light into the relationship between complexity and fractional calculus.…”

Front propagation in anomalous diffusive media governed by time-fractional diffusion

“…However, these two formulations are equivalent if standard initial condition is used [41]. Recently, in place of this operator replacement, the emergence of fractional kinetics in complex media has been explained within the standard Gaussian di↵usion framework by the randomization due to the complexity of the environment of certain 180 characteristic parameters of the medium [7]. Let 0 < 6 1/2 be a real parameter, then the time-fractional di↵usion equation in the Caputo sense reads…”

“…Nevertheless in 5 some complex media di↵usion processes show a non-linear relationship to time, as for example in biological systems [1,2,3], in fluids and plasmas [4] and in other systems [5]. In the latter case, the di↵usion process is known as anomalous di↵usion [6,2,1,7].…”