Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case. (20th May 2022)

• Record Type: Journal Article
• Authors: Awad, Emad; Metzler, Ralf
• Is Part Of: Journal of physics. Volume 55:Number 20(2022)
• Publication Date: 2022-05-20
• Subjects: multidimensional fractional diffusion equation -- continuous time random walks -- crossover anomalous diffusion dynamics -- non-Gaussian probability density; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ac5a90
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: Anomalous diffusion with a power-law time dependence ⟨ | R | 2 ( t ) ⟩ ≃ t α i of the mean squared displacement occurs quite ubiquitously in numerous complex systems. Often, this anomalous diffusion is characterised by crossovers between regimes with different anomalous diffusion exponents α i . Here we consider the case when such a crossover occurs from a first regime with α 1 to a second regime with α 2 such that α 2 > α 1, i.e., accelerating anomalous diffusion. A widely used framework to describe such crossovers in a one-dimensional setting is the bi-fractional diffusion equation of the so-called modified type, involving two time-fractional derivatives defined in the Riemann–Liouville sense. We here generalise this bi-fractional diffusion equation to higher dimensions and derive its multidimensional propagator (Green's function) for the general case when also a space fractional derivative is present, taking into consideration long-ranged jumps (Lévy flights). We derive the asymptotic behaviours for this propagator in both the short- and long-time as well the short- and long-distance regimes. Finally, we also calculate the mean squared displacement, skewness and kurtosis in all dimensions, demonstrating that in the general case the non-Gaussian shape of the probability density function changes.