Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case
Emad Awad, Ralf Metzler
Abstract
Abstract Anomalous diffusion with a power-law time dependence <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">⟨</mml:mo> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi mathvariant="bold">R</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">⟩</mml:mo> </mml:mrow> <mml:mo>≃</mml:mo> <mml:msup> <mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msup> </mml:math> 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.