Closed-form multi-dimensional solutions and asymptotic behaviours for subdiffusive processes with crossovers: II. Accelerating case

Abstract

Anomalous diffusion with a power-law time dependence $⟨|\mathbf{R}{|}^2(t)⟩\simeq t^{{\alpha }_i}$ 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.