Monte Carlo evaluation of FADE approach to anomalous kinetics

In a wide range of transport phenomena in complex systems, the mean squared displacement of a particles plume has been often found to follow a non-linear relationship of the kind 〈x2(t) ∝ tα〉, where α may be greater or smaller than 1: these evidences have been described under the generic term of anomalous diffusion. In this paper we focus on subdiffusion, i.e. the case 0 < α < 1, in presence of an external advective field. Widely adopted models to describe anomalous kinetics are continuous time random walk (CTRW) and its fractional advection–dispersion equation (FADE) asymptotic approximation, which accurately account for experimental results, e.g. in the transport of contaminant particles in porous or fractured media. FADE approximated equations, in particular, admit elegant analytical closed-form solutions for the particle concentration P(x, t). To evaluate the relevance of the approximations which allow to derive the asymptotic FADE equations, we resort to Monte Carlo simulation (which may be regarded as an exact solution of the CTRW model): this comparison shows that the FADE equations represent a less and less accurate asymptotic description of the exact CTRW model as α becomes close to 1. We propose higher-order corrections which lead to modified integral–differential equations and derive new expressions for the moments of P(x, t). These results are validated through comparison with those of Monte Carlo simulation, assumed as reference curves.