The Monte Carlo and fractional kinetics approaches to the underground anomalous subdiffusion of contaminants

It is nowadays recognized that the experimental evidences of many transport phenomena must be interpreted in the framework of the so-called anomalous diffusion: this is the case e.g. of the spread of contaminant particles in porous media, whose mean squared displacement (MSD) has been experimentally reported to grow in time as t α (with 0 < α < 2). This is in deep contrast with the linear increase which characterizes the standard Fickian diffusive processes. Anomalous transport is currently tackled within the fractional diffusion equation (FDE) analytical model, which has blown new life into the concept of fractional derivatives, dormant since Leibniz's first discovery, about 300 years ago. In this paper we focus on subdiffusion, i.e., the case α < 1, which provides a suitable explanation of the observed non-Fickian persistence of the contaminant particles near the source in various transport experiences in heterogeneous media. Unfortunately, the FDE approach requires several approximations to overcome its analytical complexities. To evaluate the relevance of these approximations, we propound the use of the Monte Carlo simulation as a suitable reference for the analytical FDE results. This comparison shows that the FDE results are less and less reliable as α becomes closer to 1. The approach herein proposed can be easily applied to various fields of science where anomalous diffusion phenomena may occur, from physics and chemistry to biology and finance