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

Abstract

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 nonFickian 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.

Monte Carlo evaluation of FADE approach to anomalous kinetics

“…This is intimately connected to the exponential pdf assumed for the waiting times, as for this distribution the transition rate, i.e. the probability per unit time to effectuate a jump in a dt after time t -given that up to t no jump has been elapsed -, is a constant t ∀ and does not depend on the past history [18,21]. We remark that, in this case, the Galilei variant and the We give now a proof of the expansion (B5) and present some remarks on its physical meaning.…”

“…In this paper we will focus on the case of subdiffusion (0<α<1) on a 1D infinite support, in presence of an external (constant) advective field. In the subdiffusive case [3][4][5][6][8][9][10]16,[19][20][21]22,25], an algebraically decaying distribution is assumed for the waiting times of the particles in the surrounding medium (instead of the traditional exponential one): this physically means that the particles will have a non-vanishing probability of extremely long sojourn times in the visited locations, due to so-called "trapping events" (see e.g. [36]).…”

“…In order to evaluate the effects of the approximations introduced in the transformed space as to obtain the FADE equations, we resort to Monte Carlo simulation as a mean to accurately solve CTRW equation (as proposed e.g. in [3,10,21]): it turns out that relevant deviations of the analytical FADE results with respect to the Monte Carlo solutions are evident when 1 → α . In particular, our analysis focuses on the moments of the distribution P(x,t), which constitute accurate estimators of the solution accuracy [19,21].…”

Some insights in superdiffusive transport

“…In this description, the pdf of flight lengths and waiting times are decoupled and Markovianity of the stochastic process is ensured by the fact that +∞ < t [10,16]. It can be shown that the pdf (3) obeys the generalized Lévy-Gnedenko's Theorem [23,1,28] and asymptotically converges to a Lévy stable law [1,23]: this means that, analogously as for the Central Limit Theorem, the sum of many successive jump lengths has the same distribution as that of a single jump, up to a scaling factor. Assuming that the distribution (3)…”

Pre-asymptotic corrections to fractional diffusion equations

“…With the help of Monte Carlo simulations, building on the discussion in Refs. [17] and [25], we show that pre-asymptotic corrections to the fractional diffusion equations play a significant role, depending on the microscopic dynamics of the particles [42]. Neglect of these corrections would lead to gross errors in the estimation of model parameters from measured data.…”