Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Levy alpha-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter Mittag-Leffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Levy alpha-stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space- and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.Comment: 7 pages, 5 figures, 1 table. Presented at the Conference on Computing in Economics and Finance in Montreal, 14-16 June 2007; at the conference "Modelling anomalous diffusion and relaxation" in Jerusalem, 23-28 March 2008; et

Overview of ¹⁴C release from irradiated zircaloys in geological disposal conditions

Carbon-14 (radiocarbon, 14C) is a long-lived radionuclide (5730 yr) of interest regarding the safety for the management of intermediate level wastes (ILW). The present study gives an overview of the release of 14C from irradiated Zircaloy cladding in alkaline media. 14C is found either in the alloy part of Zircaloy cladding due to the neutron activation of 14N impurities by 14N(n,p)14C reaction, or in the oxide layer (ZrO2) formed at the metal surface by the neutron activation of 17O from UO2 or (U-Pu)O2 fuel and water from the primary circuit in the reactor by 17O(n,α)14C reaction. Various irradiated and unirradiated Zircaloys have been studied. The total 14C inventory has been determined both experimentally and by calculations. The results seem to be in good agreement. Leaching experiments were conducted in alkaline media for several time durations. 14C was mainly released as carboxylic acids. Further, corrosion measurements were performed by using both hydrogen measurements and electrochemical measurements. The corrosion rate (CR) ranges from a few nm/yr to 100 nm/yr depending on the surface conditions and the method used for measurement. From a safety assessment point of view, the instant release fraction (IRF) was determined on irradiated Zircaloy-2. The results showed that the 14C inventory in the oxide was significantly below the 20% commonly used in safety case assessments

Semi-Markov Graph Dynamics

In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.Comment: 25 pages, 4 figures, submitted to PLoS-ON

Overview OF 14 C release from irradiated zircaloys in geological disposal conditions

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