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

Leray and LANS-α\alpha modeling of turbulent mixing

Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations provides a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modelled system of equations. We will consider the large eddy interpretation of two such mathematical regularisation principles, i.e., Leray and LANS$-\alpha$ regularisation. The Leray principle introduces a {\bfi smoothed transport velocity} as part of the regularised convective nonlinearity. The LANS$-\alpha$ principle extends the Leray formulation in a natural way in which a {\bfi filtered Kelvin circulation theorem}, incorporating the smoothed transport velocity, is explicitly satisfied. These regularisation principles give rise to implied subgrid closures which will be applied in large eddy simulation of turbulent mixing. Comparison with filtered direct numerical simulation data, and with predictions obtained from popular dynamic eddy-viscosity modelling, shows that these mathematical regularisation models are considerably more accurate, at a lower computational cost.Comment: 42 pages, 12 figure

The Ehrenfest urn revisited: Playing the game on a realistic fluid model

The Ehrenfest urn process, also known as the dogs and fleas model, is realistically simulated by molecular dynamics of the Lennard-Jones fluid. The key variable is Delta z, i.e. the absolute value of the difference between the number of particles in one half of the simulation box and in the other half. This is a pure-jump stochastic process induced, under coarse graining, by the deterministic time evolution of the atomic coordinates. We discuss the Markov hypothesis by analyzing the statistical properties of the jumps and of the waiting times between jumps. In the limit of a vanishing integration time-step, the distribution of waiting times becomes closer to an exponential and, therefore, the continuous-time jump stochastic process is Markovian. The random variable Delta z behaves as a Markov chain and, in the gas phase, the observed transition probabilities follow the predictions of the Ehrenfest theory.Comment: Accepted by Physical Review E on 4 May 200

Prognostic relevance of symptoms versus objective evidence of coronary artery disease in diabetic patients

Aim Little is known about the prognostic significance of silent versus symptomatic coronary artery disease (CAD) in diabetic patients. We therefore assessed the incidence of scintigraphic evidence of CAD in diabetic patients without known CAD and the impact of symptoms and scintigraphic findings on prognosis. Methods and results A consecutive series of 1737 diabetic patients without known CAD underwent dual-isotope myocardial perfusion SPECT (MPS) and 1430 were followed-up for a median of 2 (1-8.5) years. Critical events were defined as myocardial infarction or cardiac death. Objective evidence of CAD was found in 39% of 826 asymptomatic diabetic patients, in 51% of 151 diabetic patients with shortness of breath (SOB), and in 44% of 760 diabetic patients with angina. During follow-up, 98 critical events occurred. Annual critical event rates were 2.2% in asymptomatic, 3.2% in angina, and 7.7% in diabetic patients with shortness of breath (\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $p{<}0.001$ \end{document} versus other groups). With MPS evidence of CAD, critical event rates increased to 3.4% (asymptomatic), 5.6% (angina), and 13.2% (SOB) (\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $p{\leqslant}0.009$ \end{document} versus no evidence of CAD). Age, hypertension, shortness of breath, scarring and ischaemia were independent predictors of critical events. MPS findings added incremental information to prescan information regarding outcome prediction. Conclusions In asymptomatic diabetic patients, the rate of objective evidence of CAD and annual critical events were similar to those found in diabetic patients with angina. The outcome was three times worse in diabetic patients with shortness of breath. MPS findings were strongly predictive of outcome and proved valuable for risk stratificatio

Smectic ordering in liquid crystal - aerosil dispersions II. Scaling analysis

Liquid crystals offer many unique opportunities to study various phase transitions with continuous symmetry in the presence of quenched random disorder (QRD). The QRD arises from the presence of porous solids in the form of a random gel network. Experimental and theoretical work support the view that for fixed (static) inclusions, quasi-long-range smectic order is destroyed for arbitrarily small volume fractions of the solid. However, the presence of porous solids indicates that finite-size effects could play some role in limiting long-range order. In an earlier work, the nematic - smectic-A transition region of octylcyanobiphenyl (8CB) and silica aerosils was investigated calorimetrically. A detailed x-ray study of this system is presented in the preceding Paper I, which indicates that pseudo-critical scaling behavior is observed. In the present paper, the role of finite-size scaling and two-scale universality aspects of the 8CB+aerosil system are presented and the dependence of the QRD strength on the aerosil density is discussed.Comment: 14 pages, 10 figures, 1 table. Companion paper to "Smectic ordering in liquid crystal - aerosil dispersions I. X-ray scattering" by R.L. Leheny, S. Park, R.J. Birgeneau, J.-L. Gallani, C.W. Garland, and G.S. Iannacchion