A probabilistic approach to Zhang's sandpile model

The current literature on sandpile models mainly deals with the abelian sandpile model (ASM) and its variants. We treat a less known - but equally interesting - model, namely Zhang's sandpile. This model differs in two aspects from the ASM. First, additions are not discrete, but random amounts with a uniform distribution on an interval $[a,b]$. Second, if a site topples - which happens if the amount at that site is larger than a threshold value $E_c$ (which is a model parameter), then it divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the infinite volume limit, this model tends to behave like the ASM in the sense that the stationary measure for the system in large volumes tends to be peaked narrowly around a finite set. This belief is supported by simulations, but so far not by analytical investigations. We study the stationary distribution of this model in one dimension, for several values of $a$ and $b$. When there is only one site, exact computations are possible. Our main result concerns the limit as the number of sites tends to infinity, in the one-dimensional case. We find that the stationary distribution, in the case $a \geq E_c/2$, indeed tends to that of the ASM (up to a scaling factor), in agreement with Zhang's conjecture. For the case $a=0$, $b=1$ we provide strong evidence that the stationary expectation tends to $\sqrt{1/2}$.Comment: 47 pages, 3 figure

Simulation study of the inhomogeneous Olami-Feder-Christensen model of earthquakes

Statistical properties of the inhomogeneous version of the Olami-Feder-Christensen (OFC) model of earthquakes is investigated by numerical simulations. The spatial inhomogeneity is assumed to be dynamical. Critical features found in the original homogeneous OFC model, e.g., the Gutenberg-Richter law and the Omori law are often weakened or suppressed in the presence of inhomogeneity, whereas the characteristic features found in the original homogeneous OFC model, e.g., the near-periodic recurrence of large events and the asperity-like phenomena persist.Comment: Shortened from the first version. To appear in European Physical Journal

Cell scale self-organisation in the OFC model for earthquake dynamics

64.60.av Cracks, sandpiles, avalanches, and earthquakes, 64.60.De Statistical mechanics of model systems, 64.70.qj Dynamics and criticality, 64.60.Cn Order-disorder transformations,

Water Droplet Avalanches

We analyze the statistics of water droplet avalanches in a continuously driven system. Distributions are obtained for avalanche size, lifetime, and time between successive avalanches, along with power spectra and return maps. For low flow rates and different water viscosities, we observe a power-law scaling in the size and lifetime distributions of water droplet avalanches, indicating that a state with no characteristic time and length scales was reached. Higher flow rates resulted in an exponential behavior with characteristic scales.Comment: 12-pages, LaTe

Efficacy of perindopril in reduction of cardiovascular events among patients with stable coronary artery disease: randomised, double-blind, placebo-controlled, multicentre trial (the EUROPA study)

BACKGROUND: Treatment with angiotensin-converting-enzyme (ACE) inhibitors reduces the rate of cardiovascular events among patients with left-ventricular dysfunction and those at high risk of such events. We assessed whether the ACE inhibitor perindopril reduced cardiovascular risk in a low-risk population with stable coronary heart disease and no apparent heart failure. METHODS: We recruited patients from October, 1997, to June, 2000. 13655 patients were registered with previous myocardial infarction (64%), angiographic evidence of coronary artery disease (61%), coronary revascularisation (55%), or a positive stress test only (5%). After a run-in period of 4 weeks, in which all patients received perindopril, 12218 patients were randomly assigned perindopril 8 mg once daily (n=6110), or matching placebo (n=6108). The mean follow-up was 4.2 years, and the primary endpoint was cardiovascular death, myocardial infarction, or cardiac arrest. Analysis was by intention to treat. FINDINGS: Mean age of patients was 60 years (SD 9), 85% were male, 92% were taking platelet inhibitors, 62% beta blockers, and 58% lipid-lowering therapy. 603 (10%) placebo and 488 (8%) perindopril patients experienced the primary endpoint, which yields a 20% relative risk reduction (95% CI 9-29, p=0.0003) with perindopril. These benefits were consistent in all predefined subgroups and secondary endpoints. Perindopril was well tolerated. INTERPRETATION: Among patients with stable coronary heart disease without apparent heart failure, perindopril can significantly improve outcome. About 50 patients need to be treated for a period of 4 years to prevent one major cardiovascular event. Treatment with perindopril, on top of other preventive medications, should be considered in all patients with coronary heart disease