Nonlinear Volatility of River Flux Fluctuations

We study the spectral properties of the magnitudes of river flux increments, the volatility. The volatility series exhibits (i) strong seasonal periodicity and (ii) strongly power-law correlations for time scales less than one year. We test the nonlinear properties of the river flux increment series by randomizing its Fourier phases and find that the surrogate volatility series (i) has almost no seasonal periodicity and (ii) is weakly correlated for time scales less than one year. We quantify the degree of nonlinearity by measuring (i) the amplitude of the power spectrum at the seasonal peak and (ii) the correlation power-law exponent of the volatility series.Comment: 5 revtex pages, 6 page

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

Distribution of epicenters in the Olami-Feder-Christensen model

We show that the well established Olami-Feder-Christensen (OFC) model for the dynamics of earthquakes is able to reproduce a new striking property of real earthquake data. Recently, it has been pointed out by Abe and Suzuki that the epicenters of earthquakes could be connected in order to generate a graph, with properties of a scale-free network of the Barabasi-Albert type. However, only the non conservative version of the Olami-Feder-Christensen model is able to reproduce this behavior. The conservative version, instead, behaves like a random graph. Besides indicating the robustness of the model to describe earthquake dynamics, those findings reinforce that conservative and non conservative versions of the OFC model are qualitatively different. Also, we propose a completely new dynamical mechanism that, even without an explicit rule of preferential attachment, generates a free scale network. The preferential attachment is in this case a ``by-product'' of the long term correlations associated with the self-organized critical state. The detailed study of the properties of this network can reveal new aspects of the dynamics of the OFC model, contributing to the understanding of self-organized criticality in non conserving models.Comment: 7 pages, 7 figure

Testing Logselfsimilarity of Soil Particle Size Distribution: Simulation with Minimum Inputs

Particle size distribution (PSD) greatly influences other soil physical properties. A detailed textural analysis is time-consuming and expensive. Soil texture is commonly reported in terms of mass percentages of a small number of size fractions (typically, clay, silt and sand). A method to simulate the PSD from such a poor description or even from the poorest description, consisting in the mass percentages of only two soil size fractions, would be extremly useful for prediction purposes. The goal of this paper is to simulate soil PSDs from the minimum number of inputs, i.e., two and three textural fraction contents, by using a logselfsimilar model and an iterated function system constructed with these data. High quality data on 171 soils are used. Additionally, the characterization of soil texture by entropy-based parameters provided by the model is tested. Results indicate that the logselfsimilar model may be a useful tool to simulate PSD for the construction of pedotransfer functions related to other soil properties when textural information is limited to moderate textural data

Precursors of catastrophe in the BTW, Manna and random fiber bundle models of failure

We have studied precursors of the global failure in some self-organised critical models of sand-pile (in BTW and Manna models) and in the random fiber bundle model (RFB). In both BTW and Manna model, as one adds a small but fixed number of sand grains (heights) to any central site of the stable pile, the local dynamics starts and continues for an average relaxation time (\tau) and an average number of topplings (\Delta) spread over a radial distance (\xi). We find that these quantities all depend on the average height (h_{av}) of the pile and they all diverge as (h_{av}) approaches the critical height (h_{c}) from below: (\Delta) (\sim (h_{c}-h_{av}))(^{-\delta}), (\tau \sim (h_{c}-h_{av})^{-\gamma}) and (\xi) (\sim) ((h_{c}-h_{av})^{-\nu}). Numerically we find (\delta \simeq 2.0), (\gamma \simeq 1.2) and (\nu \simeq 1.0) for both BTW and Manna model in two dimensions. In the strained RFB model we find that the breakdown susceptibility (\chi) (giving the differential increment of the number of broken fibers due to increase in external load) and the relaxation time (\tau), both diverge as the applied load or stress (\sigma) approaches the network failure threshold (\sigma_{c}) from below: (\chi) (\sim) ((\sigma_{c}) (-)(\sigma)^{-1/2}) and (\tau) (\sim) ((\sigma_{c}) (-)(\sigma)^{-1/2}). These self-organised dynamical models of failure therefore show some definite precursors with robust power laws long before the failure point. Such well-characterised precursors should help predicting the global failure point of the systems in advance.Comment: 13 pages, 9 figures (eps

Roughness of Sandpile Surfaces

We study the surface roughness of prototype models displaying self-organized criticality (SOC) and their noncritical variants in one dimension. For SOC systems, we find that two seemingly equivalent definitions of surface roughness yields different asymptotic scaling exponents. Using approximate analytical arguments and extensive numerical studies we conclude that this ambiguity is due to the special scaling properties of the nonlinear steady state surface. We also find that there is no such ambiguity for non-SOC models, although there may be intermediate crossovers to different roughness values. Such crossovers need to be distinguished from the true asymptotic behaviour, as in the case of a noncritical disordered sandpile model studied in [10].Comment: 5 pages, 4 figures. Accepted for publication in Phys. Rev.

Scaling in a Nonconservative Earthquake Model of Self-Organised Criticality

We numerically investigate the Olami-Feder-Christensen model for earthquakes in order to characterise its scaling behaviour. We show that ordinary finite size scaling in the model is violated due to global, system wide events. Nevertheless we find that subsystems of linear dimension small compared to the overall system size obey finite (subsystem) size scaling, with universal critical coefficients, for the earthquake events localised within the subsystem. We provide evidence, moreover, that large earthquakes responsible for breaking finite size scaling are initiated predominantly near the boundary.Comment: 6 pages, 6 figures, to be published in Phys. Rev. E; references sorted correctl

Topological self-similarity on the random binary-tree model

Asymptotic analysis on some statistical properties of the random binary-tree model is developed. We quantify a hierarchical structure of branching patterns based on the Horton-Strahler analysis. We introduce a transformation of a binary tree, and derive a recursive equation about branch orders. As an application of the analysis, topological self-similarity and its generalization is proved in an asymptotic sense. Also, some important examples are presented

Self-organized model of cascade spreading

We study simultaneous price drops of real stocks and show that for high drop thresholds they follow a power-law distribution. To reproduce these collective downturns, we propose a minimal self-organized model of cascade spreading based on a probabilistic response of the system elements to stress conditions. This model is solvable using the theory of branching processes and the mean-field approximation. For a wide range of parameters, the system is in a critical state and displays a power-law cascade-size distribution similar to the empirically observed one. We further generalize the model to reproduce volatility clustering and other observed properties of real stocks.Comment: 8 pages, 6 figure