A stochastic theory for temporal fluctuations in self-organized critical systems

A stochastic theory for the toppling activity in sandpile models is developed, based on a simple mean-field assumption about the toppling process. The theory describes the process as an anti-persistent Gaussian walk, where the diffusion coefficient is proportional to the activity. It is formulated as a generalization of the It\^{o} stochastic differential equation with an anti-persistent fractional Gaussian noise source. An essential element of the theory is re-scaling to obtain a proper thermodynamic limit, and it captures all temporal features of the toppling process obtained by numerical simulation of the Bak-Tang-Wiesenfeld sandpile in this limit.Comment: 9 pages, 4 figure

Modeling temporal fluctuations in avalanching systems

We demonstrate how to model the toppling activity in avalanching systems by stochastic differential equations (SDEs). The theory is developed as a generalization of the classical mean field approach to sandpile dynamics by formulating it as a generalization of Itoh's SDE. This equation contains a fractional Gaussian noise term representing the branching of an avalanche into small active clusters, and a drift term reflecting the tendency for small avalanches to grow and large avalanches to be constricted by the finite system size. If one defines avalanching to take place when the toppling activity exceeds a certain threshold the stochastic model allows us to compute the avalanche exponents in the continum limit as functions of the Hurst exponent of the noise. The results are found to agree well with numerical simulations in the Bak-Tang-Wiesenfeld and Zhang sandpile models. The stochastic model also provides a method for computing the probability density functions of the fluctuations in the toppling activity itself. We show that the sandpiles do not belong to the class of phenomena giving rise to universal non-Gaussian probability density functions for the global activity. Moreover, we demonstrate essential differences between the fluctuations of total kinetic energy in a two-dimensional turbulence simulation and the toppling activity in sandpiles.Comment: 14 pages, 11 figure

Is there long-range memory in solar activity on time scales shorter than the sunspot period?

The sunspot number (SSN), the total solar irradiance (TSI), a TSI reconstruction, and the solar flare index (SFI), are analyzed for long-range persistence (LRP). Standard Hurst analysis yields $H \approx 0.9$, which suggests strong LRP. However, solar activity time series are non-stationary due to the almost periodic 11 year smooth component, and the analysis does not give the correct $H$ for the stochastic component. Better estimates are obtained by detrended fluctuations analysis (DFA), but estimates are biased and errors are large due to the short time records. These time series can be modeled as a stochastic process of the form $x(t)=y(t)+\sigma \sqrt{y(t)}\, w_H(t)$, where $y(t)$ is the smooth component, and $w_H(t)$ is a stationary fractional noise with Hurst exponent $H$. From ensembles of numerical solutions to the stochastic model, and application of Bayes' theorem, we can obtain bias and error bars on $H$ and also a test of the hypothesis that a process is uncorrelated ($H=1/2$). The conclusions from the present data sets are that SSN, TSI and TSI reconstruction almost certainly are long-range persistent, but with most probable value $H\approx 0.7$. The SFI process, however, is either very weakly persistent ($H<0.6$) or completely uncorrelated. Some differences between stochastic properties of the TSI and its reconstruction indicate some error in the reconstruction scheme.Comment: 23 pages, 12 figure

Multifractal modeling of short-term interest rates

We propose a multifractal model for short-term interest rates. The model is a version of the Markov-Switching Multifractal (MSM), which incorporates the well-known level effect observed in interest rates. Unlike previously suggested models, the level-MSM model captures the power-law scaling of the structure functions and the slowly decaying dependency in the absolute value of returns. We apply the model to the Norwegian Interbank Offered Rate with three months maturity (NIBORM3) and the U.S. Treasury Bill with three months maturity (TBM3). The performance of the model is compared to level-GARCH models, level-EGARCH models and jump-diffusions. For the TBM3 data the multifractal out-performs all the alternatives considered.Comment: 16 pages, 3 figures, 7 table

Scale-free vortex cascade emerging from random forcing in a strongly coupled system

The notions of self-organised criticality (SOC) and turbulence are traditionally considered to be applicable to disjoint classes of phenomena. Nevertheless, scale-free burst statistics is a feature shared by turbulent as well as self-organised critical dynamics. It has also been suggested that another shared feature is universal non-gaussian probability density functions (PDFs) of global fluctuations. Here, we elucidate the unifying aspects through analysis of data from a laboratory dusty plasma monolayer. We compare analysis of experimental data with simulations of a two-dimensional (2D) many-body system, of 2D fluid turbulence, and a 2D SOC model, all subject to random forcing at small scales. The scale-free vortex cascade is apparent from structure functions as well as spatio-temporal avalanche analysis, the latter giving similar results for the experimental and all model systems studied. The experiment exhibits global fluctuation statistics consistent with a non-gaussian universal PDF, but the model systems yield this result only in a restricted range of forcing conditions

Approximated maximum likelihood estimation in multifractal random walks

We present an approximated maximum likelihood method for the multifractal random walk processes of [E. Bacry et al., Phys. Rev. E 64, 026103 (2001)]. The likelihood is computed using a Laplace approximation and a truncation in the dependency structure for the latent volatility. The procedure is implemented as a package in the R computer language. Its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments. The method is applied to estimate parameters for various financial stock indices.Comment: 8 pages, 3 figures, 2 table

Variation in carbon footprint of milk due to management differences between Swedish dairy farms

To identify mitigation options to reduce greenhouse gas (GHG) emissions from milk production (i.e. the carbon footprint (CF) of milk), this study examined the variation in GHG emissions among dairy farms using data from previous CF studies on Swedish milk. Variation between farms in these production data, which were found to have a strong influence on milk CF were obtained from existing databases of e.g. 1051 dairy farms in Sweden in 2005. Monte Carlo analysis was used to analyse the impact of variations in seven important parameters on milk CF concerning milk yield (energy corrected milk (ECM) produced and delivered), feed dry matter intake (DMI), enteric methane emissions, N content in feed DMI, N-fertiliser rate and diesel used on farm. The largest between farm variation among the analysed production data were N-fertiliser rate (kg/ha) and diesel used (l/ha) on farm (coefficient of variation (CV) 31-38%). For the parameters concerning milk yield and feed DMI the CV was approx. 11 and 8%, respectively. The smallest variation in production data was found for N content in feed DMI. According to the Monte Carlo analysis, these variations in production data led to a variation in milk CF of between 0.94 and 1.33 kg CO2 equivalents (CO2e) per kg ECM, with an average value of 1.13 kg/CO2e kg ECM. We consider that this variation of ±17% that was found based on the used farm data would be even greater if all Swedish dairy farms were included, as the sample of farms in this study was not totally unbiased. The variation identified in milk CF indicates that a potential exists to reduce GHG emissions from milk production on both national and farm level through changes in management. As milk yield and feed DMI are two of the most influential parameters for milk CF, feed conversion efficiency (i.e. units ECM produced per unit DMI) can be used as a rough key performance indicator for predicting CF reductions. However, it must be borne in mind that feeds have different CF due to where and how they are produced

Critical fluctuations and anomalous transport in soft Yukawa-Langevin systems

Simulation of a Langevin-dynamics model demonstrates emergence of critical fluctuations and anomalous grain transport which have been observed in experiments on "soft" quasi-two-dimensional dusty plasma clusters. It has been suggested that these anomalies derive from particular non-equilibrium physics, but our model does not contain such physics: the grains are confined by an external potential, interact via static Yukawa forces, and are subject to stochastic heating and dissipation from neutrals. One remarkable feature is emergence of leptokurtic probability distributions of grain displacements $\xi(\tau)$ on time-scales $\tau<\tau_{\Delta}$, where $\tau_{\Delta}$ is the time at which the standard deviation $\sigma(\tau)\equiv ^{1/2}$ approaches the mean inter-grain distance $\Delta$. Others are development of humps in the distributions on multiples of $\Delta$, anomalous Hurst exponents, and transitions from leptokurtic towards Gaussian displacement distributions on time scales $\tau>\tau_{\Delta}$. The latter is a signature of intermittency, here interpreted as a transition from bursty transport associated with hopping on intermediate time scales to vortical flows on longer time scales.Comment: 12 pages, 9 figure