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We have studied large deviations of the height of the pile from its mean value in the Oslo ricepile model. We sampled these very rare events with probabilities of order $10^{-100}$ by Monte Carlo simulations using importance sampling. These simulations check our qualitative arguement [Phys. Rev. E, {\bf 73}, 021303, 2006] that in steady state of th...
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... Later, using the ricepile model proposed in Christensen et al. [18], it was argued [68] that the power-law tail of the distribution P(T, L) should be dominated by the grains, which are deeply buried in the pile and take a very long time to come out due to the rare height fluctuations. Indeed, the cut-off time T max to the trapping-time distribution is related to the hight fluctuation of the pile and was estimated to be exponentially large T max ∼ exp(κL 3 ) [69], where L is the system size. Moreover, the mean residence time T was exactly shown to be equal to the average mass of the pile, i.e., T ∼ L 2 . ...
Bak, Tang, and Wiesenfeld (BTW) proposed the theory of self-organized criticality (SOC), and sandpile models, to connect “1/f” noise, observed in systems in a diverse natural setting, to the fractal spatial structure. We review some of the existing works on the problem of characterizing time-dependent properties of sandpiles and try to explore if the BTW's original ambition has really been fulfilled. We discuss the exact hydrodynamic structure in a class of conserved stochastic sandpiles, undergoing a non-equilibrium absorbing phase transition. We illustrate how the hydrodynamic framework can be used to capture long-ranged spatio-temporal correlations in terms of large-scale transport and relaxation properties of the systems. We particularly emphasize certain interesting aspects of sandpiles—the transport instabilities, which emerge through the threshold-activated nature of the dynamics in the systems. We also point out some open issues at the end.
Giving a detailed overview of the subject, this book takes in the results and methods that have arisen since the term 'self-organised criticality' was coined twenty years ago. Providing an overview of numerical and analytical methods, from their theoretical foundation to the actual application and implementation, the book is an easy access point to important results and sophisticated methods. Starting with the famous Bak-Tang-Wiesenfeld sandpile, ten key models are carefully defined, together with their results and applications. Comprehensive tables of numerical results are collected in one volume for the first time, making the information readily accessible to readers. Written for graduate students and practising researchers in a range of disciplines, from physics and mathematics to biology, sociology, finance, medicine and engineering, the book gives a practical, hands-on approach throughout. Methods and results are applied in ways that will relate to the reader's own research.
Glossary
Definition of the Subject
Introduction
Simulation Techniques
Scaling
Histogram Data Representation
Future Directions
Bibliography
The following chapter provides an overview of the techniques used to
understand Self-Organised Criticality (SOC) by performing computer simulations.
Those are of particular significance in SOC, given its very paradigm, the BTW
(Bak-Tang-Wiesenfeld) sandpile, was introduced on the basis of a process that
is conveniently implemented as a computer program. The chapter is divided into
three sections: In the first section a number of key concepts are introduced,
followed by four brief presentations of SOC models which are most commonly
investigated or which have played an important part in the development of the
field as a whole. The second section is concerned with the basics of scaling
with particular emphasis of its role in numerical models of SOC, introducing a
number of basic tools for data analysis such as binning, moment analysis and
error estimation. The third section is devoted to numerical methods and
algorithms as applied to SOC models, addressing typical computational questions
with the particular application of SOC in mind. The present chapter is rather
technical, but hands-on at the same time, providing practical advice and even
code snippets (in C) wherever possible.
