Waves in the sandpile model on fractal lattices

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Abstract

The scaling properties of waves of topplings of the sandpile model on a fractal lattice are investigated. The exponent describing the asymptotics of the distribution of last waves in an avalanche is found. Predictions for scaling exponents in the forward and backward conditional probabilities for two consecutive waves are given. All predictions were examined by simulations on the Sierpinski gasket and were found to be in a reasonable agreement with the numerical data.

Introduction

Sandpile models form the paradigmatic examples of the concept of self-organised criticality (SOC) [1], [2]. This is the phenomenon in which a slowly driven systems with many degrees of freedom evolves spontaneously into a critical state, characterised by long range-correlations in space and time.

In the past decade much progress has been made in the theoretical understanding of sandpile models. This is especially true for the Bak–Tang–Wiesenfeld (BTW) model [1], [2], where, following the original work of Dhar [3], a mathematical formalism was developed [4], [5], [6], [7], [8] that allows an exact calculation of several properties of the model such as height probabilities [4], [5], the upper critical dimension [8] and so on.

Despite all this work, it has however not been possible yet to give an exact characterisation of the scaling properties of the avalanches in the BTW-model. In recent years it has become increasingly clear that, especially in two dimensions, avalanches are to be described by a multifractal set of scaling exponents [9], [10]. This spectrum of exponents has been calculated with high numerical precision, but at this moment there is no clue how it can be determined by an analytical approach.

Avalanches can be decomposed into simpler objects called waves [7]. The probability distribution of waves seems to obey simple scaling [11] and the exponent describing that scaling is known exactly, both for the general wave [7], [11] and for the last wave of each avalanche [6]. Waves have been mainly studied on Euclidean lattices.

In the present paper we study the properties of waves on fractal lattices and in particular on the Sierpinski gasket, continuing previous work [12], [13]. We correct an earlier prediction for the exponent of the last wave. Our main new results concern the statistical properties of pairs of two consecutive waves. Moreover, we use for the first time the moment analysis (introduced in the study of sandpile models in Ref. [9]) to determine some of the exponents describing wave scaling. This technique turns out to be extremely useful for fractals where log-periodic oscillations make an accurate determination of critical exponents extremely difficult.

This paper is organised as follows. In Section 2, we introduce the sandpile model and focus especially on the behaviour on a fractal. In Section 3, we discuss the behaviour of the conditional probabilities of wave sizes using heuristic arguments. Such arguments lead to a number of scaling relations among various critical exponents. In Section 4, we present our numerical results on the Sierpinski gasket. Finally, in Section 5 we present some conclusions.

Section snippets

The sandpile model on a fractal

The BTW sandpile model can be defined on any graph, but for definiteness we will introduce it in the context of the Sierpinski gasket (see Fig. 1). Each vertex (apart from the three boundary sites) of this graph has four nearest neighbours. To each such vertex i we associate a height variable zi which can take on any positive integer number. We also introduce a critical height zc, which we will take equal to four for all vertices. The number of sites in the lattice, N, is trivially related to

Distribution of consecutive waves

We now turn to a study of pairs of two consecutive waves, a topic which has received considerable attention recently [16], [17], [18].

The conditional probability that the (k+1)th wave has size sk+1 given that the previous wave had size sk, P(sk+1|sk), is the first quantity to study when one is interested in correlation effects in waves. These correlations make waves and avalanches different. Paczuski and Boettcher [16] proposed, on the basis of simulations on the square lattice, that P(sk+1|sk)

Numerical results

We have performed extensive simulations of the sandpile model on a Sierpinski gasket and have calculated both the forward and backward conditional probabilities.

In Fig. 3a, we show our data for the forward conditional probability for a Sierpinski gasket with n=9(N=29526). All our data were obtained by studying at least 1000×N avalanches. The figure shows the best fit of our data to the scaling from (11) proposed by Paczuski and Boettcher. Unfortunately, it is not possible to obtain very

Conclusions

In this paper we investigated the properties of waves in the sandpile model on a fractal lattice. We gave predictions for the exponent describing the last wave in an avalanche and for the scaling exponents occuring in forward and backward conditional probabilities for consecutive waves. These predictions were tested by extensive simulations and were found to be in good agreement with the numerics. We can therefore conclude that the wave statistics at the levels of individual waves and pairs of

Acknowledgements

One of us (VBP) thanks the Limburgs Universitair Centrum for hospitality. FD and CV would like to thank the IUAP for financial support.

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