Mean field theory for ordinary and hot sandpiles

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Abstract

A mean field theory is discussed for a sandpile model, a cellular automaton prototype of systems showing self-organized criticality. The previous formulation of the mean field does not take into account the dissipation effects that take place on boundaries. This gives rise to some inconsistencies that are eliminated by carefully considering the boundaries effects, as it is shown in this paper. We present here a revised version of the MF equations. The main result is that criticality arises in the thermodynamic limit for sandpile systems, confirming numerical observations on the behavior of the order parameter.

The mean field approach is also generalized by applying it to the more general case of sandpiles in thermal equilibrium where a temperature-like parameter T is introduced. In this case we show that criticality is not destroyed at T> 0.

Introduction

Since Bak et al. [1] introduced the concept of self-organized criticality (SOC), to explain the widespread occurrence in Nature of long-range space and time correlations, several models have been proposed to mimic very different physical situations, ranging from fractures 2, 3 to river-networks evolution 4, 5 and from ecology of species 6, 7, 8 to sand aggregation 1, 9 and memory effects [10]. Due to its simplicity, the cellular automaton introduced to describe the behavior of real sandpiles can be considered as a prototype for this class of phenomena.

In this work, we want to investigate one of the few theoretical tools introduced to explain the properties of self-organization 11, 12, 13, 14. We discuss here the mean field (MF) approach to sandpile systems introduced by Tang and Bak 11, 12.

In their formulation of the MF 11, 12, the main prediction is that criticality is ensured by the divergence of susceptibility (defined above). This behavior is not confirmed by numerical simulations [15] where the order parameter is proportional to the external field (e.g. the susceptibility does not change at the critical point). Furthermore, we show that the steady state described by the MF equations cannot be reached with the sandpile ordinary dynamics.

Due to these reasons, we present here a revised version of the MF where dynamics and then dissipation on boundaries is taken into account. The new ingredient of dissipation introduces new terms in the MF equations. It also changes the expression of the susceptibility in such a way as to agree with the numerical data, where the order parameter is proportional to the external field. However, we are able to show (due to the behavior of the prefactor of the susceptibility) that the long-range distribution of Eq. (1)still holds, as expected.

The main result is that dissipation on boundaries determines the behavior of a system and it has to be taken into account for a correct derivation of the MF equations. Furthermore, we extend the known results also to the case of systems in presence of an external field. The results in this case show that the criticality is not destroyed at T> 0.

The MF approach is based on the analogy with traditional critical phenomena. This analogy is merely formal, because in statistical mechanics the critical exponents describe the behavior of equilibrium properties, while in SOC systems they describe non-equilibrium dynamical properties. Since the most remarkable features of SOC systems is the capability to drive themselves toward a critical state, the definition of a state “out of criticality” is rather different from the usual one. In fact, for these systems there cannot exist a stationary non-critical state.

In sandpile models, one can imagine to build a sandpile adding particles randomly and very slowly. At the beginning the pile is flat and the sand addition causes small local rearrangement, due to the pile toppling on pile neighbors. After a certain period of time those rearrangements (avalanches) become larger and larger and eventually can bring the sand out of the system through toppling on boundaries. In this way the system reaches a statistically stationary state, where the avalanches (defined as the number of consecutive toppling between two external sand additions) take place at any length scale. The average height of sand remains constant, due to the balance between sand added and sand lost from boundaries at the stationary state. The probability distribution P(s) for avalanche of size s behaves asP(s)≃s−τ.

The microscopic dynamics is ruled by the height instability. This instability causes the toppling process. In fact, if the height z of a column exceeds a certain threshold zc at time t, then at time t+1 the column falls to the ground giving its sand to nearest-neighbor (n.n.) sites.

One can try to capture this behavior by setting a suitable cellular automaton. The rules for a simple cubic lattice in d-dimension, update the height on site i whenever zi(t+1)> zc=2d, according tozi(t+1)=zi(t)−zc,zk(t+1)=zk(t)+1,where k’s represent n.n. of i.

The behavior of such a model, known as Abelian sandpile model [16], can be easily checked with computer simulations. The values found for the exponent τ in Eq. (1)are τ=1.21±0.03 in dimension d=2 and τ=1.40±0.03 in d=3 [17].

The paper is organized as follows: First, the original formulation by Tang and Bak 11, 12 is presented. Second, we show how that the correct stationary state cannot be reached from the sandpile dynamical evolution without dissipation. In the same section, we propose the new formulation for the MF equations and we show how the numerical results are in fair agreement with the theoretical predictions. Third, we extend the MF formulation to a more general case of thermal sandpiles, while in the last section a summary of the results is presented.

Section snippets

The original mean field approach by Tang and Bak

The analogy with ordinary critical phenomena is related to the behavior of the system with respect to the average height θ(t)=1/Ldi=1Ld zi(t). In fact, at the stationary state (where θ(t)=θ), if θ is very small (below a “critical value” θc) no activity occurs in the system without external flux of sand. If the system is kept at θ sufficiently large (above θc) also without external additions one observes a “spontaneous” flux of sand from the boundaries. Notwithstanding the difficulties related

Problems of the original formulation

We present here some problems related to the original formulation. In fact, the above solution represents a stationary state of the following master equation:Pi(t+1)=Pi(t)+k=04[Pk(t)Tki(t)−Pi(t)Tik(t)],where Pi(t)’s represent the fraction of sites at height i at time t. and Tik’s are the transition rates between different states i and k, previously introduced and they are displayed in the Fig. 1.

However, it turns out from a simple numerical analysis that stationary solutions Pi=Pi(∞) differ

Hot sandpiles

We have tested the robustness of MF also for the more general case of hot Abelian sandpiles [9]. In fact, all the previous results can be extended also in the case of sandpile in contact with a reservoir at “real” temperature T. Our purpose is to demonstrate that this model is closely related to the original one previously introduced. We mean that the introduction of temperature (as confirmed by numerical simulations) has the only role to tune the average value of sand height in the system.

To

Results

In conclusion, we have introduced in a known MF approach 11, 12 for self-organized systems the condition of dynamical stationary equilibrium that characterizes such systems. Under this condition the role of the dissipation from boundaries arises quite naturally, and assumes a primary role in the rules definition of the model. Furthermore, the presence of the boundaries affects the behavior of the order parameter explaining some numerical results. We find that the critical exponent δ is

Acknowledgements

It is a pleasure to acknowledge A. Maritan, and A. Vespignani for helpful discussions.

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