Sandpiles

When sand is added on the top of a sand heap (also known as sandpile problem) two phenomena are observed:

• the slope of the pile grows (with little flowing of sand on it) until a critical angle is reached; after that (if sand is still poured on the top) the slope stays almost constant and the sand flows along it;

• at the critical angle the flowing of sand is made of ``avalanches'' of different sizes and durations;

Figure 1.5: Possible configurations of a sandpile

Starting from this qualitative observation, P. Bak et al. [9,10] have introduced a cellular automaton model (see Fig. fig_sandpile) where each site (for example in two dimensions $(i,j)$) of the lattice has associated a slope $z(i,j)$. If the slope exceeds a critical value $z_c$ a rearrangement of the neighboring sites is performed, e.g. $4$ is subtracted to the exceeded value and $1$ is added to its $4$ neighboring sites. The automaton may be executed in two different ways:

1. the field at time 0 has an average slope greater than $z_c$, and the system evolves freely;

2. the field at time 0 is everywhere equal to zero and at every step a randomly chosen site is incremented;

In both cases the systems reaches a stable configuration corresponding to the critical slope: every successive perturbation (i.e. increasing the field on some site) generates an avalanche that involves the rearrangement of a certain number of sites of the lattice. The authors show that the distribution of the extension of the avalanche follows a power law:

$$D(s) \sim s^{-\tau}$$ (1.12)

with $\tau \simeq 0.98$ for three or four logarithmic decades. In another work [206] the same authors define and study a set of critical exponents similar to those used in statistical mechanics of phase transitions.

The novelty of the work of Bak and coworkers is represented by the fact that a model was found that showed a critical behavior (i.e. power law relaxations, correlations at all sizes) without any fine tuning of the external parameter, whereas usual critical phase transitions need a precise tuning of the temperature to the critical temperature $T_c$. This self-organized critical behavior was intriguing as it seemed to be a key concept to understand the ubiquity of power laws in nature (e.g. $1/f$ noise, self-similar structures like fractals, turbulence and so on). The sandpile model is still studied, with all its variants (for a review see for example [101])), but it has been recognized to be not a good paradigm for the self-organized criticality: it was seen in fact [201,216] that the driving rate (i.e. the rate of falling of grains on the top of the pile) acts exactly as a control parameter that has to be fine tuned to zero in order to observe criticality. However many important issues are still open: the interplay between the self-organization into a stationary state and the dynamical developing of correlations, the (numerical) measure of critical exponents, universality classes, upper critical dimensions and so on.

Figure 1.6: The experiment of Nagel and coworkers to measure avalanches in sandpiles

More remarkably, it has been pointed out that the sandpile model has little to do with sand (and granular matter) in general. In 1992 Nagel has published [169] the results of a series of experiments on sand in order to verify the predictions of Bak and coworkers. He has initially shown the difficulty of performing an exactly constant rate in the pouring of grains from above the top of a pile. Therefore he has introduced a different kind of experiment (see Fig. fig_sandpile2), where the sand fills partially a rotating drum: the constant angular velocity of the drum guarantees the constant driving needed to reach the critical slope and the avalanche regime. The statistical analysis of the avalanches has clearly demonstrated that sand does not reproduce the critical behavior expected in self-organized criticality. The sandpile has two critical slopes: $\Theta_r$ is the rest angle (when the slope is less than $\Theta_r$ the pile is stable), $\Theta_m$ is maximum angle (when the slope is larger than $\Theta_m$ avalanches form and reduce the slope to an angle less than $\Theta_r$). If the slope is comprised in between $\Theta_r$ and $\Theta_m$ there is bistability, i.e. the sand can rest or can produce avalanches, based on its previous history. The avalanches have a typical duration and the hysteretic cycle between the two angles has a well defined average frequency. No power laws have been observed.