Project 6

Self-organized criticality in the sandpile cellular automaton

Introduction

Sandpile cellular automata were introduced almost twenty years ago as a paradigmatic example of self-organized criticality (SOC) [1,2], the tendency of slowly driven dissipative systems to display a scale free avalanche response. Such ideas have had an enormous impact in different fields, ranging from magnetic systems, superconductors and mechanics, to geophysics and plasma physics, including in particular the magnetosphere.

The influence also extends beyond physics, to for example biology, human (heart) physiology and cognitive processes or neuroscience. The reason for this success lies in the wide variety of non-equilibrium systems displaying an avalanche response to an external driving. One of the primary aims of SOC was, originally [1], to explain the wide occurrence of $1/f^{\alpha }$ ($\alpha <2$) noise in natural phenomena, through a direct relation between avalanche scaling and spectral properties [3] (this idea was soon refuted by several groups who published works independently claiming that sandpile models should lead instead to a Lorentzian spectrum that is decaying as $1/f^{2}$ at large frequencies; however, their theoretical arguments were supported by numerical simulations on relatively small system sizes, see historical account in Ref. [3]).

The criticality of this state, obtained without fine tuning of parameters (such as the tuning of temperature in the case of conventional thermal critical phenomena, as explored in Project 4), is manifested as the absence of characteristic spatial and temporal scales of these avalanches. In the less fancy language: large avalanches occur rather often (there is no exponential decay of avalanche sizes, which would result in a characteristic avalanche size), and there is a variety of power laws without cutoffs in various properties of the system.

The paradigm model for this type of behavior is the celebrated sandpile cellular automaton also know as the Bak-Tang-Wiesenfeld (BTW) model. The local rules for the so-called Bak-Tang-Wiesenfeld (BTW) cellular automaton update the integer field $z(i,j)$ defined on a square lattice synchronously according to:

$\mathrm {If} \ z(i,j)>3\ \mathrm {then}$

$z(i,j)\rightarrow z(i,j)-4$
$z(i\pm 1,j)\rightarrow z(i\pm 1,j)+1$
$z(i,j\pm 1)\rightarrow z(i,j\pm 1)+1$

Thus, if there are more than 3 "grains of sand" on a particular lattice site $(i,j)$, then that site relaxes sending one grain of sand to each of four neighbors. If the unstable site happens to be at the boundary, the grains of sand simply leave the system (imagine that they fall off the edge of the table and we are not concerned with them any longer).

Part I for both PHYS460 and PHYS660 students

To build the BTW "sandpile" that self-organizes into a critical state, start from a $128\times 128$ lattice far from equilibrium where each site has a random value $z(i,j)>3$, and update the whole lattice according to the above rules until the sandpile self-organizes into a statistically stationary (time-independent) critical state with average $\langle z\rangle _{c}<3$.

(a) Find explicitly the average height $\langle z\rangle _{c}={\frac {1}{128^{2}}}\sum _{i,j=1}^{128}z(i,j)$ in the SOC state.

(b) Add a single grain to this configuration at a randomly chosen site. This will trigger an avalanche of any size (up to the system size), which is why these state is critical. Record the size of the avalanche (number of sites that participate in the toppling rules) and time it takes for avalanche to end. If you collect many avalanches you should find power laws for the distribution function of avalanche sizes $s$ and times $t$ it take for the avalanche to cease. Plot these distributions on a log-log scale and from the slope of the line extract the critical exponents $P(s)\sim s^{-\sigma }$ and $P(t)\sim t^{-\tau }$.

Although you can check all sites for stability, you can speed up the update by keeping two arrays which contain coordinates of those sites that need to be checked for stability (each time a site topples, add its neighbors' positions to these arrays; then remove the last site from the arrays and check its stability; continue this process of removing sites and checking their stability, and adding neighbors to the array until there are no more sites to check).

(c) Select few interesting avalanches and show graphically the shape of the avalanche (i.e., the sites which participate in the avalanche). Mark the sites which have changed their $z(i,j)$ as well as those which after the avalanche is over retain the same value $z(i,j)$.

Part II for PHYS660 students only

Consider the time series $V(t)$, which records the number of "topplings" (local relaxation events) taking place in the sandpile during each parallel update of the whole lattice, one such update defining the unit of time. As in Part I, an avalanche is defined here as a connected sequence of non-zero values of $V(t)$. The slow driving condition implies that grains are added only when all the sites are below the threshold.

"Measure" the power spectra $P(f)$ generated by the dynamics of the self-organized critical state of BTW cellular automaton by computing the absolute square of the fast Fourier transform (FFT) of the signal produced by the number of toppling events $V(t)$ as a function of time. To check that $P(f)$ reflects only the correlations within avalanches, compute also the power spectrum $P_{s}(f)$ of the avalanche size $s$ time series, which is expected to exhibit white noise character (see the inset of Fig. 1 in Ref. [3]).

By fitting the slope of high-frequency portion of $P(f)\ \mathrm {vs.} \ f$ graph on the log-log scale, extract the critical exponent $\alpha$ defined by the scaling of scaling of the power spectrum, $P(f)\sim f^{\alpha }$. Compare your result with Fig. 3 in Ref. [3], both qualitatively and quantitatively (they obtained $\alpha =1.59\pm 0.05$ for sufficiently large lattices).

References

• [1] P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A 38, 364 (1988). [PDF]

• [2] S. K. Grumbacher, K. M. McEwen, D. A. Halverson, D. T. Jacobs, and J. Lindner, Self‐organized criticality: An experiment with sandpiles, Am. J. Phys. 61, 329 (1993). [PDF]

• [3] L. Laurson, M. J. Alava, and S. Zapperi, Power spectra of self-organized critical sandpiles, J. Stat. Mech. L11001, (2005). [PDF]