October 31, 2007

In their famous 1988 paper, Per Bak *et al.*
first introduced the concept of self-organized criticality to explain
the 1/f noise and scale-invariant structure which occur in many
spatially extended dynamical systems. [1] To illustrate this idea, they
analyzed a cellular automaton model of a sandpile. This study brought
appealing prospect to the study of complexity in nature. While a broad
range of phenomena have received promising explanation from this
self-organized criticality concept, people's curiosity on the original
picture of sandpile is not fulfilled or worn down over twenty years.
Numerous experiments on the avalanches in a sandpile have been done
under different conditions using different kinds of "sand" and with
different focus of measurement.[2-5] These experiments suggest that the
existence of this self-organized criticality is not as universal as
first considered but rather depends on the characteristics of the
granular system. And no general result on characterizing a system with
self-organized criticality is given so far.

In their paper, Bak *et al.* argue that
spatially extended dynamical systems naturally evolve into
self-organized critical states regardless of the initial conditions, and
that the critical states are insensitive to randomness. In their
sandpile model, they start from scratch and randomly add sand, one grain
at a time. While the pile grows, the sand will slide off if the slope
grows beyond a critical value. The sliding will also happen if the pile
is started from a too steep state. The sliding is spreading out until a
system-wide critical state is reached. This critical state is barely
stable that any further perturbation will result in more sliding until
the critical state is reached again.

Consider a one-dimensional sandpile of length N with a
boundary condition that the sand can only leave the system at the N side.
Use an array {z_{n}=h(n)-h(n+1)} to represent the height
difference between neighboring columns of sand. Each grain added at the
nth column is expressed as

When the height difference gets larger than a critical
value z_{c}, one grain of sand on the left will slide to the lower
level, expressed as

Boundary conditions are

The condition for stability is
z_{n}_{n}=z_{c}.

In analog to one-dimensional case, the shape of an N by N sandpile is represented by a matrix z(x,y). Not as straightforward as one-dimensional case, the sand columns are represented by the bonds between the neighbors of (x,y) to avoid next nearest neighborhood interaction. Adding two grains of sand at the upper and left bond of (x,y) follows

Sliding of one grain from upper bond to lower bond and one grain from left bond to right bond follows

The minimally stable state as in one-dimensional sandpile is no longer an attracting state. Because from a two dimensional minimally stable state, any fluctuation will spread out through the entire lattice via the neighbors in a chain reaction. As the system evolves, minimally stable clusters will form and eventually prevent the communication through infinite distances. So the result of a single perturbation can be on any length scale and thus any time scale. Meanwhile it can be shown that with randomly superimposed pulses of a physical quantity, the power-law distribution for lifetime leads to the 1/f power spectrum. Thus to understand the 1/f noise, it is important to study the distribution of lifetime.

Simulations with closed boundary condition, i.e. z(0,y)=z(x,0)=z(N+1,y)=z(x,N+1)=0 or equivalent for three dimension, give quite precise power-law distribution of slide size (the number of sliding induced by one perturbation), except for small slide sizes due to the discreteness effects of the model lattice and for large slide sizes due to the finite-size effect. A power-law behavior is also found in the distribution of lifetimes weighted by the average response s/T .

For an energy transportation system, the sliding in
the sandpile model could be interpreted as energy dissipation *
f(x,t)* at site *x* and time t. Dissipation rate
*F(t)*=**∫***f(x,t) dx*. Total energy dissipated
(cluster size) *s*=**∫***F(t) dt*. The power spectrum
is defined by

Take a linear process as example, the correlation
function for a single perturbation *c(t) =
(s/T)*exp*(-t/T)*

Take a high frequency cutoff for
*1/(1+(2πfT) ^{2}*.

Thus the power-law distribution of lifetime found in the simulation leads to a 1/f power spectrum.

Jaeger et al. first investigated a real sandpile to
explore the idea of self-organized criticality in a realistic system.[2]
They measured the lifetime of avalanches at the rim of a sandpile.
Disappointingly they found frequency-independent power spectrum instead
of power-law spectrum. However Jensen et al. soon argued that the
behavior of sand flow over the rim, which was measured by Jaeger et al.,
is different from that down the slope, which was studied by Bak *et
al.* [3] Thus Jaeger's result is in expectation but the existence of
self-organized criticality and 1/f noise in a sandpile remains
mysterious.

Among the later on experiments with a pile of granular materials, one of the most noticeable ones is Frette et al.'s ricepile. [5] Rice grains of different anisotropy factors are studies and found to have different behavior; the more elongated the grains are, the more self-organized criticality behavior is observed. Thus the self-organized criticality is further proved to be not universal but dependent on the characteristics of the system.

© 2007 Y. Jiang. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. A
**38**, 364 (1988).

[2] H. M. Jaeger, C. Liu and S. R. Nagel, Phys. Rev.
Lett. **62**, 40 (1989).

[3] H. J. Jensen, K. Christensen and H. C. Fogedby,
Phys. Rev. B **40**, 7425 (1989).

[4] G. A. Held *et al.*, Phys. Rev. Lett.
**65**, 1120 (1990).

[5] V. Frette *et al.*, Nature **379**, 49
(1996).