Sandpile ModelBeta

About Sandpile Model

This is a one-dimensional sandpile seen from the side. The floor is divided into columns, and the number of grains in a column is its height. When the height difference (the local slope) between a column and its neighbour exceeds a threshold, one grain slides toward the lower side; if that grain pushes the neighbour past its own threshold, the chain continues. The whole chain is one avalanche, and its size is counted as the number of individual grain slides.

The threshold is the “base threshold” plus a random 0 or 1, redrawn every time a grain leaves that column. With the defaults (base 1, irregularity 0.5) the threshold is 1 or 2 with equal odds — the rule of the Oslo model, which researchers at the University of Oslo built in 1996 to explain their rice-pile experiment. The difference is that grains in the Oslo model only move one way; here they slide toward the steeper side, and either end of the pile can be opened or walled off.

The 1987 Bak–Tang–Wiesenfeld sandpile

The term self-organized criticality (SOC) comes from a 1987 paper by Per Bak, Chao Tang and Kurt Wiesenfeld. They proposed a simple cellular automaton: drop grains one at a time onto a grid, and whenever a cell holds more than a limit, it hands grains to its neighbours. Without anyone tuning a parameter, the system drives itself into a critical state in which avalanches of every size occur, and the avalanche sizes follow a power law with no typical scale. As the paper's title says, the original aim was to explain the 1/f noise seen throughout nature.

Critical behaviour in physics usually needs a parameter such as temperature to be set exactly right. The sandpile is interesting because slow pouring alone keeps the system at that point. If the slope is too gentle, grains pile up and steepen it; if it is too steep, avalanches cut it back down.

Why real sand disappointed and rice worked

Experiments looking for SOC in actual sand mostly failed to find clean power laws. In 1996, Frette and colleagues dropped rice grains one by one into a narrow gap between two glass plates. Elongated rice showed power-law avalanches; rounder rice did not. Long grains lock together in many different ways, so the slope a patch can hold keeps changing.

The same group introduced the Oslo model to capture this: instead of a fixed threshold, each site's critical slope is redrawn as 1 or 2 every time it topples. They reported an avalanche size exponent of about 1.55 (±0.10). Setting this tool's irregularity slider to 0 gives every grain the same threshold, and the pile settles into a regular sawtooth. Comparing the avalanche statistics before and after shows why the randomness matters.

Reading the log–log plot

Each dot on the plot is one size bin, and the bins double in width: [1, 2), [2, 4), [4, 8) and so on. Because the bins differ in width, the tool divides each count by the bin width and plots a density. That makes the dashed line's slope an estimate of −τ directly. Had the raw counts per bin been plotted, the slope would be −(τ − 1) and look much flatter.

Scatter and a drop-off at the right end are expected. With a finite number of columns, no avalanche can be larger than the pile allows (a finite-size cutoff), and the largest bins contain only a handful of events. You need hundreds to thousands of avalanches before the slope settles, which is why the rice-pile scenario asks for 1,000.

The “angle of repose” readout comes from the steady-state slope measured by running this model for a long time. With the defaults the mean threshold is 1.5, but the pile settles at a steeper slope of about 1.71. Sites that draw a low threshold (1) topple right away and redraw, while sites that draw a high one (2) hold the slope up for longer — a known property of the Oslo model, whose slope approaches about 1.73 for large piles. One grain height is half a column width, so the tool shows arctan(1.71 × 0.5) ≈ 40.5°. Pouring 120,000 grains onto a 100-column ramp gives an average slope that matches this, 3–4 degrees steeper than the 37° you would get from the mean threshold. The “Slope” readout measures the pile at this instant, so it swings a degree or two either side just before and after avalanches.

What each scenario is testing

“Hourglass” drops one grain at a time on one spot and asks you to witness a big avalanche of 200 or more slides. “Snowy slope” and “Construction stockpile” reverse the goal: keep material from spilling onto the road. A fence (a column with a raised floor) props up the slope above it, and wetting a column raises its threshold so it holds a steeper face — the same reason damp sand makes a sandcastle.

The “Poke” tool briefly drops the threshold to zero within a radius, forcing loose grains to slide. It is the same idea as the controlled avalanches mountain road crews trigger on purpose so that a small release happens before a large one can build up.

Worked examples

How to use it

1. Choose a scenario and, if allowed, place fences, water or the pour point first.
2. Press ▶ Start pouring and watch avalanche sizes and the log–log plot.
3. In Free play, change irregularity and base threshold and compare the distributions.

Key terms

Self-organized criticality

The tendency of a slowly driven system to settle into a critical state on its own, with event sizes following a power law.

Avalanche size

Here, the total number of grain slides between dropping one grain and the pile coming to rest.

Exponent τ

If avalanches of size s occur with frequency proportional to s^−τ, τ is the exponent — the magnitude of the slope on a log–log density plot.

Oslo model

A one-dimensional rice-pile model in which each site's threshold is randomly redrawn as 1 or 2 after every toppling (Christensen et al., 1996).

Frequently asked questions