Forest Fire
Power Law

Most fires on the canvas above die in one cell. Lightning hits, burns a single tree, and it's over. Every few minutes, though, the same lightning lands on the wrong tree and takes half the screen with it. Same trigger, completely different outcome. What you're watching is a textbook case of self-organized criticality: a system that slowly tunes itself toward an unstable edge, then releases energy in bursts whose sizes follow a power law. Drossel and Schwabl wrote the rules for this model in 1992, and physicists have been pointing at it ever since as the cleanest visual proof that the trigger isn't the cause. The forest's connected state is. The simulation above is the same rules, stripped to the bone.

What is self-organized criticality in a forest fire model?

Three states and four rules. A cell is empty, a tree, or burning. Empty cells regrow into trees with a small probability each step. Trees catch fire if a neighbor is already burning. Trees also ignite directly from lightning at a much rarer rate. Burning cells turn empty after a short burn duration. That's the whole engine.

The Growth rate slider sets how often an empty cell turns green. Push it high and the forest fills in fast. Lightning rate sets how often sparks land on an existing tree. Burn duration is how many steps a burning cell takes to collapse back to ash. Spread is four-neighbor by default; diagonal spread is a toggle. That's it. No wind, no fuel moisture, no embers launched on updrafts. Everything you see emerges from those four rules running in a loop.

How simplified is this model?

Very, on purpose. A real wildfire has wind, terrain, fuel moisture, crown versus ground fire, species mixes, and embers launched kilometers ahead of the front. This simulation has none of that. The grid is flat, the trees are binary (present or absent), and the spread is local. That's a deliberate design choice, not a bug.

The simplification is the point. What survives it is the one thing the model is trying to isolate: the relationship between forest density and event size. Strip a wildfire of its physics and the shape of its event-size distribution stays the same. That shape is the power law. The source notes above list the specific omissions.

Where the same shape shows up outside forests

Per Bak's insight in the late 1980s was that lots of systems tune themselves to this edge without anyone setting the balance. When energy builds up slowly and releases through a connected network, the size distribution of releases is heavy-tailed: many small events, a few enormous ones, no typical size. The forest-fire model is one of the cleanest illustrations. The same math fits other systems:

• Power grids. Most outages are local and brief; a few cascade across continents. The 2003 Northeast blackout hit 50 million people from one sag in Ohio.
• Earthquakes. The Gutenberg-Richter law says small quakes are common, huge ones rare, and the frequency-to-magnitude relationship is a clean power law across many orders of magnitude.
• Financial markets. Daily returns have fat tails. Most days are quiet, rare days are catastrophic, and you can't forecast which from the trigger alone.
• Avalanches and landslides. Same shape, same mechanism: slow buildup, local threshold, connected release.

The common pattern: slow input, local rules, a connected substrate. When those three line up, you get a power law whether the substrate is trees, copper wire, rock, or order books. The ant colony simulation on this site shows a related story from the opposite direction: the same kind of local rules producing coordination instead of collapse.

Things to try

Start with the Critical regime preset. Let the forest fill for thirty seconds before touching anything. Watch what happens when a fire finally lands: most are small, some are large, occasionally one erases a whole quadrant. Now push Growth rate up toward Dense overload. Fires happen less often but nearly every one becomes a giant. Drop the rate toward Sparse regrowth and fires are everywhere but stay small; the forest never densifies enough to carry them. The interesting band is narrow, and that narrowness is the whole phenomenon. Turn on Use diagonal spread to see how a small increase in connectivity changes cascade sizes. Use Spark burst to trigger fires manually and notice: the size of what follows depends on the forest's current state, not on you. Browse the full simulation library for other systems where local rules produce global shapes.