Hodgepodge and forest fire

Two models from the late 1980s and early 1990s that brought cellular automata into chemistry and statistical physics. The hodgepodge machine imitates a catalytic chemical reaction and produces rotating spiral waves. The forest-fire model grows trees and burns them, and became a standard example of self-organised criticality: a system that tunes itself, without outside adjustment, to a state where events of every size happen.

What it is

The hodgepodge machine. Each cell holds a number from 0 to n. State 0 is healthy, n is ill, and everything between is infected, more so for larger numbers. Ill cells recover; healthy cells are infected by their neighbours; infected cells get worse, faster when their neighbours are worse. The parameters k1 and k2 set how easily healthy cells catch the infection, and g how fast infection grows. With suitable values the grid fills with spiral waves.

JETCARP’s version of the rule, per generation, is:

• An ill cell (state n) becomes healthy (0).
• A healthy cell becomes ⌊a / k1⌋ + ⌊b / k2⌋, where a is the number of infected neighbours and b the number of ill neighbours.
• An infected cell becomes ⌊S / (a + b + 1)⌋ + g, where S is the sum of the states of the cell and its neighbours.
• Results above n are capped at n.

This follows the commonly published form of the rule; we have not yet checked it line by line against Gerhardt and Schuster’s paper.

The forest fire. Each cell is empty, a tree or burning. Each generation a burning tree burns out and leaves empty ground; a tree catches fire if a neighbour is burning, or is struck by lightning with a small probability f; and empty ground grows a tree with probability p. When lightning is much rarer than growth, trees build up into large forests, and a single strike can burn a forest of any size. The sizes of the fires then follow a power law, with no typical scale.

Where it came from

Hodgepodge. The hodgepodge machine was designed by Martin Gerhardt and Heike Schuster of the University of Bielefeld, inspired by catalytic chemical reactions. A. K. Dewdney’s Scientific American column “The hodgepodge machine makes waves” introduced it to the public in August 1988 (Dewdney 1988), and the CelLab manual credits it as invented by Gerhardt and Schuster (Rucker and Walker 1989). Their own paper appeared in Physica D in 1989 (Gerhardt and Schuster 1989). With John J. Tyson they went on to a more quantitative automaton model of excitable media, published in Science in 1990 (Gerhardt, Schuster and Tyson 1990).

Forest fire. Per Bak, Kan Chen and Chao Tang introduced a forest-fire automaton in 1990: trees grow, fire spreads from burning trees to their neighbours, and fires never start on their own (Bak, Chen and Tang 1990); Chen, Bak and Jensen published a deterministic variant the same year (Chen, Bak and Jensen 1990). Barbara Drossel and Franz Schwabl added lightning in 1992, and showed that with lightning much rarer than growth the model organises itself into a critical state with fires of all sizes (Drossel and Schwabl 1992). Theirs became the standard forest-fire model; the review by Clar, Drossel and Schwabl tells the story and notes that C. L. Henley independently invented the limit in which clusters burn instantly (Clar, Drossel and Schwabl 1996; Henley 1993). The model descends from Bak, Tang and Wiesenfeld’s idea of self-organised criticality, introduced with the sandpile (Bak, Tang and Wiesenfeld 1987). Later work showed that its scaling is not simple at very large sizes (Grassberger 2002).

How JETCARP runs it

Notation

States are 0 to n for the hodgepodge machine (so n + 1 states) and 0 (empty), 1 (tree) and 2 (burning) for the forest fire.

Randomness that replays

The forest fire is random, but JETCARP’s random numbers are a fixed function of the scene’s seed, the generation and the cell, so a scene replays to exactly the same state every time, on any number of threads. The hodgepodge machine is deterministic.

On any tiling

Both rules count neighbours in particular states, so they run on any neighbour graph with touch or edge neighbours (see How JETCARP generalises automata). The hodgepodge machine can also be stepped on the GPU on lattice tilings. On tilings where tiles have different numbers of neighbours, the hodgepodge machine’s averages and the forest’s spreading vary from tile to tile, but the spirals and fires still appear.

What is new here

Our prior-art search found no earlier version of either model on aperiodic tilings such as Penrose, hat or spectre, and no hodgepodge machine off the square grid at all, though forest fires on triangular and hexagonal lattices probably exist in the physics literature. Either way this is a direct application of known rules to known graphs, so we do not count it as a discovery.

Try it

• Quick Start: Hodgepodge machine and Forest fire.
• jetcarp run docs/scenes/automata/hodgepodge-penrose.json --gens 1000 --png hodge.png
• Lower the forest fire’s lightning probability while keeping growth the same: fires become rarer and larger.

References

• Dewdney 1988; Gerhardt and Schuster 1989; Gerhardt, Schuster and Tyson 1990; Rucker and Walker 1989
• Bak, Chen and Tang 1990; Chen, Bak and Jensen 1990; Drossel and Schwabl 1992; Clar, Drossel and Schwabl 1996; Henley 1993; Grassberger 2002
• Bak, Tang and Wiesenfeld 1987