In statistical physics and probability theory, Percolation Theory investigates how fluid, fire, or electricity flows through a random porous medium.
Imagine a grid representing a forest, where every plot has a probability $p$ of containing a tree (the tree density $p$):
Green: Living Tree (state 1)
Orange: Burning Fire (state 2)
Gray: Burned Ash (state 3)
Dark: Empty land (state 0)
A spark starts a wildfire on the left border of the forest. The propagation rules are simple:
A tree catches fire if any adjacent neighbor (up, down, left, right) is currently burning.
A burning tree burns out in the next generation and turns to ash.
Empty ground cannot burn.
The Phase Transition: The Critical Percolation Threshold $p_c$
Does the fire burn completely across the forest to the other side?
Subcritical ($p < 0.59$): Trees form isolated, fragmented clusters. The fire quickly runs out of fuel and dies out locally!
Supercritical ($p > 0.59$): Trees connect into a giant spanning cluster. The wildfire spreads uncontrollably all the way across the forest!
On a 2D square lattice, mathematicians proved that the critical percolation threshold is:
$$p_c \approx 0.592746$$
Near $p \approx 0.59$, we observe a critical phase transition — a tiny $2\%$ change in tree density flips the forest from fire-resistant to fully combustible!
ca_forest(0.62) ca_run()
Move the mouse over a dotted box for more information.
Phase Transitions in Nature: Critical thresholds appear throughout nature: water freezing into ice at $0^\circ\text{C}$, iron becoming magnetic at its Curie temperature, and epidemic transmission thresholds ($R_0 > 1$) in epidemiology!
Fractal Burning Fronts: Right at the critical threshold $p \approx 0.59$, the leading edge of the burning fire forms an intricate fractal curve!
Now you try. Change density = 0.52 in the editor and click Run to watch the fire die out.
Type your code here:
See your results here:
The code has ???? for tree density. Replace ???? with 0.62 and click Run to watch the fire spread eastward across the forest!
1. At density = 0.62, there are enough connected trees to form a highway of fuel, carrying the fire all the way across the screen.
2. Try changing density = 0.52 (see Example 1). Because $p < 0.59$, the fire will sputter out in isolated pockets, leaving the right side untouched!
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