A raging inferno

Classical Mechanics Level 2

What is the average number of trees that are consumed by flames per unit time?

Recall that new fires are initiated by lightning striking a random site on the lattice; if it's occupied by a tree then it will ignite; if it's empty then nothing will happen. The average fire rages until ⟨ s ⟩ \langle s\rangle trees are burnt down.

f × r × ρ empty × L 2 f\times r\times \rho_\textrm{empty}\times L^2 f × ρ tree × ⟨ s ⟩ × L 2 f\times \rho_\textrm{tree}\times \langle s\rangle\times L^2 f × ( 1 − r ) × ρ tree × ⟨ s ⟩ × L 2 f\times (1-r)\times \rho_\textrm{tree}\times \langle s\rangle \times L^2 ρ tree × ⟨ s ⟩ × L 2 \rho_\textrm{tree}\times \sqrt{\langle s\rangle}\times L^2

Josh Silverman by Josh Silverman

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1 solution

Josh Silverman Staff
Oct 16, 2017

To avoid double counting, we can associate each tree burning with the lightning strike that sparked the firewall that consumed it.

We expect lightning to strike the lattice f L 2 fL^2 times per time step. Of these, a fraction ρ tree \rho_\textrm{tree} will contain an unburned tree, prone to incineration. Thus we expect a total of f ρ tree L 2 f\rho_\textrm{tree}L^2 fire events to be initiated per unit time.

By definition, each of these strikes will lead to the destruction of ⟨ s ⟩ \langle s\rangle trees. Thus, in each time step we expect f × ρ tree × L 2 × ⟨ s ⟩ f\times\rho_\textrm{tree}\times L^2\times\langle s\rangle trees to be engulfed by ravenous wildfire.

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