Walkers That Solve Equations

Random walks and steady heat

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• The probability that a walker first exits through a part of the boundary obeys the same equation as steady heat.
• Steady temperatures are averages of their neighbours, so extremes lie on the boundary.
• Symmetry and superposition give exact answers: a square with one hot side is 25° at its centre.

Steps

1. Gambler’s ruin, again
2. A hot edge
3. Why exactly 25°?
4. The average of the neighbours
5. Walking is solving
6. Two hot edges