# Random Walks and the Heat Equation

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This Demonstration illustrates the steady-state temperature distribution on a square with prescribed boundary temperatures on each of the four sides, along with random walks that can be used to compute the temperature distribution experimentally. This is done as follows. Choose a point in the square using the locator and then generate many random walks starting at this point and ending at one of the four sides of the square. The average of the temperatures at the endpoints of these random walks is approximately equal to the steady-state temperature at the given point.

Contributed by: Yirui Luo (February 2019)

Based on an undergraduate research project at the Illinois Geometry Lab by Yuheng Chang, Baihe Duan, Yirui Luo, Yitao Meng, Cameron Nachreiner and Yiyin Shen and directed by A. J. Hildebrand.

Open content licensed under CC BY-NC-SA

## Details

The random walks are generated using steps of the following types:

Gaussian: The steps are given by a two-dimensional normal distribution with standard deviation equal to the "scale" parameter.

Discrete 1: The steps are of length "scale" and in directions chosen uniformly from the angles 0°, 120° and 240°.

Discrete 2: The steps are of length "scale" and in directions chosen uniformly from the angles 0°, 90°, 180° and 270°.

Discrete 3: The steps are of length "scale" and in directions chosen uniformly from the angles 0°, 60°, 120°, 180°, 240° and 300°.

These random walks approximate a Brownian motion path that is stopped at the boundary. Kakutani showed that the expected temperature at the endpoint of such a path is equal to the steady-state temperature at the given point. See [1].

Reference

[1] R. Hersh and R. J. Griego, "Brownian Motion and Potential Theory," *Scientific American*, 220(3), 1969 pp. 66–74. doi:10.1038/scientificamerican0369-66.

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