A Student's Guide Through the Great Physics Texts pp 143-162 | Cite as

# The Diffusion Equation

## Abstract

In his *Analytical Theory of Heat*, Joseph Fourier developed a mathematical equation which governs the diffusion of heat—and the resulting distribution of temperatures—in bodies of various shapes, sizes and compositions.^{1} Essential to Fourier’s analysis is the precise stipulation of the thermal boundary conditions of the body, such as the initial temperature within the body and the rate of heat flow through the exterior surfaces of the body. Fourier’s treatment of thermal diffusion was both exhaustive and mathematically sophisticated. On the other hand, in Chap. 18 of his popular *Theory of Heat*, Maxwell presents Fourier’s method in a simpler and more intuitive manner. This is the text included below. Maxwell begins by defining the *thermal conductivity* of a substance. This includes a discussion of *dimensional analysis* (another topic which was pioneered by Fourier). He then explains how Fourier’s diffusion equation can be used to calculate the time-evolution of the temperature distribution within a heated substance. In so doing he recalls Fourier’s method of describing the temperature within a body as a series of harmonic temperature distributions of various spatial frequencies.^{2} The high-frequency (short wave-length) terms in the series die out most rapidly as heat diffuses through the substance, leaving the low-frequency (long wave-length) components as the most long-lived. Maxwell then goes on to explain how the diffusion equation can be used to solve geophysical problems such as seasonal variations of Earth’s subsurface temperature, and perhaps even to determine the maximum age of Earth’s crust.