Fourier's law of heat conduction

Why does heat flow at all, and which way and how fast? Fourier's law answers the everyday observation that heat moves from hot to cold, and the bigger the temperature difference over a short distance, the harder it flows. It is the constitutive law — the physics input — that, combined with bookkeeping, gives the heat equation.

Fourier's law says the heat flux (the amount of thermal energy crossing a unit area per unit time) is proportional to minus the temperature gradient: flux = -kappa grad u, where kappa > 0 is the thermal conductivity of the material. The minus sign is the whole physical idea: heat flows in the direction in which temperature decreases, i.e. downhill on the temperature landscape, and it flows faster where the temperature drops more steeply (where grad u is large). In one dimension this is just flux = -kappa u_x. To get the heat equation, combine Fourier's law with conservation of energy: the rate at which heat builds up in a small region equals minus the net flux out of it, which is -div(flux) = kappa div(grad u) = kappa Laplacian u. Dividing by the heat capacity per unit volume turns kappa into the thermal diffusivity k and yields u_t = k Laplacian u.

Fourier introduced this law (and the trigonometric series now named after him) in his 1822 study of heat. The law is empirical and linear: it assumes the flux responds to the local gradient instantly and proportionally. That assumption is excellent for ordinary solids over ordinary temperature ranges, but it is also the source of the heat equation's unphysical infinite propagation speed; more refined models (e.g. the telegrapher / hyperbolic heat equation) add a relaxation time to fix that.