thermal diffusivity

Touch a metal spoon and a wooden spoon left in the same hot pot: the metal one burns you and the wood does not, even though both reached the same temperature. The difference is how quickly heat travels through them. Thermal diffusivity is the single number that measures that speed of spreading in the heat equation.

In the heat equation u_t = k u_xx, the constant k is the thermal diffusivity. Physically it bundles three material properties: k = kappa / (rho c), where kappa is the thermal conductivity (how easily heat is conducted), rho is the density, and rho c is the heat capacity per unit volume (how much heat it takes to raise the temperature). So a high k means the material conducts heat well relative to how much heat it stores — metals have high k, wood and water have low k. Its units are length squared over time, and that is the deep clue to its meaning: a disturbance spreads over a distance of order sqrt(k t) in time t. Want to know how long heat takes to penetrate a depth d? Roughly t ~ d^2 / k. Doubling the depth quadruples the time.

Because k sets the only time-and-length scale in the bare heat equation, it controls everything quantitative: how fast Fourier modes decay (the n-th mode decays like e^(-k (n pi / L)^2 t)), how wide the heat kernel spreads, and how quickly a body reaches steady state. A common confusion: do not mix up diffusivity k (governs the speed of the smoothing) with conductivity kappa (governs the steady-state heat flux). Two materials can conduct equally but diffuse at very different speeds if their heat capacities differ.