Heat equation

Summary

The heat equation $\partial_t u = \Delta u$ describes how temperature, or the concentration of a diffusing substance, evolves in time. Here $\Delta$ is the Laplacian in the space variables. It is the model parabolic partial differential equation. Its solutions become smooth at once, spread information at infinite speed, and decay toward equilibrium. It is linked to probability through the Wiener process, and to geometry through the spectrum of the Laplacian.