the diffusion equation
Open a bottle of perfume in the corner of a still room and the scent slowly fills the space, moving from where it is concentrated toward where it is not. The diffusion equation is the law of that spreading-out of a substance. It is the same equation as the heat equation, just told in the language of concentration of stuff rather than temperature.
Let c(x,t) be the concentration of a dissolved substance (or the density of a gas, a pollutant, a dye). The diffusion equation is c_t = D c_xx, where D > 0 is the diffusion coefficient. It comes from two physical facts. First, Fick's law of diffusion: the flux of substance is proportional to minus the gradient of concentration, flux = -D c_x, meaning stuff flows downhill from high concentration to low. Second, conservation of mass: stuff is neither created nor destroyed, so the rate of change of concentration equals minus the divergence of the flux. Put those together and you get c_t = D c_xx. The structure is identical to the heat equation because heat conduction and molecular diffusion obey the same two principles (a gradient-driven flux plus a conservation law).
Calling it the diffusion equation rather than the heat equation is a matter of emphasis and field: chemists, biologists, and probabilists usually say diffusion; physicists and engineers studying conduction usually say heat. Mathematically there is no difference, and the qualitative lessons transfer exactly — concentration profiles smooth out, total mass is conserved on the whole line, the process cannot be run backward, and the spreading reaches everywhere instantly (infinite propagation speed). Honest caveat: real diffusion of fast particles is not literally instantaneous; the equation is an excellent macroscopic model, not a microscopic truth.
A drop of ink placed in still water spreads into a fuzzy cloud whose width grows like the square root of time, exactly as c_t = D c_xx predicts.
Width grows like sqrt(D t): four times longer means only twice as wide.
Diffusion and heat conduction are the same equation; the diffusion coefficient D plays the role of the thermal diffusivity k. The width-grows-like-sqrt(time) law is the signature of diffusion, not of straight-line transport (where width would grow like time).