Fick's law of diffusion

If one side of a room is full of perfume molecules and the other side has almost none, molecules drift, on average, toward the empty side until things even out. Fick's law is the quantitative statement of that: the net flow of a substance is proportional to how steeply its concentration changes from place to place, and points from crowded toward empty.

Fick's first law says the diffusive flux is flux = -D grad c, where c is the concentration and D > 0 is the diffusion coefficient. The minus sign means the substance flows down the concentration gradient, from high concentration to low. In one dimension it is flux = -D c_x. There is nothing mysterious driving this: it is the statistical result of a huge number of particles each jiggling randomly (Brownian motion). Random jiggling has no preferred direction, yet where there are more particles, more of them happen to wander into the sparse region than the other way around, producing a net downhill flow. Combining Fick's first law with conservation of mass (c_t = -div(flux)) gives Fick's second law, the diffusion equation c_t = D Laplacian c.

Fick wrote this down in 1855 by direct analogy with Fourier's law of heat — concentration plays the role of temperature, D plays the role of thermal diffusivity. The analogy is exact at the level of the equations, which is why heat and diffusion share all their qualitative features. Caveat: Fick's law assumes ordinary (Fickian) diffusion in a uniform medium; in crowded, porous, or biological media you can get anomalous diffusion where the spreading does not grow like sqrt(time) and the simple law needs modification.