Diffusion

Diffusion is the net transport of particles from regions of high concentration to regions of low concentration, arising not from any directed force but from the random thermal motion of the particles themselves. Although each molecule wanders aimlessly, more start where they are crowded, so on average the crowd disperses. Fick's law states that the particle flux is proportional to the concentration gradient, with the diffusion coefficient D as the constant of proportionality.

The hallmark of a diffusive process is the square-root law: a diffusing front, or the typical displacement of a tagged particle, grows as √(Dt) rather than linearly in time. This is why diffusion is fast over micrometres but agonisingly slow over centimetres, and why stirring — which adds bulk flow — is so much more effective than waiting for sugar to sweeten a cup of tea on its own.

Diffusion and Brownian motion are two views of the same physics: Brownian motion is the random walk of one particle, diffusion the collective spreading of many. Einstein's 1905 analysis derived the diffusion coefficient from molecular collisions, linking the macroscopic transport coefficient to k_BT.

Described phenomenologically by Adolf Fick in 1855; given its molecular foundation by Einstein and Smoluchowski (1905–06), who connected the diffusion coefficient to thermal energy and molecular size.