Diffusion

Diffusion is the net movement of particles from a region of higher concentration to a region of lower concentration, driven by random thermal motion. It is the macroscopic expression of Brownian motion — the accumulated effect of countless microscopic collisions that produce a systematic drift toward equilibrium.

Fick's laws describe diffusion phenomenologically: the flux is proportional to the concentration gradient, and the concentration evolves according to the heat equation. But the deeper structure is statistical: diffusion is what happens when a large number of independent random displacements accumulate, and the central limit theorem guarantees that the resulting concentration field will be Gaussian.

Diffusion is not limited to chemistry. It appears in ecology as the spread of populations, in finance as the dispersal of price information, in neuroscience as the propagation of action potentials, and in sociology as the adoption of innovations. The mathematical structure is identical because the underlying process — many small, independent perturbations accumulating — is substrate-independent. This is why the Einstein relation connects diffusion to mobility in every physical domain.