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Diffusion is net transport associated with random microscopic motion. A localized patch of a dilute solute can spread through a stationary solvent: individual molecules move in both directions, but more cross from the crowded side of an imaginary surface than from the less crowded side. This imbalance gives a net diffusive flux down the concentration gradient; the molecules do not need to sense which direction has lower concentration. At uniform concentration, microscopic exchanges continue even though the net diffusive flux vanishes. ^[ref]

The distinction from advection is easiest to see in a dye patch. Uniform bulk motion carries its centre downstream; diffusion broadens it relative to that centre. Both can happen together. A moving or spreading patch by itself does not identify the mechanism: a nonuniform velocity field can also stretch a patch. ^[ref]

For a dilute nonreacting solute in a uniform medium at fixed temperature, Fick's first law models the relative diffusive flux as $\mathbf{j}=-D\nabla C$, with scalar diffusion coefficient $D>0$ of units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. Here $C$ is the amount of substance per unit volume, so $\mathbf{j}$ has units $\mathrm{mol}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. With carrier velocity field $\mathbf{u}$, the total flux is $C\mathbf{u}+\mathbf{j}$. The continuity equation, with no sources or chemical reactions, givesFor constant $D$ and incompressible flow, this reduces to $C_t+\mathbf{u}\cdot \nabla C=D{\nabla }^{2}C$, where ${\nabla }^2$ is the Laplacian. In a stationary carrier it is the diffusion equation. These assumptions matter: variable $D$ must remain inside the divergence, and changing sources require extra terms. ^[ref]

The useful surprise is the common mathematics: thermal conduction smooths temperature through the analogous heat equation, while molecular diffusion smooths composition. This analogy already motivated the original formulation of Fick's laws. ^[ref] However, the thermal diffusivity and the solutal diffusivity need not be equal. A diffusive time scale $t_D\sim L^2/D$ means that crossing ten times the distance requires roughly one hundred times the time. This is why molecular diffusion is especially effective across thin layers, and why chaotic advection can assist mixing by making those layers thinner. Physical diffusion is a transport mechanism, distinct from the broader probabilistic class of Markov diffusion.

Fick's laws describe diffusion through a flux law and conservation of mass. The first law says that diffusive flux points down the concentration gradient:Combining it with local conservation gives
$C_t=\nabla \cdot (D(C)\nabla C)$.