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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.

At long wavelengths and on a diffusive time scale, a rapidly varying periodic advection-diffusion equation can have a constant effective diffusivity. A multiple-scale expansion solves periodic cell equations for the fast variables. The solvability condition at the next order gives the macroscopic equation. For a periodic sinusoidal shear this is the effective diffusivity of a periodic sinusoidal shear; its transverse zero-mean cell modes decay, leaving the slow conserved concentration. This use of homogenization concerns differential equations, rather than polynomial homogenization.

Put $a=\kappa m^2$, with $\kappa >0$ and $m\ne 0$. The relevant homogenization of a periodic advection-diffusion equation limit is a long horizontal wavelength observed on its diffusive time scale. Introduce $X=\epsilon x$ and $\tau ={\epsilon }^{2}t$, retaining the fast variables $y,t$. Since the velocity field is independent of $x$, no fast horizontal cell variable is needed. Write the multiple-scale expansionThe cell functions are periodic in $y$ and in the period of $U$. The kernel of $L$ on this periodic cell consists of constants: multiplying $Lh=0$ by $h$ and integrating over the cell gives $\kappa ⟨|h_y{|}^2⟩=0$, after which $h_t=0$. Thus the leading concentration really is independent of the fast variables. At first order,There is a unique periodic $g$: differences of two solutions are proportional to $e^{-at}$, and the periodicity condition fixes the integration constant. Equivalently,At second order the advection-diffusion equation givesIntegration over the fast cell annihilates the left side. Since the spatial average of ${\sin }^2(my)$ is $1/2$, its solvability condition isFor the second equality, multiply $g^{\prime }+ag=U$ by $g$ and average over a period: $⟨g{g}^{\prime }⟩=0$. In particular the enhancement of diffusivity is nonnegative. Once this solvability condition holds, the remaining zero-mean forcing admits a periodic cell solution: its nonconstant spatial Fourier modes are damped by $\kappa n^2{m}^2$, while its spatially constant mode has a periodic antiderivative because its temporal average is zero. This also explains the relaxation of initial transverse inhomogeneity. The first corrector has zero spatial mean, so $\bar{\chi }=C+O({\epsilon }^2)$.

This derivation describes the large-scale, long-time limit, after times comparable to the period and to $a^{-1}$. It does not give an exact closed equation for the spatial average of arbitrary finite-time data: that average contains a correlation between the velocity field and concentration. In physical coordinates the leading long-wavelength concentration therefore obeys ${\bar{\chi }}_t={\kappa }_e{\bar{\chi }}_{xx}$ asymptotically. The cell expansion, its bounded periodic correctors, and the decay of transverse cell transients justify this effective diffusivity of a periodic sinusoidal shear.

For constant $U_0$, the periodic cell solution is $g=U_0/a$. ConsequentlyThis is Taylor dispersion: molecular diffusion moves particles between different horizontal velocities and turns their accumulated horizontal displacement into large-scale diffusion.

For the impulsive flow, $g$ jumps by $U_1$ at each kick and decays exponentially between kicks. If $G$ denotes its value just after a kick, periodicity requires $G=e^{-aT}G+U_1$. Thus, within one period,Using the mean-square expression, which has no ambiguous product of a Dirac delta with a discontinuous function, givesThe impulsive model can also be obtained by shrinking smooth pulses of fixed integrated strength. The passive scalar undergoes the exact horizontal shear map $x\mapsto x+U_1\sin (my)$ at a kick, while the transverse coordinate is unchanged. The cell solution above is the corresponding limit of the smooth cell solutions. Using a one-sided value in $Ug$ instead would miss the contribution from the jump of $g^2$.

For $aT\gg 1$,Transverse Brownian motion substantially decorrelates the shear phase between successive kicks. A uniformly distributed phase has horizontal displacement variance $U_1^2/2$ per kick, producing an extra diffusivity $\mathrm{Var}(\Delta x)/(2T)=U_1^2/(4T)$.

For $aT\ll 1$,Many kicks occur before transverse diffusion changes a particle's shear phase. Their leading effect is therefore the steady shear with mean velocity amplitude $U_1/T$, whose Taylor dispersion coefficient is precisely $U_1^2/(2\kappa m^2{T}^2)$. The divergence as $\kappa \to 0$ describes a very long decorrelation time; it does not imply infinite finite-time spreading. At $\kappa =0$ the phase never changes and, for a uniform phase, $n$ kicks give variance $n^2{U}_1^2/2$. The spreading is then ballistic transport, rather than diffusion. The long-time diffusive limit must precede the zero-diffusivity limit, with $t\gg a^{-1}$.

The thermal diffusion time $t_D=L^2/\alpha$, with $\alpha =\kappa$ the thermal diffusivity, is a characteristic scale for smoothing a temperature variation over length $L$. It is the thermal instance of the diffusive time scale. Rescaling $x=L\hat{x}$ and $t=(L^2/\alpha )\hat{t}$ removes $L$ and $\alpha$ from the constant-coefficient heat equation. Doubling length therefore multiplies comparable internal equilibration times by four; increasing thermal diffusivity by four divides them by four.

This scale is not a universal time of complete thermal equilibrium. A nonzero Fourier mode decays exponentially and never becomes exactly zero at finite time. The geometry, boundary conditions, initial pattern and accepted residual determine the numerical factor. For example, an insulated bar's slowest nonuniform Fourier mode has decay time $L^2/({\pi }^2\alpha )$, and reducing its amplitude to a fraction $\epsilon$ requiresExternal heat transfer can impose a longer cooling time even when internal temperature differences smooth quickly. Thermal diffusivity alone does not determine cooling to the room's temperature.