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A diffusion tensor describes direction-dependent diffusive flux. In the local, linear Fickian model,where $C$ is concentration and the entries of $\mathbf{D}$ have the units of a diffusion coefficient. A scalar coefficient is recovered when $\mathbf{D}=D\mathbf{I}$. Material models can instead specify a diagonal matrix or symmetric matrix to represent anisotropic transport.^[ref]

For an original two-dimensional example, chooseA concentration gradient directed along $x$, with components $(g,0)$, produces diffusive flux $(-2g,-g)$ in consistent units. The flux need not be antiparallel to the gradient. Nevertheless,for every nonzero gradient in this positive-definite matrix example.

The eigenvectors $(1,1)/\sqrt{2}$ and $(1,-1)/\sqrt{2}$ have eigenvalues 3 and 1. They identify the principal directions and coefficients. Rotating the coordinate basis changes the matrix entries according to ${\mathbf{D}}^{\prime }=R\mathbf{D}{R}^T$; it does not change these principal coefficients. In an unbounded, homogeneous, drift-free Gaussian model, displacement covariance is $2t\mathbf{D}$, so spreading also differs along those directions.

Symmetry and positive definiteness are assumptions of this ordinary reciprocal diffusion model, not a definition covering every transport law called a tensor. Boundaries also matter: a reflecting boundary condition constrains $\mathbf{n}\cdot \mathbf{D}\nabla C$, which need not be proportional to the normal derivative alone.

Restricted diffusion is diffusion whose displacement statistics are constrained by boundaries that particles cannot cross. A reflecting boundary condition models an impermeable wall: particles continue moving, but there is no normal diffusive flux through it. Finite permeability would require a different boundary model.

A physical demonstration uses a liquid confined between solid lamellae. In his first-person account, John E. Tanner described constructing 99 mica layers separated by aluminium-foil strips, removing wrinkles and burrs to obtain regular spacing; Edward O. Stejskal inspected the stack with a stereomicroscope. Pulsed magnetic-field gradients encode displacements in the nuclear magnetic resonance signal, allowing motion normal to the barriers to be compared with less confined directions.^[ref] This is a material confinement experiment, not a change in the liquid's chemistry.

For an original one-dimensional model, let particles diffuse in $0\le x\le L$ with constant diffusion coefficient $D>0$:The wall conditions are homogeneous Neumann boundary conditions for the diffusion equation. Start in equilibrium, so $X(0)$ has the uniform distribution on $[0,L]$. Its variance isAt long time separation, the positions decorrelate. Consequently the mean squared displacement approachesAt short times relative to $L^2/D$, most displacements have not encountered a wall, and the leading scale is the free-diffusion result $2D\tau$. For a hypothetical gap $L=10\mu \mathrm{m}$, the long-time plateau is $100/6\mu {\mathrm{m}}^2$ and its root is about $4.08\mu \mathrm{m}$.

The initial condition matters: fixing $X(0)=L/2$ instead gives the long-time value $L^2/12$. Likewise, echo attenuation in a confined sample must not automatically be converted into a free-diffusion coefficient. Confinement geometry, timing and the selected boundary model all affect the inference.