Diffusion tensor ($\mathbf{D}$)

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.