Diffusion tensor
= Diffusion tensor
{title2=$\mathbf D$}
A \b[diffusion tensor] describes direction-dependent <diffusive flux>. In the local, linear Fickian model,
$$
\mathbf j=-\mathbf D\nabla C,\qquad
\partial_t C=\nabla\cdot(\mathbf D\nabla C),
$$
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.<Diffusion in solids in the COMSOL 6.4 documentation>[Solid, Diffusion: isotropic, diagonal and symmetric diffusion coefficients]{ref}
For an original two-dimensional example, choose
$$
\mathbf D=\begin{pmatrix}2&1\\1&2\end{pmatrix}\,\mathrm{\mu m^2/s}.
$$
A <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,
$$
\nabla C\cdot\mathbf j=-(\nabla C)^T\mathbf D\nabla C<0
$$
for every nonzero gradient in this <positive-definite matrix> example.
The <eigenvectors> $(1,1)/\sqrt2$ and $(1,-1)/\sqrt2$ have <eigenvalues> 3 and 1. They identify the principal directions and coefficients. Rotating the coordinate basis changes the matrix entries according to $\mathbf D'=R\mathbf D R^T$; 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.