  Widely applicable, modified Green-Kubo expressions for the local diffusion
coefficient ($D_l$) are obtained using linear response theory. In contrast to
past definitions in use, these expressions are statistical mechanical results.
Molecular simulations of systems with anisotropic diffusion and an
inhomogeneous density profile confirm the validity of the results. Diffusion
coefficients determined from different expressions in terms of currents and
velocity correlations agree in the limit of large systems. Furthermore, they
apply to arbitrarily small local regions, making them readily applicable to
nanoscale and inhomogeneous systems where knowledge of $D_l$ is important.
