  We consider the transport of conserved charges in spatially inhomogeneous
quantum systems with a discrete lattice symmetry. We analyse the retarded two
point functions involving the charge and the associated currents at long
wavelengths, compared to the scale of the lattice, and, when the DC
conductivity is finite, extract the hydrodynamic modes associated with charge
diffusion. We show that the dispersion relations of these modes are related to
the eigenvalues of a specific matrix constructed from the DC conductivity and
certain thermodynamic susceptibilities, thus obtaining generalised Einstein
relations. We illustrate these general results in the specific context of
relativistic hydrodynamics where translation invariance is broken using
spatially inhomogeneous and periodic deformations of the stress tensor and the
conserved $U(1)$ currents. Equivalently, this corresponds to considering
hydrodynamics on a curved manifold, with a spatially periodic metric and
chemical potential.

High Energy Physics - Theory;Strongly Correlated Electrons