Diffusion in inhomogeneous media

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.