Diffusion in inhomogeneous media.
2017
We consider the transport of conserved charges in spatially inhomogeneous quantum systems with a
discrete lattice symmetry. We analyze the retarded two-point functions involving the charges and the
associated currents at long wavelengths, compared to the scale of the lattice, and, when the dc
conductivities are finite, extract the hydrodynamic modes associated with diffusion of the charges. We
show that the dispersion relations of these modes are related to the eigenvalues of a specific matrix
constructed from the dc conductivities and certain thermodynamic susceptibilities, thus obtaining
generalized 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 Uð1Þ currents. Equivalently, this corresponds to
considering hydrodynamics on a curved manifold, with a spatially periodic metric and chemical potential,
and we obtain the dispersion relations for the heat and charge diffusive modes.
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