Diffusion in inhomogeneous media.

摘要

We consider the transport of conserved charges in spatially inhomogeneous quantum systems with auddiscrete lattice symmetry. We analyze the retarded two-point functions involving the charges and theudassociated currents at long wavelengths, compared to the scale of the lattice, and, when the dcudconductivities are finite, extract the hydrodynamic modes associated with diffusion of the charges. Weudshow that the dispersion relations of these modes are related to the eigenvalues of a specific matrixudconstructed from the dc conductivities and certain thermodynamic susceptibilities, thus obtainingudgeneralized Einstein relations. We illustrate these general results in the specific context of relativisticudhydrodynamics where translation invariance is broken using spatially inhomogeneous and periodicuddeformations of the stress tensor and the conserved Uð1Þ currents. Equivalently, this corresponds toudconsidering hydrodynamics on a curved manifold, with a spatially periodic metric and chemical potential,udand we obtain the dispersion relations for the heat and charge diffusive modes.