Tracer diffusivity in a time- or space-dependent temperature field

The conventional assumption that the self-diffusion coefficient of a small tracer can be obtained by a local and instantaneous application of Einstein's relation in a temperature field with spatial and temporal heterogeneity is revisited. It is shown that hydrodynamic fluctuations contribute to the self-diffusion tensor in a universal way, i.e. independent of the size and shape of the tracer. The hydrodynamic contribution is anisotropic it reflects knowledge of the global anisotropy in the temperature pro le, leading to an anisotropic self-diffusion tensor for a spherical tracer. It is also retarded it creates memory effects during the diffusion process due to hydrodynamic interactions.