  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 profile, leading to
anisotropic self-diffusion tensor for a spherical tracer. It is also
retarded--it creates memory effects during the diffusion process due to
hydrodynamic interactions.
