Brownian motion in the presence of temperature gradients. Sedimentation-equilibrium phenomena in single-component fluids

Einstein's theory of Brownian motion, which addresses only isothermal fluids, is here extended to situations in which the fluid is subject to an externally imposed temperature gradient. This extension involves adding a temperature-gradient animated ``drift velocity'' ${\rm {\bf U}}_D $ to the diffusive Brownian contribution $D$ appearing in the Fokker-Planck equation governing the coarse-grained conditional probability density. The \textit{ansatz }underlying the theory is derived by elementary sedimentation-equilibrium-type arguments of the type invoked by Einstein in his classic 1905 paper. The underlying theory is supported by experimental thermophoretic data, as well as by a recent theory of diffusive volume transport.