  The classical theory of Brownian motion rests on fundamental laws of
statistical mechanics, such as the equipartition theorem and the
fluctuation-dissipation theorem, which are not applicable in non-isothermal
situations. We derive the generalized fluctuation-dissipation relations and
Langevin equations governing such non-isothermal Brownian motion, including
explicit results for the frequency-dependent noise temperature and Brownian
thermometry far from equilibrium.
