Generalised Einstein Relation for Hot Brownian Motion
  The Brownian motion of a hot nanoparticle is described by an effective Markov theory based on fluctuating hydrodynamics. Its predictions are scrutinized over a wide temperature range using large-scale molecular dynamics simulations of a hot nanoparticle in a Lennard-Jones fluid. The particle positions and momenta are found to be Boltzmann distributed according to distinct effective temperatures $T_\mathrm{HBM}$ and $T_\mathrm{k}$ . For $T_\mathrm{HBM}$ we derive a formally exact theoretical prediction and establish a generalised Einstein relation that links it to directly measurable quantities. 
D. Chakraborty, M. V. Gnann, D. Rings, J. Glaser, F. Otto, F. Cichos   and K. Kroy
cond-mat.soft