We derive generalized Langevin equations for the translational and rotational motion of a heated Brownian particle from the fluctuating hydrodynamics of its nonisothermal solvent. The temperature gradient around the particle couples to the hydrodynamic modes excited by the particle itself so that the resulting noise spectrum is governed by a frequency-dependent temperature. We show how the effective temperatures at which the particle coordinates and (angular) velocities appear to be thermalized emerge from this central quantity.

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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 THBM and Tk. For T HBM we derive a formally exact theoretical prediction and establish a generalised Einstein relation that links it to directly measurable quantities.@en