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_HBM and T_k. For T _HBM we derive a formally exact theoretical prediction and establish a generalised Einstein relation that links it to directly measurable quantities.