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 $T_{\mathrm{H}\mathrm{B}\mathrm{M}}$ and $T_{\mathrm{k}}$ . For $T_{\mathrm{H}\mathrm{B}\mathrm{M}}$ we derive a formally exact theoretical prediction and establish a generalised Einstein relation that links it to directly measurable quantities.