We derive the markovian description for the nonequilibrium brownian motion of a heated nanoparticle in a simple solvent with a temperature-dependent viscosity. Our analytical results for the generalized fluctuation-dissipation and Stokes-Einstein relations compare favorably with measurements of laser-heated gold nanoparticles and provide a practical rational basis for emerging photothermal tracer and nanoparticle trapping and tracking techniques.

We introduce a new technique to measure tracer dynamics, which is sensitive to single metal nanoparticles down to a radius of 2.5 nm with a time resolution of a few microseconds. It is based on a fluctuation analysis of a heterodyne photothermal scattering signal emanating from the hot halo around the laser-heated tracer. A generalized Stokes-Einstein relation for "hot brownian motion" is developed and verified. Exploiting the excellent photostability of gold nanoparticles, the developed method promises broad applications especially in the field of quantitative biomedical screening.

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. * * *We gratefully acknowledge helpful discussions with Jean-Louis Barrat (Grenoble), Ramin Golestanian (Oxford), and Markus Selmke (Leipzig), and thank Hugo Brandao for a careful reading of the manuscript. This work was supported by the Alexander von Humboldt foundation, the Deutsche Forschungsgemeinschaft (DFG) via FOR 877 and, within the German excellence initiative, p-5

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.

We establish an effective Markov theory for the rotational Brownian motion of hot nanobeads and nanorods. Compact analytical expressions for the effective temperature and friction are derived from the fluctuating hydrodynamic equations of motion. They are verified by comparison with recent measurements and with parallel molecular dynamics simulations over a wide temperature range. This provides unique insights into the physics of hot Brownian motion and an excellent starting point for further experimental tests and applications involving laser-heated nanobeads, nanorods and Janus particles.The popular Markovian theory of Brownian motion, as developed by Einstein, Langevin and Smoluchowski a century ago, has been the starting point and inspiration for innumerable applications [1,2]. However, the usual convenient formulation in terms of the centre-of-mass coordinates of particles only pertains to the special case of an isolated spherical particle. In the general case of interacting and/or anisotropic particles, both translational and rotational degrees of freedom couple, calling for a more elaborate mathematical description. This is most obvious for rod-shaped particles that have different mobilities for the movement parallel and perpendicular to their long axis [3], but in fact also holds for interacting spherical particles [4]. Due to the associated technical complications, the present theoretical understanding is still relatively incomplete [6], in particular with regard to micro-swimmers and other active or selfpropelled colloidal particles [7][8][9], for which the proper hydrodynamic description is even more subtle than for passive particles in external fields [10,11]. The directed motion for such selfpropelled particles from sperms [12] to Janus particles running on chemical fuel [13] is usually limited by (equilibrium or nonequilibrium) rotational Brownian motion. Besides, rotational Brownian motion is undoubtedly of interest for its own sake. It is accessible to spectroscopy [14] and has been the basis for the development of new microrheological techniques [15] and nanoscopic heat engines [16].In this paper, we are concerned with a specific type of rotational Brownian motion that occurs whenever the colloidal particles have an elevated temperature with respect to their solvent. In this case, we speak of rotational hot Brownian motion, in analogy to the better understood translational case [17]. Both intended [18][19][20][21][22] and unintentional [23,24] realizations of (rotational) hot Brownian motion are nowadays widespread in biophysical and nanotechnological applications, which often employ nanoparticles exposed to laser light as tracers, anchors and localized heat sources. Deliberate heating of nanoparticles is, for instance, common in photothermal therapy [25,26], but it also helps to enhance the optical contrast for detection [27] or in photothermal correlation spectroscopy [28]. Laser-heating is also a convenient way of supplying the energy for the self-thermophoretic propulsion of anisotropic parti...