Effective temperatures of hot Brownian motion

Abstract

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.

Mesoscopic virial equation for nonequilibrium statistical mechanics

“…Indeed, recent experiments with heatconducting metals show intriguing deviations from equipartition, related to enhancements of low-frequency vibrational modes that may become even 'hotter' than the highest boundary temperature [3]. Similar deviations from equipartition are observed for strongly heated cantilevers [4] and Brownian particles [5,6]. These are some out of many manifestations of nontrivial effects characterizing systems driven far from thermodynamic equilibrium.…”

Nonisothermal fluctuating hydrodynamics and Brownian motion

“…The consequences of this result are more fully explored in Ref. [27], where the generalized Langevin equation (43) is derived by direct application of linear response theory to the momentum of a Brownian particle and to (locally equilibrated) distant coarse-grained solvent volume elements.…”

“…IV, we summarize our results, leaving a more thorough discussion of the consequences on the level of the coarse-grained Langevin dynamics to Ref. [27].…”

“…where in (30) we used the divergence theorem and in (31) we defined the generalized dissipation tensor…”

“…Nevertheless, in most practical cases the thermal conductivities of particle and solvent (κ p and κ, respectively) will be such that the feedback of the particle motion onto the temperature field (which is of the maximum relative strength O(κ p /κ − 1) close to the particle surface) can be treated as a small correction to the overall temperature field in the solvent. If the particle itself acts as the heat source, like in hot Brownian motion [26,30], the temperature field can be calculated once and for all in the particle frame -advection terms arising from the change of frame can again be neglected. Therefore, Eqs (41), (42) and (43) alone can be taken to entirely describe the particle dynamics.…”