The effective temperature for the thermal fluctuations in hot Brownian motion

TL;DR

This paper investigates the effective temperature in hot Brownian motion, deriving exact and approximate formulas for different temperature measures using fluctuating hydrodynamics and extending previous models with frequency-dependent corrections.

Contribution

It provides a detailed calculation of the effective temperature in hot Brownian motion, including first-order frequency corrections and analytical expressions in various limits, advancing the theoretical understanding of non-equilibrium thermal fluctuations.

Findings

Effective temperature can be exactly calculated using fluctuating hydrodynamics.

Different degrees of freedom have distinct effective temperatures, with configurational temperature predicted accurately.

Analytical formulas for effective temperature in small and high frequency limits are derived.

Abstract

We revisit the effective parameter description of hot Brownian motion -- a scenario where a colloidal particle is kept at an elevated temperature than the ambient fluid. Due to the time scale separation between heat diffusion and particle motion, a stationary halo of hot fluid is carried along with the particle, resulting in a spatially varying comoving temperature and viscosity profile. The resultant Brownian motion in the overdamped limit can be well described by a Langevin equation with effective parameters such as effective temperature $T_{\rm HBM}$ and friction coefficient $\zeta_{\rm HBM}$ that quantifies the thermal fluctuations and the diffusivity of the particle. These parameters can exactly be calculated using the framework of fluctuating hydrodynamics. Additionally, it was also observed that configurational and the kinetic degrees of freedom admits to different effective temperatures, $T^{\mathbf{x}}_{\rm HBM}$ and $T^{\mathbf{v}}_{\rm HBM}$, respectively, with the former predicted accurately from fluctuating hydrodynamics. A more rigorous calculation by Falasco et. al. Physical Review E , 90, $032131(2014)$ extends the overdamped description to a generalized Langevin equation where the effective temperature becomes frequency dependent and consequently, for any temperature measurement from a Brownian trajectory requires the knowledge of this frequency dependence. We use this framework to expand on this earlier work and look at the first order correction to the effective temperature. The effective temperature is calculated from the weighted average of the temperature field with the dissipation function. Further, we provide a closed form analytical result for effective temperature in the small as well high frequency limit and using this we determine the kinetic temperature from the generalized Langevin equation and the Wiener-Khinchine theorem.