Exact time-average distribution for a stationary non-Markovian massive Brownian particle coupled to two heat baths

TL;DR

This paper derives an exact probability distribution for a Brownian particle influenced by two heat baths at different temperatures, incorporating inertial effects and contributions from both fast and slow noise components.

Contribution

It provides an exact analytical solution for the position and velocity distribution of a non-Markovian Brownian particle with inertial effects coupled to two heat baths.

Findings

Exact distribution for position and velocity obtained

Includes contributions from both fast and slow noise

Accounts for inertial effects beyond over-damped approximation

Abstract

Using a time-averaging technique we obtain exactly the probability distribution for position and velocity of a Brownian particle under the influence of two heat baths at different temperatures. These baths are expressed by a white noise term, representing the fast dynamics, and a colored noise term, representing the slow dynamics. Our exact solution scheme accounts for inertial effects, that are not present in approaches that assume the Brownian particle in the over-damped limit. We are also able to obtain the contribution associated with the fast noise that are usually neglected by other approaches.