Brownian Dynamics, Time-averaging and Colored Noise

TL;DR

This paper introduces a method combining time-averaging and Laplace transforms to derive equilibrium distributions for particles under various noise conditions, including colored noise, with exact solutions demonstrated for a damped harmonic oscillator.

Contribution

It presents a novel analytical approach to determine equilibrium states for systems influenced by Gaussian colored noise, extending beyond traditional white noise assumptions.

Findings

Exact equilibrium distribution for a damped harmonic oscillator with colored noise

Method applicable to Gaussian noise exponentially correlated in time

Insights into properties of the equilibrium solution

Abstract

We propose a method to obtain the equilibrium distribution for positions and velocities of a one-dimensional particle via time-averaging and Laplace transformations. We apply it to the case of a damped harmonic oscillator in contact with a thermal bath. The present method allows us to treat, among other cases, a Gaussian noise function exponentially correlated in time, e.g., Gaussian colored noise. We obtain the exact equilibrium solution and study some of its properties.