Dynamical behavior of passive particles with harmonic, viscous, and correlated Gaussian forces

TL;DR

This paper investigates the stochastic dynamics of passive particles under harmonic, viscous, and correlated Gaussian forces, deriving probability densities and analyzing their short- and long-time behaviors.

Contribution

It introduces a method to derive the joint probability density for passive particles influenced by correlated Gaussian forces using Fourier transforms.

Findings

Short-time motion exhibits super-diffusive behavior.

Long-time distribution becomes Gaussian.

Numerical moments are calculated from derived equations.

Abstract

In this paper, we study the Navier-Stokes equation and the Burgers equation for the dynamical motion of a passive particle with harmonic and viscous forces, subject to an exponentially correlated Gaussian force. As deriving the Fokker-Planck equation for the joint probability density of a passive particle, we find obviously the important solution of the joint probability density by using double Fourier transforms in three-time domains, and the moments from derived moment equation are numerically calculated. As a result, the dynamical motion of a passive particle with respect to the probability density having two variables of displacement and velocity in the short-time domain has a super-diffusive form, whereas the distribution in the long-time domain is obtained to be Gaussian by analyzing only from the velocity probability density.