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

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 importantly find the solution of the joint probability density by using double Fourier transforms in three-time domains, and the moments from the derived moment equation are numerically calculated. As a result, the dynamical motion of a passive particle for the probability density having two variables of the displacement and the velocity in the short-time domain takes the super-diffusive form, whereas the distribution in the long-time domain is attained to be Gaussian by analyzing only from the velocity probability density. The moment${\mu }_{2,2}$ particularly scales as $\sim t^5$ in the Navier-Stokes equation for $\tau =0$ ($\tau =$correlation time), while the moment in Burgers equation only with correlated Gaussian force reduces to ${\mu }_{2,2}\sim t^4$ in $t>>\tau$ and$\tau =0$.