On the motion of passive and active particles with harmonic and viscous forces

In this paper, we solve the joint probability density for the passive and active particles with harmonic, viscous, and perturbative forces. After deriving the Fokker-Planck equation for a passive and a run-and-tumble particles, we approximately get and analyze the solution for the joint distribution density subject to an exponential correlated Gaussian force in three kinds of time limit domains. Mean squared displacement (velocity) for a particle with harmonic and viscous forces behaviors in the form of super-diffusion, consistent with a particle having viscous and perturbative forces. A passive particle with both harmonic, viscous forces and viscous, perturbative forces has the Gaussian form with mean squared velocity ~t. Particularly, In our case of a run-and-tumble particle, the mean squared displacement scales as super-diffusion, while the mean squared velocity has a normal diffusive form.In addition, the kurtosis, the correlation coefficient, and the moment from moment equation are numerically calculated.