Inhomogeneity of the phase space of the damped harmonic oscillator under Levy noise

TL;DR

This paper investigates how symmetric Lévy white noise causes inhomogeneous phase space in a damped harmonic oscillator, revealing the combined effects of damping and heavy-tailed noise on the system's dynamics.

Contribution

It uncovers the physical origin of phase space inhomogeneity under Lévy noise, highlighting the roles of damping and heavy tails in the noise distribution.

Findings

Phase space becomes inhomogeneous under Lévy noise, unlike Gaussian noise.

Inhomogeneity arises from the interplay of damping and heavy-tailed noise.

Coordinate and velocity show anti-association due to these effects.

Abstract

The damped harmonic oscillator under symmetric L\'{e}vy white noise shows inhomogeneous phase space, which is in contrast to the homogeneous one of the same oscillator under the Gaussian white noise, as shown in a recent paper [I. M. Sokolov, W. Ebeling, and B. Dybiec, Phys. Rev. E \textbf{83}, 041118 (2011)]. The inhomogeneity of the phase space shows certain correlation between the coordinate and the velocity of the damped oscillator under symmetric L\'{e}vy white noise. In the present work we further explore the physical origin of these distinguished features and find that it is due to the combination of the damped effect and heavy tail of the noise. We demonstrate directly this in the reduced coordinate $\tilde{x}$ versus velocity $\tilde{v}$ plots and identify the physics of the anti-association of the coordinate and velocity.