Underdamped stochastic harmonic oscillator driven by Lévy noise

We investigate the distribution of potential and kinetic energy in stationary states of the linearly damped stochastic oscillator driven by Lévy noises. In the long time limit distributions of kinetic and potential energies of the oscillator follow the power-law asymptotics and do not fulfill the equipartition theorem. The partition of the mechanical energy is controlled by the damping coefficient. In the limit of vanishing damping a stochastic analog of the equipartition theorem can be proposed, namely, the statistical properties of potential and kinetic energies attain distributions characterized by the same widths. For larger damping coefficient the larger fraction of energy is stored in its potential form. In the limit of very strong damping the contribution of kinetic energy becomes negligible. Finally, we demonstrate that the ratio of instantaneous kinetic and potential energies, which signifies departure from the mechanical energy equipartition, follows universal power-law asymptotics, regardless of the symmetric $\alpha$-stable noise parameters. Altogether our investigations clearly indicate strongly nonequilibrium character of Lévy-stable fluctuations with the stability index $\alpha <2$.

Figure 1

The ratio of distributions widths $D=\frac{w_v^{\alpha }}{w_x^{\alpha }}$ for various values of the stability index $\alpha$. Note the double logarithmic scale in the main plot and the linear scale in the inset.

Figure 2

Distributions of the potential ${\mathcal{E}}_p$ (full symbols) and kinetic ${\mathcal{E}}_k$ (empty symbols) energies for $\gamma =1,2,10$ (from top to bottom). Various curves correspond to various values of the stability index $\alpha$.

Figure 3

Distribution of the energy ratio $r={\mathcal{E}}_k/{\mathcal{E}}_p$ for $\gamma =1,2,10$ (from top to bottom).