Anomalous diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise

The diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise is studied. Using the Laplace analysis we derive exact expressions for the relaxation functions of the particle in terms of generalized Mittag-Leffler functions and its derivatives from a generalized Langevin equation. Our results show that the oscillator displays an anomalous diffusive behavior. In the strictly asymptotic limit, the dynamics of the harmonic oscillator corresponds to an oscillator driven by a noise with a pure power-law autocorrelation function. However, at short and intermediate times the dynamics has qualitative difference due to the presence of the characteristic time of the noise.

Figure 1

(Color online) Relaxation function $I\left(t\right)$ vs time $t$, for $\lambda =1/2$, ${\gamma }_{\lambda }=1$, $\tau =1$, and $\omega =1$. The solid line corresponds to the exact expression (30) (for a Mittag-Leffler noise); the dashed line corresponds to the exact expression (40) (for a pure power-law noise).

Figure 2

(Color online) Relaxation function $I\left(t\right)$ vs time $t$, for $\lambda =3/2$, ${\gamma }_{\lambda }=1$, $\tau =1$, and $\omega =1$. The solid line corresponds to the exact expression (30) (for a Mittag-Leffler noise); the dashed line corresponds to the exact expression (40) (for a pure power-law noise).

Figure 3

(Color online) Relaxation function $g\left(t\right)$ vs time $t$, for $\lambda =1/2$, ${\gamma }_{\lambda }=1$, $\tau =1$, and $\omega =1$. The solid line corresponds to the exact expression (35) (for a Mittag-Leffler noise); the dashed line corresponds to the exact expression (42) (for a pure power-law noise).

Figure 4

(Color online) Relaxation function $g\left(t\right)$ vs time $t$, for $\lambda =3/2$, ${\gamma }_{\lambda }=1$, $\tau =1$, and $\omega =1$. The solid line corresponds to the exact expression (35) (for a Mittag-Leffler noise); the dashed line corresponds to the exact expression (42) (for a pure power-law noise).