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

TL;DR

This paper investigates the anomalous diffusion of a harmonic oscillator influenced by Mittag-Leffler noise, deriving exact relaxation functions and revealing complex dynamics that differ from classical models due to noise's unique autocorrelation properties.

Contribution

It provides exact analytical expressions for relaxation functions of a harmonic oscillator driven by Mittag-Leffler noise, highlighting the noise's impact on anomalous diffusion behavior.

Findings

Oscillator exhibits anomalous diffusive behavior.

Short and intermediate times show qualitative differences from classical models.

Asymptotic behavior aligns with power-law autocorrelation noise.

Abstract

The diffusive behavior of a harmonic oscillator driven by a Mittag-Leffler noise is studied. Using 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.