# Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential

## TL;DR

This paper develops a theoretical framework to analyze the relaxation dynamics of various anomalous diffusion processes confined in harmonic potentials, linking mean square displacements to velocity correlation functions.

## Contribution

It introduces a general theory connecting relaxation behavior of confined anomalous diffusion to the velocity correlation function, applicable to multiple models.

## Key findings

- Derived explicit expressions for stationary and relaxation behaviors.
- Applicable to fractional Brownian motion, scaled Brownian motion, and Lévy walks.
- Provides a unified approach to analyze confined anomalous diffusion processes.

## Abstract

Optical tweezers setup is often used to probe the motion of individual tracer particle, which promotes the study of relaxation dynamics of a generic process confined in a harmonic potential. We uncover the dependence of ensemble- and time-averaged mean square displacements of confined processes on the velocity correlation function $C(t,t+\tau)$ of the original process. With two different scaling forms of $C(t,t+\tau)$ for small $\tau$ and large $\tau$, the stationary value and the relaxation behaviors can be obtained immediately. The gotten results are valid for a large amount of anomalous diffusion processes, including fractional Brownian motion, scaled Brownian motion, and the multi-scale L\'{e}vy walk with different exponents of running time distribution.