# Fractional Langevin equation from damped bath dynamics

## TL;DR

This paper derives a fractional Langevin equation from a hierarchical bath model with slow and fast components, showing how power-law memory effects emerge based on spectral properties.

## Contribution

It introduces a model combining explicit slow bath dynamics with fast bath effects, demonstrating the emergence of fractional Langevin equations from hierarchical thermal baths.

## Key findings

- Fractional Langevin equation with power-law memory kernel emerges.
- The fractional exponent depends on low-frequency spectral properties.
- The model bridges microscopic bath dynamics with macroscopic anomalous diffusion.

## Abstract

We consider the stochastic dynamics of a system linearly coupled to a hierarchical thermal bath with two well-separated inherent timescales: one slow, and one fast. The slow part of the bath is modeled as a set of harmonic oscillators and taken into account explicitly, while the effects of the fast part of the bath are simulated by dissipative and stochastic Langevin forces, uncorrelated in space and time, acting on oscillators of the slow part of the bath. We demonstrate for this model the robust emergence of a fractional Langevin equation with a power-law decaying memory kernel. The conditions of such an emergence and the specific value of the fractional exponent depend only on the asymptotic low-frequency spectral properties of the slow part of the bath.