Fractional Langevin equation from damped bath dynamics

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.