Generalized dynamics and fluctuation-dissipation theorem for a parabolic potential

This article revisits the fluctuation-dissipation relationship of a generalized Brownian particle constrained in a harmonic potential and immersed in a thermal bath whose degrees of freedom also interact with the external field. The generalized Langevin equation put forth herein keeps its original form. The different terms now depend on a compound strength related to those of the field and bath oscillators. The resulting theorem keeps Kubo's original form with the memory kernel depending on the frequency of the field. It has the consequence that friction is not a constant but is a function of the field frequency as predicted by molecular dynamics simulations. The position particle distribution and the Fokker-Planck equation associated with the generalized Langevin equation are also derived.