Harmonic velocity noise: Non-Markovian features of noise-driven systems at long times

We propose a harmonic velocity noise with a broadband feature, which is the time derivative of the harmonic noise. If this noise is regarded as a thermal one, the system has a vanishing effective friction and it should induce ballistic diffusion of a free particle at long times. The effective temperature of the system coupled to such a structured heat bath represented by the harmonic velocity noise is introduced. This means that any initial preparation will approach asymptotically a preparation-dependent variance and mean value for velocity variable. Thus the fluctuation-dissipation theorem does not hold as there is no unique stationary state being connected with a breakdown of ergodicity. This noise can show greenness when it is taken as an external noise source to drive a correlation ratchet.