Dissipation in a partially coherent flux-driven ring

We have studied a mesoscopic ring threaded by a magnetic flux that increases linearly with time. The ring is partially coherent such that conduction electrons in the ring will encounter incoherent scatterings. We have treated both the incoherent scatterings and the coherent inelastic processes on the same footing. A S-matrix model, as proposed by B\"uttiker12 for incoherent scatterings, has been adopted for our situation. This allows us to solve exactly, and analytically, the coherent inelastic processes caused by the time-varying magnetic flux. Our results demonstrate unequivocally that, for the electrons emanating out of incoherent scatterings, the lower the energies of these electrons the greater will be their net contribution to the dc component of the induced current. A physical explanation is presented.