  The Microbunched Electron Cooling (MBEC) is a promising cooling technique
that can find applications in future hadron and electron-ion colliders to
counteract intrabeam scattering that limits the maximum achievable luminosity
of the collider. To minimize the cooling time, one would use amplification
cascades consisting of a drift section followed by a magnetic chicane. In this
paper, we first derive and optimize the gain factor in an amplification section
for a simplified one-dimensional model of the beam. We then deduce the cooling
rate of a system with one and two amplification cascades. We also analyze the
noise effects that counteract the cooling process through the energy diffusion
in the hadron beam. Our analytical formulas are confirmed by numerical
simulations for a set of model parameters.
