  We demonstrate the approach towards a Gibbs-like equilibrium state, with a
common temperature and a chemical potential, of two finite metallic grains,
prepared with a different number of noninteracting electrons, connected by a
weak link that is susceptible to incoherent and inelastic processes. By
developing an analytic method and by using an exact numerical approach, the
quantum time evolution of the electrons in the metallic grains is followed. In
the absence of decoherring and inelastic effects, equilibration is never
reached. Introducing dephasing effects on the link only, using a dephasing
probe, the two quantum systems equilibrate, but do not evolve towards a
Gibbs-like state. In contrast, by mimicking inelastic interactions with a
voltage probe, the metal pieces evolve towards a common Gibbs-like equilibrium
state, with the probe.
