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We demonstrate the approach towards a Gibbs-like equilibrium state, with acommon temperature and a chemical potential, of two finite metallic grains,prepared with a different number of noninteracting electrons, connected by aweak link that is susceptible to incoherent and inelastic processes. Bydeveloping an analytic method and by using an exact numerical approach, thequantum time evolution of the electrons in the metallic grains is followed. Inthe absence of decoherring and inelastic effects, equilibration is neverreached. Introducing dephasing effects on the link only, using a dephasingprobe, the two quantum systems equilibrate, but do not evolve towards aGibbs-like state. In contrast, by mimicking inelastic interactions with avoltage probe, the metal pieces evolve towards a common Gibbs-like equilibriumstate, with the probe.

Towards equilibration and thermalization between finite quantum systems: The role of dephasing effects and inelastic interactions