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We present in detail a method we recently introduced (PRL. 103, 176404(2009)) to describe finite systems in and out of equilibrium, where theevolution in time is performed via the Kadanoff-Baym Equations (KBE) withinMany-Body Perturbation Theory (MBPT). The main property we analyze is thetime-dependent density. We also study is the exchange-correlation potential ofTDDFT, obtained via reverse engineering from the time-dependent density. Oursystems consist of small, strongly correlated clusters, described by a HubbardHamiltonian within the Hartree-Fock, second Born, GW and T-matrixapproximations. We compare the results from the KBE dynamics to those fromexact numerical solutions. The outcome of our comparisons is that, among themany-body schemes considered, the T-matrix approximation is overall superior atall electron densities. Such comparisons permit a general assessment of thewhole idea of applying MBPT, in the KBE sense, to finite systems. A strikingoutcome of our analysis is that when the system evolves under a strong externalfield, the KBE develop a steady-state solution as a consequence of acorrelation-induced damping. This damping is present both in isolated (finite)systems, where it is purely artificial, as well as in clusters contacted to(infinite) macroscopic leads. To illustrate this point we present selectedresults for a system coupled to contacts within the T-matrix and second Bornapproximation. The extensive characterization we performed indicates that thisbehavior is present whenever approximate self energies, based upon infinitepartial summations, are used. A second important result is that, for isolatedclusters, the steady state reached is not unique but depends on how oneswitches on the external field. This may also true for clusters connected toleads.

Kadanoff-Baym description of Hubbard clusters out of equilibrium:  performance of many-body schemes, correlation-induced damping and multiple  steady states