Systems in which the interparticle interactions are important present a major challenge in solid state physics. In this talk, I will discuss the non- equilibrium behavior of the 1D and 3D Hubbard models, as described within Time Dependent Density Functional Theory, TDDFT. The talk is divided in two parts:

In the first part, I will present time and space-resolved simulations of the density of trapped ultracold fermions; the out-of-equilibrium dynamics of the Mott insulator is found to differ profoundly from that of the band insulator and the metallic phase, displaying the self-induced stability of the Mott insulator.

In the second part, I will introduce a new method for obtaining exchange-correlation potentials for 3D lattices using Dynamical Mean Field Theory, DMFT. The potential obtained via DMFT is shown, within TDDFT, to be able to describe strongly correlated electrons in presence of time dependent fields, out of equilibrium.