The equilibrium and nonequilibrium quantum evolution of extended observables and entanglement properties are discussed in systems of Fermi gases/liquids of any dimension. In particular I focus on the entanglement entropy of extended space regions at equilibrium and off-equilibrium during the expansion after the drop of a trapping potential, which generally shows multiplicative logarithmic corrections to the leading power-law behaviors, analogously to the logarithmic corrections to the area law in lattice fermions.