  We investigate the entanglement properties of the equilibrium and
nonequilibrium quantum dynamics of 2D and 3D Fermi gases, by computing
entanglement entropies of extended space regions, which generally show
multiplicative logarithmic corrections to the leading power-law behaviors,
corresponding to the logarithmic corrections to the area law.
  We consider 2D and 3D Fermi gases of N particles constrained within a limited
space region, for example by a hard-wall trap, at equilibrium at T=0, i.e. in
their ground state, and compute the first few terms of the asymptotic large-N
behaviors of entanglement entropies and particle fluctuations of subsystems
with some convenient geometries, which allow us to significantly extend their
computation. Then, we consider their nonequilibrium dynamics after
instantaneously dropping the hard-wall trap, which allows the gas to expand
freely. We compute the time dependence of the von Neumann entanglement entropy
of space regions around the original trap. We show that at small time it is
characterized by the relation $S \approx \pi^2 V/3$ with the particle variance,
and multiplicative logarithmic corrections to the leading power law, i.e. $S
\sim t^{1-d}\ln(1/t)$.
