  We consider a non-interacting Fermi gas in $d$ dimensions, both in the
non-relativistic and relativistic case. The system of size $L^{d}$ is initially
prepared into two halves $\mathcal{L}$ and $\mathcal{R}$, each of them
thermalized at two different temperatures, $T_{\mathcal{L}}$ and
$T_{\mathcal{R}}$ respectively. At time $t=0$ the two halves are put in contact
and the entire system is left to evolve unitarily. We show that, in the
thermodynamic limit, the time evolution of the particle and energy densities is
perfectly described by a semiclassical approach which permits to analytically
evaluate the correspondent stationary currents. In particular, in the case of
non-relativistic fermions, we find a low-temperature behavior for the particle
and energy currents which is independent from the dimensionality $d$ of the
system, being proportional to the difference
$T_{\mathcal{L}}^{2}-T_{\mathcal{R}}^{2}$. Only in one spatial dimension
($d=1$), the results for the non-relativistic case agree with the massless
relativistic ones.
