  We consider an ideal Fermi gas confined to a geometric structure consisting
of a central region -- the sample -- connected to several infinitely extended
ends -- the reservoirs. Under physically reasonable assumptions on the
propagation properties of the one-particle dynamics within these reservoirs, we
show that the state of the Fermi gas relaxes to a steady state. We compute the
expected value of various current observables in this steady state and express
the result in terms of scattering data, thus obtaining a geometric version of
the celebrated Landauer-B\"uttiker formula.
