A Geometric Approach to the Landauer-B\"uttiker Formula

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.