  In the independent electron approximation, the average
(energy/charge/entropy) current flowing through a finite sample S connected to
two electronic reservoirs can be computed by scattering theoretic arguments
which lead to the famous Landauer-B\"uttiker formula. Another well known
formula has been proposed by Thouless on the basis of a scaling argument. The
Thouless formula relates the conductance of the sample to the width of the
spectral bands of the infinite crystal obtained by periodic juxtaposition of S.
In this spirit, we define Landauer-B\"uttiker crystalline currents by extending
the Landauer-B\"uttiker formula to a setup where the sample S is replaced by a
periodic structure whose unit cell is S. We argue that these crystalline
currents are closely related to the Thouless currents. For example, the
crystalline heat current is bounded above by the Thouless heat current, and
this bound saturates iff the coupling between the reservoirs and the sample is
reflectionless. Our analysis leads to a rigorous derivation of the Thouless
formula from the first principles of quantum statistical mechanics.
