Geometry, statistics, and asymptotics of quantum pumps

We give a pedestrian derivation of a formula of Büttiker, Pretre, and Thomas [Z. Phys. B 94, 133 (1994)] (BPT) relating the adiabatically pumped current to the S-matrix and its (time) derivatives. We relate the charge in BPT to Berry's phase and the corresponding Brouwer pumping formula to curvature. As applications we derive explicit formulas for the joint probability density of pumping and conductance when the S-matrix is uniformly distributed; and derive a formula that describes hard pumping when the S-matrix is periodic in the driving parameters.