  Pumping of charge (Q) in a closed ring geometry is not quantized even in the
strict adiabatic limit. The deviation form exact quantization can be related to
the Thouless conductance. We use Kubo formalism as a starting point for the
calculation of both the dissipative and the adiabatic contributions to Q. As an
application we bring examples for classical dissipative pumping, classical
adiabatic pumping, and in particular we make an explicit calculation for
quantum pumping in case of the simplest pumping device, which is a 3 site
lattice model.

Mesoscale and Nanoscale Physics