Quantum pumping: the charge transported due to a translation of a scatterer.

The amount of charge that is pushed by a moving scatterer is dQ= -GdX , where dX is the displacement of the scatterer. The question is: what is G ?. Does it depend on the transmission g(0) of the scatterer? Does the answer depend on whether the system is open (with leads attached to reservoirs) or closed? In the latter case what are the implications of having "quantum chaos" and/or coupling to the environment? The answers to these questions illuminate some fundamental aspects of the theory of quantum pumping. For the analysis we take a network (graph) as a model system, and use the Kubo formula approach.