Quantum pumping: The charge transported due to a translation of a scatterer

The amount of charge which is pushed by a moving scatterer is $dQ = -G dX$, 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 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.