temperature limit it yields Here is a projector that restrict the trace operations to the open channels of the lead where the current is measured. This BPT formula has been originally derived using a scattering approach, but later its relation to the Kubo formula has been worked out. The effect of interactions A very recent work considers the role of interactions in the stirring of Bose condensed particles. Otherwise the rest of the literature concerns primarily electronic devices. Typically the pump is modeled as a quantum dot. The effect of electron–electron interactions within the dot region is taken into account in the Coulomb blockade regime or in the Kondo regime. In the former case charge transport is quantized even in the case of small backscattering. Deviation from the exact quantized value is related to dissipation. In the Kondo regime, as the temperature is lowered, the pumping effect is modified. There are also works that consider interactions over the whole system (including the leads) using the Luttinger liquid model. Quantum pumping in deformable mesoscopic systems A quantum pump, when coupled to classical mechanical degrees of freedom, may also induce cyclic variations of the mechanical degrees of freedom coupled to it. In such a configuration, the pump works similarly to an Adiabatic quantum motor. A paradigmatic example of this class of systems is a quantum pump coupled to an elastically deformable quantum dot. The mentioned paradigm has been generalized to include non-linear effects and stochastic fluctuations. Non-periodic quantum pumping In most of the proposed examples of quantum pumping, there are one or more parameters that vary cyclically over time. However, it has been theoretically showed that the damped oscillations of a set of parameters can also be used to generated non-vanishing pumped charges at long times. The phenomenon is named geometric rectification of the vibrationally-induced currents. The quantity of interest in such a case is the asymptotic pumped charge, i.e. the total charge pumped from or to a reservoir at infinite time, instead of the instantaneous current, which changes its sign quasi-periodically, or the pumped charge per cycle, a quantity that loses its meaning due to the lack of defined cycles. Here, is the charge of the electron, is the asymptotic pumped charge from the reservoir , where ``asymptotic'' refers to the long-time limit of the pumped charge , i.e. . The modes of the mechanical part of the system are labeled