Local and effective temperatures of quantum driven systems

We introduce thermometers to define the local temperature of an electronic system driven out of equilibrium by local ac fields. We also define the effective temperature in terms of a local fluctuation-dissipation relation. We show that within the weak driving regime these two temperatures coincide. We also discuss the behavior of the local temperature along the sample. We show that it exhibits spatial fluctuations following an oscillatory pattern. For weak driving, regions of the sample become heated, while others become cooled as a consequence of the driving.