  A theory of local temperature measurement of an interacting quantum electron system far from equilibrium via a floating thermoelectric probe is developed. It is shown that the local temperature so defined is consistent with the zeroth, first, second, and third laws of thermodynamics, provided the probe-system coupling is weak and broad band. For non-broad-band probes, the local temperature obeys the Clausius form of the second law and the third law exactly, but there are corrections to the zeroth and first laws that are higher-order in the Sommerfeld expansion. The corrections to the zeroth and first laws are related, and can be interpreted in terms of the error of a nonideal temperature measurement. These results also hold for systems at negative absolute temperature. 
Local temperature of an interacting quantum system far from equilibrium