Temperature and voltage measurement in quantum systems far from equilibrium

TL;DR

This paper establishes the uniqueness and conditions for local temperature and voltage measurements in far-from-equilibrium quantum systems, providing a rigorous foundation for thermodynamic measurements in such complex states.

Contribution

It introduces a unified, noninvasive measurement framework for temperature and voltage in quantum systems far from equilibrium, with conditions for their existence and uniqueness.

Findings

Unique temperature and voltage measurement when they exist.

Positive temperature solutions exist without population inversion.

Negative temperature states can be characterized when population inversion occurs.

Abstract

We show that a local measurement of temperature and voltage for a quantum system in steady state, arbitrarily far from equilibrium, with arbitrary interactions within the system, is unique when it exists. This is interpreted as a consequence of the second law of thermodynamics. We further derive a necessary and sufficient condition for the existence of a solution. In this regard, we find that a positive temperature solution exists whenever there is no net population inversion. However, when there is a net population inversion, we may characterize the system with a unique negative temperature. Voltage and temperature measurements are treated on an equal footing: They are simultaneously measured in a noninvasive manner, via a weakly-coupled thermoelectric probe, defined by requiring vanishing charge and heat dissipation into the probe. Our results strongly suggest that a local temperature measurement without a simultaneous local voltage measurement, or vice-versa, is a misleading characterization of the state of a nonequilibrium quantum electron system. These results provide a firm mathematical foundation for voltage and temperature measurements far from equilibrium.