Concluding Remarks

Abstract

In the present book, we presented a number of interesting results pertaining to steady-state quantum fermion systems. In Chap. 2 we defined a mathematically rigorous notion of temperature and voltage for quantum systems arbitrarily far from equilibrium and having arbitrary interactions within the system. We showed that a meaningful notion of temperature and voltage for nonequilibrium systems requires the simultaneous measurement of both. This joint measurement requires that the probe be in both electrical and thermal equilibrium with the nonequilibrium system of interest. We established the notion of an ideal probe as one that operates in the broadband limit and is weakly coupled to the system of interest. The results obtained here have a deep connection with the second law of thermodynamics: We proved the uniqueness of the probe measurement and showed its close relation to the Onsager’s statement of the second law of thermodynamics (which we also proved for the case of quantum thermoelectric transport). We derived necessary and sufficient conditions for the existence of a solution and found that a solution always exists, and that one may encounter negative temperature solutions if the system is driven sufficiently far away from equilibrium. We developed also the notion of entropy for steady-state noninteracting systems by first providing an expression for the exact entropy in terms of the scattering states. We then formulated the entropy inferred by a local observer with varying amounts of information regarding the system and showed that they obey a hierarchy of inequalities. The entropy was seen to be inversely related to the available information, with the most knowledgable formulation leading to the least entropy. We showed the validity of the third law of thermodynamics for open quantum systems in equilibrium and in nonequilibrium steady states. We also proposed a novel experimental method, whose working principle rests our theoretical findings, to enhance the existing spatial resolution of scanning thermal measurements by over two orders of magnitude. In addition to their theoretical merit we believe these results will have significant practical impact, for example, in characterizing nonequilibrium device performances.