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Lecture notes in mathematics

Quantum gibbs states and the zeroth law of thermodynamics

Vittorio Gorini, Alberto Frigerio, Maurizio Verri

Published Jan 1, 1985• 0 citations
Publisher
DOI
10.1007/bfb0074476

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Ukrainian Journal of Physics•Aug 1, 2013•1 citation
Gibbs states play a central role in quantum statistical mechanics as the standard description of thermal equilibrium. Traditionally, their use is justified either by a heuristic, a posteriori reasoning, or by derivations based on notions of typicality or passivity. In this work, we show that Gibbs states are completely characterized by assuming dynamical stability of the system itself and of the system in weak contact with an arbitrary environment. This builds on and strengthens a result by Frigerio, Gorini, and Verri (1986), who derived Gibbs states from dynamical stability using an additional assumption that they referred to as the "zeroth law of thermodynamics", as it concerns a nested dynamical stability of a triple of systems. We prove that this "zeroth law" is redundant and that an environment consisting solely of harmonic oscillators is sufficient to single out Gibbs states as the only dynamically stable states.
The presented approach to incorporate the stochastic thermodynamics into the quantum theory is based on the idea, proposed earlier by the authors, to consistently consider the stochastic influence by the environment considered as the whole and described by the wave functions of arbitrary vacua. In this research, a possibility of the explicit incorporation of the zeroth law of stochastic thermodynamics into the quantum-mechanical theory in the form of the saturated Schr¨odinger uncertainty relation is realized. This allows a comparative analysis between the sets of arbitrary vacuum states, namely, squeezed coherent (SCSs) and correlated coherent (CCSs) states, to be carried out. A possibility to establish a relation between SCSs and CCSs, on the one hand, and thermal states, on the other hand, is discussed.