Gibbs state postulate from dynamical stability -- Redundancy of the zeroth law
Vjosa Blakaj, Matthias C. Caro, Anouar Kouraich, Daniel Malz, Michael M. Wolf
TL;DR
Gibbs states are the standard description of quantum thermal equilibrium, yet their dynamical justification remains debated. The authors prove that any stationary state stable of order two under arbitrary weak coupling to an environment consisting of harmonic oscillators must be Gibbs, i.e. $\rho=\rho(β)=e^{-β H}/Tr[e^{-β H}]$. This shows the zeroth-law assumption is redundant and that a bosonic bath suffices to single out the canonical form, linking dynamical stability directly to equilibrium. Moreover, while a three-mode oscillator bath suffices, the appendix notes that under commensurable energy gaps a single-mode bath can suffice, clarifying the minimal dynamical requirements for the canonical ensemble.
Abstract
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.