The Kubo–Martin–Schwinger (KMS) condition is a widely studied fundamental property in quantum statistical mechanics which characterizes the thermal equilibrium states of quantum systems. In the seventies, Gallavotti and Verboven, proposed an analogue to the KMS condition for infinite classical mechanical systems and highlighted its relationship with the Kirkwood–Salzburg equations and with the Gibbs equilibrium measures. In this paper, we prove that in a certain limiting regime of high temperature the classical KMS condition can be derived from the quantum condition in the simple case of the Bose–Hubbard dynamical system on a finite graph. The main ingredients of the proof are Golden–Thompson inequality, Bogoliubov inequality and semiclassical analysis.
We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable
H∈Md(C) with real spectrum, the biorthogonal Gibbs functional
ωbi(A)=Zbi−1∑ne−βEn⟨φn∣A∣ψn⟩ satisfies
ωbi(A†A)≥0 for all
A if and only if
H is quasi-Hermitian. The proof reconstructs the metric
η directly from the eigenprojectors of
ωbi via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the
η-Gibbs state
ωη(A)=Zη−1Tr[ηe−βHA] satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever
[η,h]=0, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the
η-Gibbs state in finite dimensions, while the construction of a compatible
C∗-norm and the proof of
σ-weak continuity remain open.
We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.