The Kubo-Martin-Schwinger boundary condition, which had first been introduced as a property characterizing the canonical ensemble average in quantum statistical mechanics, was later recoginized as a property characterizing the modular automorphisms in the so-called Tomita-Takesaki theory of von Neumann algebras. This miraculous link made it possible to apply powerful mathematical technique supplied by the Tomita-Takesaki theory to a study of some problems in mathematical formulation of statistical mechanics. More recently, the KMS condition in classical statistical mechanics has also been investigated. The aim of this exposition is to review for non-specialists those development of the “rigorous” treatment of statistical mechanics which are related to the KMS condition.
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.
Thermalization of a system when interacting with a thermal bath is an interesting problem. If a system eventually reaches a thermal state in the long time limit, it's expected that its density matrix would resemble the mean-force Gibbs state. Moreover, the correlation function must satisfy the Kubo-Martin-Schwinger (KMS) condition, or equivalently, the fluctuation-dissipation relation (FDR). In this paper, we derive a formal expression for the non-Markovian two-point function within the context of the weak coupling limit. Using this expression, we explicitly compute the two-point function for specific models, demonstrating their adherence to the KMS. In addition, we formulate a nonperturbative approach in the form of a self-consistent approximation that includes a partial resummation of perturbation theory. This approach can capture strong coupling phenomena while still relying on simple equations. Notably, we verify that the two-point function obtained through this method also satisfies the KMS condition.
Thermalization of a system when interacting with a thermal bath is an interesting problem. If a system eventually reaches a thermal state in the long time limit, it's expected that its density matrix would resemble the mean-force Gibbs state. Moreover, the correlation function must satisfy the Kubo-Martin-Schwinger (KMS) condition or equivalently the Fluctuation-Dissipation Relation (FDR). In this paper, we derive a formal expression for the non-Markovian two-point function within the context of the weak coupling limit. Using this expression, we explicitly compute the two-point function for specific models, demonstrating their adherence to the KMS. In addition, we have formulated a non-perturbative approach in the form of a self-consistent approximation that includes a partial resummation of perturbation theory. This approach can capture strong coupling phenomena while still relying on simple equations. Notably, we verify that the two-point function obtained through this method also satisfies the KMS condition.
We construct a set of equilibrium states for the ideal Bose gas at temperature β−1 and chemical potential μ. Our choice for the equilibrium states is based on the Kubo−Martin−Schwinger conditions. In particular, we define ℰβ,μ to be the set of states which satisfy the KMS conditions suitably defined. In the condensed phase a certain subset Δβ,μ of ℰβ,μ correspond to states whose mean local densities are not uniformly bounded with respect to the volume. We, therefore, propose that ℰβ,μ Δβ,μ ≡ ?β,μ corresponds to the set of equilibrium states. Then it is shown that ?β,μ contains, as special cases, the equilibrium states obtained via thermodynamical limit arguments by Araki−Woods and Lewis−Pulé. The extremal elements of the convex set ℰβ,μ are obtained explicitly.
Abstract We show that the long-time limit of the two-point correlation function obtained via the standard quantum regression theorem (QRT), a standard tool to compute correlation functions in open quantum systems, does not respect the Kubo–Martin–Schwinger equilibrium condition to the non-zero order of the system-bath coupling. We then follow the recently developed Heisenberg operator method for open quantum systems and by applying a ‘weak’ Markov approximation, derive a new modified version of the QRT that not only respects the KMS condition but further predicts exact answers for certain paradigmatic models in specific limits. We also show that in cases where the modified QRT does not match with exact answers, it always performs better than the standard QRT.