It is rather paradoxical that, although entropy is one of the most important quantities in physics, its main properties are rarely listed in the usual textbooks on statistical mechanics. In this paper we try to fill this gap by discussing these properties, as, for instance, invariance, additivity, concavity, subadditivity, strong subadditivity, continuity, etc., in detail, with reference to their implications in statistical mechanics. In addition, we consider related concepts such as relative entropy, skew entropy, dynamical entropy, etc. Taking into account that statistical mechanics deals with large, essentially infinite systems, we finally will get a glimpse of systems with infinitely many degrees of freedom.

General properties of entropy

It has been $30\phantom{\rule{0.3em}{0ex}}\text{years}$ since the discovery of the Unruh effect. It has played a crucial role in our understanding that the particle content of a field theory is observer dependent. This effect is important in its own right and as a way to understand the phenomenon of particle emission from black holes and cosmological horizons. The Unruh effect is reviewed here with particular emphasis on its applications. A number of recent developments are also commented on and some controversies are discussed. Effort is also made to clarify what seem to be common misconceptions.

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The Unruh effect and its applications

Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon

Real- and imaginary-time field theory at finite temperature and density

Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

Representations of theC*-algebra$\mathfrak{A}$ of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential μ are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of$\mathfrak{A}$ onto its commutant. This means that there is an equivalent anti-linear representation of$\mathfrak{A}$ in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.

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On the equilibrium states in quantum statistical mechanics

It is shown that K.M.S.-states are locally normal on a great number ofC*-algebras that may be of interest in Quantum Statistical Mechanics. The lattice structure and the Choquet-simplex structure of various sets of states are investigated. In this respect special attention is payed to the interplay of the K.M.S.-automorphism group with other automorphism groups under whose action K.M.S.-states are possibly invariant. A seemingly weaker notion thanG-abelianness of the algebra of observables, namelyG′-abelianness, is introduced and investigated. Finally a necessary and sufficient condition (on aC*-algebra with a sequential separable factor funnel) for decomposition of a locally normal state into locally normal states is given.

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Local normality in quantum statistical mechanics

We investigate possible ways to generalize the concept of Gibbs states. For classical lattice systems we do so by modifying the configuration space and considering the continuity of conditional probabilities thereupon. For quantum systems we are led by the structure that can be inferred from considering the correlation functions of the two-dimensional (ferromagnetic) Ising model as so-called ‘classical’ states on a quantum system. In the latter context our notion of an almost Gibbs state coincides with the notion of a state that does not satisfy the Kubo-Martin-Schwinger boundary condition but instead has only the structure that follows from Tomita’s theorem for so-called separating states on the observables.

Some Remarks on Almost Gibbs States

Spectra of the generators of time translations (“Liouville operators”) on representation spaces determined by thermodynamic equilibrium states are compared and their nature is investigated.

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Spectra of Liouville operators

We investigate the possibility to generalize the KMS-boundary condition for a thermodynamical system by following essentially the same procedure that for a finite system would amount to choosing a certain class of more general density functions on phase space (or density matrices) than the ones corresponding to the canonical or grand-canonical ensemble.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103921282

Marinus Winnink

Papers

On the equilibrium states in quantum statistical mechanics

Local normality in quantum statistical mechanics

Some Remarks on Almost Gibbs States

Spectra of Liouville operators

On generalizations of the kms-boundary condition