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

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

A general Hermitian scalar field, assumed to be an operator−valued tempered distribution, is considered. A theorem which relates certain complex Lorentz transformations to the TCP transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the field. A theorem analogous to Borchers’ theorem on relatively local fields is stated and proved. Local internal symmetries are discussed, and it is shown that any such symmetry commutes with the Poincare

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On the duality condition for a Hermitian scalar field

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

A theorem is derived giving sufficient conditions for a factor to be either finite or purely infinite. These conditions are:

i.

In the Hilbert space$\mathfrak{H}$ exists a conjugation operatorJ transforming the factor ℜ into its commutant ℜ′.




ii.

There exists a one parameter abelian group of automorphisms of ℜ implemented by unitary operatorsUt weakly continuous int and commuting withJ.




iii.

There is a cyclic and separating vector Ω, which is invariant forJ and which is the only vector in$\mathfrak{H}$ invariant forUt.

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On the factor type of equilibrium states in quantum Statistical Mechanics

We study automorphisms of the CAR algebra which map the family of gauge-invariant, quasi-free states of the CAR algebra onto itself and show (Theorem 3.1) that they are one-particle automorphisms.

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Automorphisms and quasi-free states of the CAR algebra

A sufficient condition is given in order that a von Neumann algebra with cyclic vector is quasi-standard. With the help of this result it is proved that a locally normal state with a cyclic and separating vector in the representation space gives rise to a quasi-standard von Neumann algebra. Furthermore it is proved that the representation space determined by a locally normal state in the G.N.S. construction is separable.

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On locally normal states in quantum statistical mechanics

For quantum spin systems it is known that for a suitable space of potentials the equilibrium states areW*-dense in the set of all translation invariant states. The problem discussed in this paper is how to recognize such equilibrium states and how to find the corresponding potential. A necessary and sufficient condition for a state to be an equilibrium state for some potential is given in Sect. 3.

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N. M. Hugenholtz

Papers

On the equilibrium states in quantum statistical mechanics

On the factor type of equilibrium states in quantum Statistical Mechanics

Automorphisms and quasi-free states of the CAR algebra

On locally normal states in quantum statistical mechanics

On the inverse problem in statistical mechanics