Abstract
In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space\(\mathfrak{B}\) of translationally invariant interactions. We prove that each tangent functional to the graph ofP defines a set of translationally invariant thermodynamic expectation values. More precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra\(\mathfrak{A}\) of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and it is shown that they form a dense set in the set of all invariant states over\(\mathfrak{A}\). With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Schwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under time-translations.
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Lanford, O.E., Robinson, D.W. Statistical mechanics of quantum spin systems. III. Commun.Math. Phys. 9, 327–338 (1968). https://doi.org/10.1007/BF01654286
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DOI: https://doi.org/10.1007/BF01654286