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Communications in Mathematical Physics

, Volume 9, Issue 4, pp 327–338 | Cite as

Statistical mechanics of quantum spin systems. III

  • Oscar E. LanfordIII
  • Derek W. Robinson
Article

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.

Keywords

Boundary Condition Entropy Neural Network Free Energy Banach Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • Oscar E. LanfordIII
    • 1
    • 2
  • Derek W. Robinson
    • 3
  1. 1.I. H. E. S.Bures-sur-Yvette
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.CERNGeneva

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