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


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.


Boundary Condition Entropy Neural Network Free Energy Banach Space 
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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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