Communications in Mathematical Physics

, Volume 13, Issue 3, pp 194–215 | Cite as

Observables at infinity and states with short range correlations in statistical mechanics

  • O. E. LanfordIII
  • D. Ruelle


We say that a representation of an algebra of local observables has short-range correlations if any observable which can be measured outside all bounded sets is a multiple of the identity, and that a state has finite range correlations if the corresponding cyclic representation does. We characterize states with short-range correlations by a cluster property. For classical lattice systems and continuous systems with hard cores, we give a definition of equilibrium state for a specific interaction, based on a local version of the grand canonical prescription; an equilibrium state need not be translation invariant. We show that every equilibrium state has a unique decomposition into equilibrium states with short-range correlations. We use the properties of equilibrium states to prove some negative results about the existence of metastable states. We show that the correlation functions for an equilibrium state satisfy the Kirkwood-Salsburg equations; thus, at low activity, there is only one equilibrium state for a given interaction, temperature, and chemical potential. Finally, we argue heuristically that equilibrium states are invariant under time-evolution.


Correlation Function Equilibrium State Statistical Mechanic Quantum Computing Classical Lattice 
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Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • O. E. LanfordIII
    • 1
    • 2
  • D. Ruelle
    • 1
  1. 1.I. H. E. S.Bures-Sur-YvetteFrance
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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