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

Abstract

Abstract. 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 equi. librium 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.

Quantum equilibrium and the origin of absolute uncertainty

“…In a Bohmian universe such knowledge is absolutely unattainable! 41 We emphasize that we do not claim that knowledge of the detailed configuration of a system is impossible, a claim that would be manifestly false. We maintain only that-as a consequence of the fact that the configuration X of a system and the configuration Y of its environment are conditionally independent given its wave function ψ-all such knowledge must be mediated by ψ.…”

“…(How otherwise would we know the temperature?) Furthermore, for a rigorous analysis of equilibrium distributions in the thermodynamic limit-i.e., of (the idealization given by) global thermodynamic equilibrium-the 52 equations of Dobrushin and Lanford-Ruelle [27,41], stipulating that (13.2)-regarded as expressing such a conditional distribution-be satisfied for all subsystems, often play a defining role. 43 Moreover, what we have just described is only a part of a deeper and broader analogy, between the scheme classical mechanics =⇒ equilibrium statistical mechanics =⇒ thermodynamics, (13.3) which outlines the (classical) connection between the microscopic level of description and a phenomenological formalism on the macroscopic level; and the scheme Bohmian mechanics =⇒ quantum equilibrium: statistical mechanics relative to the wave function =⇒ the quantum formalism, (13.4) which outlines the (quantum) connection between the microscopic level and another phenomenological formalism-the quantum measurement formalism.…”

Aspects of Ergodic, Qualitative and Statistical Theory of Motion

“…and in the East (thanks mainly to the works of Dobrushin, Minlos, Sinai, etc.). The more or less definitive formulation of the notion and of the basic properties of a Gibbs distribution can be found in the classical papers [Do68a], [Do68b], [Do68c], [Do69], [Ru69] and [LR69].…”

Mixing properties of stationary Poisson cylinder models

“…On the other hand, the reverse implication is false in general. Following the terminology in [11], a stationary RACS Ξ in R d having (non-)trivial tail-σ-algebra σ ∞ f (Ξ) is said to have (long) short range correlations or (long) short range dependences. For each k = 1, . .…”