On the equivalence between classical KMS states and the DLR formalism

Abstract:

Within the context of classical spin lattice systems, thermal equilibrium at fixed temperature and at finite volume is described in terms of Gibbs states. In the thermodynamic (i.e. infinite volume) limit the notion of thermal equilibrium can be described with the Dobrushin, Lanford and Ruelle (DLR) condition, which selects a class of physically interesting states by assigning their conditional probability on finite volumes. At the same time, the notion of (classical) thermal equilibrium can be considered also within the context of Poisson geometry: therein, the relevant class of states is identified by the Kubo, Martin, Schwinger (KMS) condition. In this talk we will prove that the DLR and KMS conditions coincide for a large class of spin lattice systems with values on a compact symplectic manifold.
Based on a joint work with C. J. F. van de Ven.