Equivalence of the grand canonical ensemble and the canonical ensemble on 1d-lattice systems

Abstract

We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show the equivalence of the grand canonical ensemble (gce) and the canonical ensemble (ce), in the sense of observables and correlations. A direct consequence is that the correlations of the ce decay exponentially plus a volume correction term. The volume correction term is uniform in the external field, the mean spin and scales optimally in the system size. This extends prior results of Cancrini & Martinelli for bounded discrete spins to unbounded continuous spins. The result is obtained by adapting Cancrini & Martinelli's method combined with authors' recent approach on continuous real-valued spin systems.

Hydrodynamic limit of the Kawasaki dynamics on the 1D-lattice with strong, finite-range interaction

“…In this section, we provide auxiliary results which were proved in [KM18], [KM19], [KM20] and [KLM19]. Recall that a generalized gce µ σ N is defined by…”

“…However, given the equivalence of ensembles (cf. [KLM19]) meaning that the canonical ensemble is equivalent to properly modified grand canonical ensemble, it is not surprising that one is able to deduce the hydrodynamic limit of Kawasaki dynamics in the case of a non-interactive Hamiltonian.…”

“…This is a culmination of a series of works done by the authors. Recently, the authors provided better understandings of the canonical ensembles with strong finite-range interactions in a series of articles (see [KM18], [KM19], [KM20] and [KLM19]). These include, but not restricted to, equivalence of ensembles, decay of correlations and uniform LSI for the canonical ensemble.…”