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

TL;DR

This paper proves the equivalence of grand canonical and canonical ensembles for 1D lattice systems with unbounded continuous spins, showing exponential decay of correlations and uniform volume corrections, extending previous bounded-spin results.

Contribution

It extends the equivalence of ensembles to unbounded continuous spins in 1D lattice systems, with uniform correlation decay and optimal volume correction scaling.

Findings

Correlations decay exponentially with volume correction

Equivalence holds uniformly in external field and mean spin

Results extend prior bounded-spin ensemble equivalence to unbounded spins

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.