We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrarystrong, quadratic, finite-range interaction. The first area of interest concerns the
equivalence of the grand canonical ensemble and the canonical ensemble: on the level of
thermodynamic functions, on the level of observables, and on the level of correlations. More
precisely, in the thermodynamic limit (size N of the system goes to infinity), we show that the
free energy, expectation of intensive observable, and correlation of two intensive functions
are the same for the grand canonical ensemble and canonical ensemble.
The second area of interest concerns the decay of correlations and uniqueness of infinite-volume
Gibbs measure of the canonical ensemble. It is shown that the correlations of the
canonical ensemble decay exponentially plus a volume correction term. As a consequence,
we verify a conjecture that the infinite-volume Gibbs measure of the canonical ensemble is
unique on the one-dimensional lattice, extending results that are known for the case of weak
interaction.
The third area of interest concerns the logarithmic Sobolev inequality (LSI). It is shown
that the canonical ensemble satisfies a uniform LSI. The LSI constant is uniform in the
boundary data, the external field and scales optimally in the system size. We deduce the
LSI by combining two different methods, the two-scale approach and the Zegarlinski method.
The last area of interest concerns the hydrodynamic limit. We deduce the hydrodynamic
limit of Kawasaki dynamics. The main ingredients are uniform LSI and decay of correlations
for the canonical ensemble. The proof is based on a method invented by Grunewald, Otto,
Villani and Westdickenberg.