Equilibrium and non-equilibrium aspects of Gibbs measures
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.