A two-scale approach to the hydrodynamic limit, part II: local Gibbs behavior

This work is a follow-up on [GOVW]. In that previous work a two-scale approach was used to prove the logarithmic Sobolev inequality for a system of spins with fixed mean whose potential is a bounded perturbation of a Gaussian, and to derive an abstract theorem for the convergence to the hydrodynamic limit. This strategy was then successfully applied to Kawasaki dynamics. Here we shall use again this two-scale approach to show that the microscopic variable in such a model behaves according to a local Gibbs state. As a consequence, we shall prove the convergence of the microscopic entropy to the hydrodynamic entropy.Comment: 31 pages, 2nd version. The proof of Theorem 1.15 has been simplifie

Modified logarithmic Sobolev inequalities for canonical ensembles

In this paper, we prove modified logarithmic Sobolev inequalities for canonical ensembles with superquadratic single-site potential. These inequalities were introduced by Bobkov and Ledoux, and are closely related to concentration of measure and transport-entropy inequalities. Our method is an adaptation of the iterated two-scale approach that was developed by Menz and Otto to prove the usual logarithmic Sobolev inequality in this context. As a consequence, we obtain convergence in Wasserstein distance $W_p$ for Kawasaki dynamics on the Ginzburg-Landau model.Comment: 19 pages v2: a mistake has been corrected in the proof of Lemma 2.3 (formerly Lemma 2.8), and the presentation has been reworke

Free Stein kernels and an improvement of the free logarithmic Sobolev inequality

We introduce a free version of the Stein kernel, relative to a semicircular law. We use it to obtain a free counterpart of the HSI inequality of Ledoux, Peccatti and Nourdin, which is an improvement of the free logarithmic Sobolev inequality of Biane and Speicher, as well as a rate of convergence in the (multivariate) entropic free Central Limit Theorem. We also compute the free Stein kernels for several relevant families of self-adjoint operators

Entropic Ricci curvature bounds for discrete interacting systems

We develop a new and systematic method for proving entropic Ricci curvature lower bounds for Markov chains on discrete sets. Using different methods, such bounds have recently been obtained in several examples (e.g., 1-dimensional birth and death chains, product chains, Bernoulli-Laplace models, and random transposition models). However, a general method to obtain discrete Ricci bounds had been lacking. Our method covers all of the examples above. In addition, we obtain new Ricci curvature bounds for zero-range processes on the complete graph. The method is inspired by recent work of Caputo, Dai Pra and Posta on discrete functional inequalities.Comment: Published at http://dx.doi.org/10.1214/15-AAP1133 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The two-scale approach to hydrodynamic limits for non-reversible dynamics

In a recent paper by Grunewald et.al., a new method to study hydrodynamic limits was developed for reversible dynamics. In this work, we generalize this method to a family of non-reversible dynamics. As an application, we obtain quantitative rates of convergence to the hydrodynamic limit for a weakly asymmetric version of the Ginzburg-Landau model endowed with Kawasaki dynamics. These results also imply local Gibbs behavior, following a method introduced in a recent paper by the second author.Comment: 26 page