88 research outputs found
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 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
- …