The quantitative hydrodynamic limit of the Kawasaki dynamics

TL;DR

This paper establishes a quantitative rate of convergence for the hydrodynamic limit of one-dimensional Kawasaki dynamics, utilizing an improved two-scale approach with spline-based mesoscopic observables.

Contribution

It introduces a novel spline-based projection in the two-scale approach, achieving better convergence rates in the hydrodynamic limit analysis.

Findings

First quantitative rate of convergence for Kawasaki dynamics hydrodynamic limit.

Spline-based mesoscopic observables improve convergence analysis.

Enhanced two-scale approach yields more natural and effective results.

Abstract

We derive for the first time in the literature a rate of convergence in the hydrodynamic limit of the Kawasaki dynamics for a one-dimensional lattice system. We use an adaptation of the two-scale approach. The main difference to the original two-scale approach is that the observables on the mesoscopic level are described by a projection onto splines of second order, and not by a projection onto piecewise constant functions. This allows us to use a more natural definition of the mesoscopic dynamics, which yields a better rate of convergence than the original two-scale approach.