FM 09-14 (arXiv:1405.4676)

Authors: M. Pulvirenti, S. Simonella

Title: The Boltzmann-Grad Limit of a Hard Sphere System: Analysis of the Correlation Error

 
Abstract: We present a quantitative analysis of the Boltzmann-Grad (low-density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order $k$ measures the event where $k$ tagged particles are connected by a chain of interactions preventing the factorization. We prove that, provided $k$ is not too large, such an error flows to zero with the hard spheres diameter $\varepsilon$, for short times, as $\varepsilon^{\gamma k}$, for some $\gamma>0$. This requires a new analysis of many-recollision events, and improves previous estimates of high order correlation functions.