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 $k$ particles are connected by a chain of interactions preventing the factorization. We show that, provided $k<{\epsilon }^{-\alpha }$, such an error flows to zero with the average density $\epsilon$, for short times, as ${\epsilon }^{\gamma k}$, for some positive $\alpha ,\gamma \in (0,1)$. This provides an information on the size of chaos, namely, $j$ different particles behave as dictated by the Boltzmann equation even when $j$ diverges as a negative power of $\epsilon$. The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.

98 pages, 12 figures. Subject of the Harold Grad Lecture at the 29th International Symposium on Rarefied Gas Dynamics (Xi'an, China). This revised version contains new results (a theorem on the convergence of high order fluctuations; estimates of integrated correlation error) and several improvements of presentation, inspired by comments of the anonymous referee

References in corpus (1)

• Wick polynomials and time-evolution of cumulants

Cited by in corpus (13)

• Propagation of chaos: a review of models, methods and applications. II. Applications
• Fluctuation theory in the Boltzmann--Grad limit
• The propagation of chaos for a rarefied gas of hard spheres in the whole space
• On the size of chaos in the mean field dynamics
• A new approach to the mean-field limit of Vlasov-Fokker-Planck equations
• The Boltzmann-Grad asymptotic behavior of collisional dynamics: a brief survey
• Propagation processes of correlations of hard spheres
• On the foundations of statistical mechanics
• From a non-Markovian system to the Landau equation
• Summability of joint cumulants of nonindependent lattice fields
• Cluster expansion for a dilute hard sphere gas dynamics
• Structure of correlations for the Boltzmann-Grad limit of hard spheres
• A Kac model for kinetic annihilation