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 < ε −α , such an error flows to zero with the average density ε, for short times, as ε γk , for some positive α, γ ∈ (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 ε. The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.

Quantitative analysis of the correlations in the Boltzmann-Grad limit for hard spheres

“…In collaboration with Simonella (see [19]) we tried to give quantitative estimates of this property. With this in mind, we introduce the following quantity, called correlation error, (denoted by E), implicitly defined by…”

A Kac Model for Kinetic Annihilation

“…The literature on the subject is extremely wide: we mention the seminal paper by O. E. Lanford [25] concerning the derivation of the spatially inhomogeneous Boltzmann equation from a particles system in the Boltzmann-Grad limit, see also the more recent contributions [15,37]. We also mention the recent analysis of the correlation error for Boltzmann equation in [38] which, as [25], works in the grand canonical ensemble formalism which is the one adopted here.…”

Structure of correlations for the Boltzmann-Grad limit of hard spheres

“…Pulvirenti and Simonella analyzed the size of higher-order correlations in the context of Lanford's theorem. [6,7] Remarkably, the authors were able to quantify the correlations even among ε −α particles for some α > 0.…”