The Propagation of Chaos for a Rarefied Gas of Hard Spheres in the Whole Space

Abstract

We discuss old and new results on the mathematical justification of Boltzmann's equation. The classical result along these lines is a theorem which was proven by Lanford in the 1970s. This paper is naturally divided into three parts. I. Classical. We give new proofs of both the uniform bounds required for Lanford's theorem, as well as the related bounds due to Illner & Pulvirenti for a perturbation of vacuum. The proofs use a duality argument and differential inequalities, instead of a fixed point iteration. II. Strong chaos. We introduce a new notion of propagation of chaos. Our notion of chaos provides for uniform error estimates on a very precise set of points; this set is closely related to the notion of strong (one-sided) chaos and the emergence of irreversibility. III. Supplemental. We announce and provide a proof (in Appendix A) of propagation of partial factorization at some phase-points where complete factorization is impossible. 1 L ∞ (without at least an assumption like factorization or exchangeability) but it is wellposed for continuous data. 8 The strong chaos result in [5] requires |xi − xj | ε log 1 ε . 9By contrast, the authors of [28] have provided a very precise but averaged (not pointwise) description of correlations.

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

“…As in [3], we will also need a condition which forces particles to disperse from one another. Hence for any η > 0 we define:…”

“…The definition of (m − 1)-nonuniform chaoticity is not exactly the same as the notion of 2-nonuniform chaoticity introduced in [3] when m = 3. Nevertheless, the two notions are almost the same, in terms of the complexity of sets involved in the definition.…”

“…However, we have avoided using their special representation formula for correlations. Instead, we employ an unsymmetric Boltzmann-Enskog hierarchy (defined in [3]) which tracks correlations between just the first m−1 particles. This is convenient because we can show that the intermediate hierarchy propagates partial factorization in an exact sense.…”

“…The theorem then follows by comparing the BBGKY and Boltzmann-Enskog pseudo-dynamics; this paper is devoted primarily to making such stability estimates precise. This paper relies heavily on the developments of [3]; indeed, all we do in this paper is refine part (ii) of Theorem 2.1 of [3] to include the case of arbitrarily many correlated particles (instead of just two correlated particles). For this reason, we will only briefly summarize the main results of [3] that are relevant here, and fill in the missing ideas and estimates that are needed to prove our main theorem.…”

“…In Section 4, we recall an unsymmetric Boltzmann-Enskog hierarchy from Appendix A of [3]. A crucial stability result is proven in Section 5; the remainder of the convergence proof proceeds as in [3]. Acknowledgements R.D.…”