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Register a new userStructure of correlations for the Boltzmann-Grad limit of hard spheres
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Ryan Denlinger
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We consider a gas of $N$ identical hard spheres in the whole space, and we enforce the Boltzmann-Grad scaling. We may suppose that the particles are essentially independent of each other at some initial time; even so, correlations will be created by the dynamics. We will prove a structure theorem for the correlations which develop at positive time. Our result generalizes a previous result which states that there are phase points where the three-particle marginal density factorizes into two-particle and one-particle parts, while further factorization is impossible. The result depends on uniform bounds which are known to hold on a small time interval, or globally in time when the mean free path is large.
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Boltzmanns equation provides a microscopic model for the evolution of dilute classical gases. A fundamental problem in mathematical physics is to rigorously derive Boltzmanns equation starting from Newtons laws. In the 1970s, Oscar Lanford provided such a derivation, for the hard sphere interaction, on a small time interval. One of the subtleties of Lanfords original proof was that the strength of convergence proven at positive times was much weaker than that which had to be assumed at the initial time, which is at odds with the idea of propagation of chaos. Several authors have addressed this situation with various notions of strong one-sided chaos, which is the true property which is propagated by the dynamics. We provide a new approach to the problem based on duality and the evolution of observables; the observables encode the detailed interaction and allow us to define a new notion of strong one-sided chaos.
This paper provides the first rigorous derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard-spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this paper introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time.
We review a virial-type estimate which bounds the strength of interaction for a gas of $N$ hard spheres (billiard balls) dispersing into Euclidean space $mathbb{R}^d$. This type of estimate has been known for decades in the context of (semi-)dispersing billiards, and is essentially trivial in that context. Our goal, however, is to write virial estimates in a way which may lend insight into the problem of rigorously deriving Boltzmanns equation (cf. Lanfords theorem). Using virial estimates, we provide a short proof of lower bounds (sharp up to powers of logarithms) on the convergence rate of the first marginal in Lanfords theorem. Such lower bounds will often, but not always, follow trivially from energy conservation, the proof we present holds assuming only that the limiting dynamics is regular enough and does not reduce to free transport.
We consider the motion of a finite though large number $N$ of hard spheres in the whole space $mathbb{R}^n$. Particles move freely until they experience elastic collisions. We use our recent theory of Compensated Integrability in order to estimate how much the particles are deviated by collisions. Our result, which is expressed in terms of hodographs, tells us that only $O(N^2)$ collisions are significant.
We discuss old and new results on the mathematical justification of Boltzmanns 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 Lanfords 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.
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