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

Abstract

Abstract. 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.

On Global-in-time Chaotic Weak Solutions of the Liouville Equation for Hard Spheres

“…Our interest in solutions of the Liouville equation associated to hard sphere dynamics stems primarily from the BBGKY hierarchy to which it is connected. The derivation of global-in-time weak solutions of the BBGKY hierarchy which governs the marginals F (1) and F (2) of F given by…”

“…1 where S denotes the space of tempered distributions on T R 3N × (∞, ∞), π j : T R 3N → T R 3 is a canonical projection operator for j = 1, ..., N, the symbol ⊗ denotes the standard tensor product operator on Schwartz functions defined on T R 3 , and R N : S → S is a reflectiontype operator on Schwartz space S which is built using classical Boltzmann scattering matrices. We term the identity (2) in this article propagation of Schwartz chaos. Note that this is a global-in-time statement about the chaotic structure of weak solutions, in that the equality (2) holds in the sense of tempered distributions on the whole set D N × (∞, ∞) and not a strict subset thereof.…”

“…We term the identity (2) in this article propagation of Schwartz chaos. Note that this is a global-in-time statement about the chaotic structure of weak solutions, in that the equality (2) holds in the sense of tempered distributions on the whole set D N × (∞, ∞) and not a strict subset thereof. In this work, we detail only the case of two hard spheres in the whole space R 3 , i.e.…”

“…, the map β is a collision crosssection and C + (n; v) ⊂ R 3 is a half-space of velocities to be defined in the sequel. Finally, the collision operator C (2) is simply C given by ( 5) above. Before we present our new method of construction and main results, let us mention briefly some of the major results in the literature which (i) tackle the construction of various notions of solution of the Liouville equation (L), (ii) tackle construction of some notion of solution of its associated BBGKY hierachy, and (iii) investigate the (in)ability of these equations to preserve the structure of so-called chaotic initial data.…”

“…It is known that the dynamics of the hiearchy (in whichever sense is appropriate) does not preserve the structure of chaotic initial data for all times. See, for instance, the recent work of Denlinger [2]. Recently, Pulvirenti and Simonella [6] have performed a detailed study of so-called correlation errors which measure the manner in which solutions of the hierarchy deviate from profiles which are in product form.…”