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Register a new userThe propagation of chaos for a rarefied gas of hard spheres in the whole space
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Authors
Ryan Denlinger
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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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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.
The inconsistency between the time-reversible Liouville equation and time-irreversible Boltzmann equation has been pointed out long ago by Loschmidt. To avoid Loschmidts objection, here we propose a new dynamical system to model the motion of atoms of gas, with their interactions triggered by a random point process. Despite being random, this model can approximate the collision dynamics of rigid spheres via adjustable parameters. We compute the exact statistical steady state of the system, and determine the form of its marginal distributions for a large number of spheres. We find that the Kullback-Leibler entropy (a generalization of the conventional Boltzmann entropy) of the full system of random gas spheres is a nonincreasing function of time. Unlike the conventional hard sphere model, the proposed random gas model results in a variant of the Enskog equation, which is known to be a more accurate model of dense gas than the Boltzmann equation. We examine the hydrodynamic limit of the derived Enskog equation for spheres of constant mass density, and find that the corresponding Enskog-Euler and Enskog-Navier-Stokes equations acquire additional effects in both the advective and viscous terms. In the dilute gas approximation, the Enskog equation simplifies to the Boltzmann equation, while the Enskog-Euler and Enskog-Navier-Stokes equations become the conventional Euler and Navier-Stokes equations.
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.
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.
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