Dynamics of dilute gases: a statistical approach

The evolution of a gas can be described by different models depending on the observation scale. A natural question, raised by Hilbert in his sixth problem, is whether these models provide consistent predictions. In particular, for rarefied gases, it is expected that continuum laws of kinetic theory can be obtained directly from molecular dynamics governed by the fundamental principles of mechanics. In the case of hard sphere gases, Lanford [44] showed that the Boltzmann equation emerges as the law of large numbers in the low density limit, at least for very short times. The goal of this survey is to present recent progress in the understanding of this limiting process, providing a complete statistical description. 1. AIM : PROVIDING A STATISTICAL PICTURE OF DILUTE GAS DYNAMICS 1.1. A very simple physical model. Even though at the time Boltzmann published his famous paper [16], the atomistic theory was still dismissed by some scientists, it is now well established that matter is composed of atoms, which are the elementary constituents of all solid, liquid and gaseous substances. The particularity of dilute gases is that their atoms are very weakly bound and almost independent. In other words, there are very few constraints on their geometric arrangement because their volume is negligible compared to the total volume occupied by the gas. If we neglect the internal structure of atoms (consisting of a nucleus and electrons) and their possible organisation into molecules, we can represent a gas as a large system of correlated interacting particles. We will also neglect the effect of long range interactions and assume strong interatomic forces at very short distance. Each particle moves freely most of the time and occasionally collides with some other particle leading to an almost instantaneous scattering. The simplest example of such a model consists in assuming that the particles are identical tiny balls of unit mass interacting only by contact (see Figure 1). We then speak of a gas of hard spheres. All the results we will present should nevertheless extend to isotropic, compactly supported stable interaction potentials [60, 55]. This microscopic description of a gas is daunting because the number of particles involved is extremely large, the individual size of these particles is tiny (of diameter ε ≪ 1) and therefore positions are very sensitive to small spatial shifts (see Figure 2). In practice, this model is not efficient for making theoretical predictions, and numerical methods are often in favour of Monte Carlo simulations. The question we would like to address here is a more fundamental one, namely the consistency of this (simplified) atomic description with the kinetic or fluid models used in applications. This question was formalised by Hilbert at the ICM in 1900, in his sixth problem: ”Boltzmann’s work on the principles of mechanics suggests the problem of developing mathematically the limiting processes, there merely indicated, which lead from the atomistic view to the laws of motion of continua”. The Boltzmann equation, mentioned by Hilbert and which we will present in more detail later, expresses that the distribution of particles evolves under the combined effect of free transport and collisions. For these two effects to be of the same order of magnitude, a simple calculation shows that, in dimension d ≥ 2, the number of particles N and their diameter size ε must satisfy the scaling relation Nεd−1 = O(1), the so-called Boltzmann-Grad scaling [38]. Indeed the regime described by the Boltzmann equation is such that the mean free path, namely the average distance covered by 1 ar X iv :2 20 1. 10 14 9v 1 [ m at h. A P] 2 5 Ja n 20 22 2 T. BODINEAU, I. GALLAGHER, L. SAINT-RAYMOND, AND S. SIMONELLA