A brief introduction to the scaling limits and effective equations in kinetic theory

These lecture notes provide the material for a short introductory course on effective equations for classical particle systems. They concern the basic equations in kinetic theory, written by Boltzmann and Landau, describing rarefied gases and weakly interacting plasmas respectively. These equations can be derived formally, under suitable scaling limits, taking classical particle systems as a starting point. A rigorous proof of this limiting procedure is difficult and still largely open. We discuss some mathematical problems arising in this context.Comment: arXiv admin note: substantial text overlap with arXiv:1611.0708

The Boltzmann-Grad Limit of a Hard Sphere System: Analysis of the Correlation Error

We present a quantitative analysis of the Boltzmann-Grad (low-density) limit of a hard sphere system. We introduce and study a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order $k$ measures the event where $k$ particles are connected by a chain of interactions preventing the factorization. We show that, provided $k < \varepsilon^{-\alpha}$, such an error flows to zero with the average density $\varepsilon$, for short times, as $\varepsilon^{\gamma k}$, for some positive $\alpha,\gamma \in (0,1)$. This provides an information on the size of chaos, namely, $j$ different particles behave as dictated by the Boltzmann equation even when $j$ diverges as a negative power of $\varepsilon$. The result requires a rearrangement of Lanford perturbative series into a cumulant type expansion, and an analysis of many-recollision events.Comment: 98 pages, 12 figures. Subject of the Harold Grad Lecture at the 29th International Symposium on Rarefied Gas Dynamics (Xi'an, China). This revised version contains new results (a theorem on the convergence of high order fluctuations; estimates of integrated correlation error) and several improvements of presentation, inspired by comments of the anonymous refere

On the evolution of the empirical measure for the Hard-Sphere dynamics

We prove that the evolution of marginals associated to the empirical measure of a finite system of hard spheres is driven by the BBGKY hierarchical expansion. The usual hierarchy of equations for $L^1$ measures is obtained as a corollary. We discuss the ambiguities arising in the corresponding notion of microscopic series solution to the Boltzmann-Enskog equation

Evolution of correlation functions in the hard sphere dynamics

The series expansion for the evolution of the correlation functions of a finite system of hard spheres is derived from direct integration of the solution of the Liouville equation, with minimal regularity assumptions on the density of the initial measure. The usual BBGKY hierarchy of equations is then recovered. A graphical language based on the notion of collision history originally introduced by Spohn is developed, as a useful tool for the description of the expansion and of the elimination of degrees of freedom

Backward Clusters, Hierarchy and Wild Sums for a Hard Sphere System in a Low-Density Regime

We study the statistics of backward clusters in a gas of hard spheres at low density. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We derive upper and lower bounds on the average size of clusters by using the theory of the homogeneous Boltzmann equation combined with suitable hierarchical expansions. These representations are known in the easier context of Maxwellian molecules (Wild sums). We test our results with a numerical experiment based on molecular dynamics simulations