Mathematical Analysis of Dilute Classical Gases
Final Report Abstract
This project was devoted to the rigorous description of nonequilibrium dynamics for interacting particle systems on large scales, and especially to the bridge between the microscopic models and kinetic theory. We took as reference the Grad approach to the classical Boltzmann theory, mostly focusing on the simplest case which is the hard sphere gas at low density. We developed a cumulant method to quantify the small correlations which separate the microscopic model (BBGKY hierarchy) from the solution to the Boltzmann equation. It is amazing how rich the structure of such correlations turns out to be. The way to detect this structure is based on an organizing principle of the error in Lanford’s proof. The method has a combinatorial part (cluster expansion), strongly inspired from equilibrium statistical mechanics. Its first future application is a complete analysis of random dynamical deviations of the hard sphere gas from the predictions of the kinetic equation. The line initiated has led to several developments, among which: a better understanding of the microscopic origin of the Boltzmann-Enskog equation for moderately dense gases; sharp estimates of correlations in a broad class of mean field models; optimality of the notion of one-sided convergence coding irreversibility. In parallel, we have developed a statistical theory of collisions for the Boltzmann gas, with connections to coalescence processes and percolation theory. We have showed that, in the rarefied gas and after a critical time, with positive probability the particles are interrelated by chains of infinitely many collisions, forming a giant (macroscopic) dynamically connected cluster. Finally, we have started an investigation of the kinetic limit for particle systems with long range interactions, focused on linear regimes.
Publications
- Kinetic Theory of Cluster Dynamics. Physica D: Nonlinear Phenomena 335, 26–32, 2016
R. I. A. Patterson, S. Simonella and W. Wagner
(See online at https://doi.org/10.1016/j.physd.2016.06.007) - A kinetic equation for the distribution of interaction clusters in rarefied gases. J. Stat. Phys., 169, 1, 126–167, 2017
R. I. A. Patterson, S. Simonella and W. Wagner
(See online at https://doi.org/10.1007/s10955-017-1865-0) - The Boltzmann–Grad limit of a hard sphere system: analysis of the correlation error. Inventiones, 207, 3, 1135-1237, 2017
M. Pulvirenti and S. Simonella
(See online at https://doi.org/10.1007/s00222-016-0682-4) - On the theory of Lorentz gases with long range interactions. Rev. Math. Phys. 30, 3, 2018
A. Nota, S. Simonella and J. J. L. Velázquez
(See online at https://doi.org/10.1142/S0129055X18500071) - One-sided convergence in the Boltzmann-Grad limit. Ann. Fac. Sci. Toulouse Math. Ser. 6 27, 5, 985-1022, 2018
T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella
(See online at https://doi.org/10.5802/afst.1589) - On the Size of Chaos in the Mean Field Dynamics. Arch. Rat. Mech. and Anal. , 231:1, 285-317, 2019
T. Paul, M. Pulvirenti and S. Simonella
(See online at https://doi.org/10.1007/s00205-018-1280-y)
