From Newton to Boltzmann: the case of short-range potentials
Year: 2012
Type: preprint
Abstract: We fill in all details in the proof of Lanford's theorem. This provides a rigorous derivation of the Boltzmann equation as the mesoscopic limit of systems of Newtonian particles interacting via a short-range potential, as the number of particles $N$ goes to infinity and the characteristic length of interaction $\varepsilon$ simultaneously goes to $0,$ in the Boltzmann-Grad scaling $N \varepsilon^{d-1} \equiv 1.$ The case of localized elastic interactions, i.e., hard spheres, is a corollary of the proof. The time of validity of the convergence is a fraction of the mean free time between two collisions, due to a limitation of the time on which one can prove the existence of the BBGKY and Boltzmann hierarchies. Our proof relies on the important contributions of King, Cercignani, Illner and Pulvirenti, and Cercignani, Gerasimenko and Petrina.
Source: HAL
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2621002144