From Newton to Boltzmann: Hard Spheres and Short-range Potentials

By Isabelle Gallagher, Laure Saint-Raymond, Benjamin Texier. 2012

223 citations recorded by the source corpus.

We provide a rigorous derivation of the Boltzmann equation as the mesoscopic limit of systems of hard spheres, or Newtonian particles interacting via a short-range potential, as the number of particles $N$ goes to infinity and the characteristic length of interaction $\e$ simultaneously goes to $0,$ in the Boltzmann-Grad scaling $N \e^{d-1} \equiv 1.$ The time of validity of the convergence is a fraction of the average time of first collision, due to a limitation of the time on which one can prove uniform estimates for the BBGKY and Boltzmann hierarchies. Our proof relies on the fundamental ideas of Lanford, and the important contributions of King, Cercignani, Illner and Pulvirenti, and Cercignani, Gerasimenko and Petrina. The main novelty here is the detailed study of pathological trajectories involving recollisions, which proves the term-by-term convergence for the correlation series expansion.

Subfields: Applied Mathematics, Ocean Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/850848361998a1b06e3cd4172a21ed2df17febe2

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In Applied Mathematics

In Ocean Engineering

In Statistical and Nonlinear Physics

Citations in the corpus

The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, background): “A rigorous derivation of the Boltzmann equation from the Newtonian dynamics has only been given for short times (of the order of the average time of ﬁrst collision), see [57] [40].”
Mean Field Limit for Stochastic Particle Systems (2017, background): “After the seminal in [53] and later [23], the rigorous derivation of the Boltzmann equation was finally achieved in [31] but only for short times (of the order of the average time between collisions).”
A review of the mean field limits for Vlasov equations (2014, background): “A full solution was however only given recently in the seminal [64], still only for a short time but with possibly more complex interactions of the type (1.”