From Newton’s law to the linear Boltzmann equation without cut-off


We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling.
The main difficulty in our context is that, due to the infinite range of the potential, a non- integrable singularity appears in the angular collision kernel, making no longer valid the single- use of Lanford’s strategy.
Our proof relies then on a combination of Lanford’s strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated to the long-range interaction, leading to some explicit weak estimates.

In Communications in Mathematical Physics