Mathematics Colloquium

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Hilbert’s sixth problem asks for the axiomatic derivation of the laws of physics. Within this broad question, Hilbert singled out the derivation of the laws of fluid mechanics (like Euler’s or Navier-Stokes’ equations) by way of rigorously justifying Boltzmann’s kinetic theory for particle systems. This will be the subject of our talk. The rigorous derivation of Boltzmann’s kinetic equation was established for short times in a classic work of Lanford (1975), following inputs from Grad and Cercignani. However, Hilbert’s sixth problem requires a long time version of this result. In a recent work with Yu Deng (Chicago) and Xiao Ma (Michigan), we extend Lanford’s theorem to long times; more precisely, for as long as the solution of Boltzmann’s equation exists. The proof relies on a novel methodology that allows us to obtain effective estimates on the probability of many recollisions.