From Hard Sphere Dynamics to the Stokes–Fourier Equations: An [InlineEquation not available: see fulltext.] Analysis of the Boltzmann–Grad Limit

We derive the linear acoustic and Stokes–Fourier equations as the limiting dynamics of a system of N hard spheres of diameter ε in two space dimensions, when N→ ∞, ε→ 0 , Nε= α→ ∞, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford’s strategy (Time evolution of large classical systems, Springer, Berlin, 1975), and on the pruning procedure developed in Bodineau et al. (Invent Math 203:493–553, 2016) to improve the convergence time to all kinetic times with a quantitative control which allows us to reach also hydrodynamic time scales. The main novelty here is that uniform L² a priori estimates combined with a subtle symmetry argument provide a weak version of chaos, in the form of a cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.