Long Time Validity of the Linearized Boltzmann Equation for Hard Spheres: A Proof Without Billiard Theory

Abstract

We study space–time fluctuations of a hard sphere system at thermal equilibrium, and prove that the covariance converges to the solution of a linearized Boltzmann equation in the low density limit, globally in time. This result was obtained previously in Bodineau et al. (Commun Pure Appl Math 76:3852–3911, 2021) by using uniform bounds on the number of recollisions of dispersing systems of hard spheres [as provided for instance in Burago et al. (Ann Math (2), 147(3):695–708, 1998)]. We present a self-contained proof with substantial differences, which does not use this geometric result. This can be regarded as the first step of a program aiming of deriving the fluctuation theory of the rarefied gas for interaction potentials different from hard spheres.