Long-time derivation at equilibrium of the fluctuating Boltzmann equation

We study a hard sphere gas at equilibrium, and prove that in the low density limit, the fluctuations converge to a Gaussian process governed by the fluctuating Boltzmann equation. This result holds for arbitrarily long times. The method of proof builds upon the weak convergence method introduced in the companion paper [8] which is improved by considering clusters of pseudo-trajectories as in [7].

[1]  S. Simonella.  Evolution of Correlation Functions in the Hard Sphere Dynamics , 2012, 1205.2789.

[2]  R. Alexander.  THE INFINITE HARD-SPHERE SYSTEM , 1975 .

[3]  Viktor Ivanovich Gerasimenko,et al.  Low-Density Asymptotic Behavior of Observables of Hard Sphere Fluids , 2018 .

[4]  R. Illner,et al.  Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum: Erratum and improved result , 1989 .

[5]  D. Stroock,et al.  Generalized Ornstein-Uhlenbeck Processes and Infinite Particle Branching Brownian Motions , 1978 .

[6]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[7]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[8]  J. Lebowitz,et al.  Equilibrium time correlation functions in the low-density limit , 1980 .

[9]  Dmitry Burago,et al.  Uniform estimates on the number of collisions in semi-dispersing billiards , 1998 .

[10]  C. Cercignani.  On the Boltzmann equation for rigid spheres , 1972 .

[11]  M. Ernst,et al.  Nonequilibrium fluctuations in μ space , 1981 .

[12]  R. Illner.  On the number of collisions in a hard sphere particle system in all space , 1989 .

[13]  H. Grad.  On the kinetic theory of rarefied gases , 1949 .

[14]  J. Norris.  Appendix: probability and measure , 1997 .

[15]  O. Lanford.  Time evolution of large classical systems , 1975 .

[16]  M. Kac,et al.  Fluctuations and the Boltzmann equation. I , 1976 .

[17]  Suren Poghosyan,et al.  Abstract cluster expansion with applications to statistical mechanical systems , 2008, 0811.4281.

[18]  G. Fitzgerald,et al.  'I. , 2019, Australian journal of primary health.

[19]  S. Méléard.  Convergence of the fluctuations for interacting diffusions with jumps associated with boltzmann equations , 1998 .

[20]  L. Saint-Raymond,et al.  Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations , 2020, 2008.10403.

[21]  M. Pulvirenti,et al.  The Boltzmann–Grad limit of a hard sphere system: analysis of the correlation error , 2014, 1405.4676.

[22]  Ryan Denlinger.  The Propagation of Chaos for a Rarefied Gas of Hard Spheres in the Whole Space , 2016, 1605.00589.

[23]  L. Saint-Raymond,et al.  One-sided convergence in the Boltzmann-Grad limit , 2016, 1612.03722.

[24]  Isabelle Gallagher,et al.  From Newton to Boltzmann: Hard Spheres and Short-range Potentials , 2012, 1208.5753.