Open Access
Translator Disclaimer
April 2016 A consistency estimate for Kac’s model of elastic collisions in a dilute gas
James Norris
Ann. Appl. Probab. 26(2): 1029-1081 (April 2016). DOI: 10.1214/15-AAP1111


An explicit estimate is derived for Kac’s mean-field model of colliding hard spheres, which compares, in a Wasserstein distance, the empirical velocity distributions for two versions of the model based on different numbers of particles. For suitable initial data, with high probability, the two processes agree to within a tolerance of order $N^{-1/d}$, where $N$ is the smaller particle number and $d$ is the dimension, provided that $d\ge3$. From this estimate we can deduce that the spatially homogeneous Boltzmann equation is well posed in a class of measure-valued processes and provides a good approximation to the Kac process when the number of particles is large. We also prove in an appendix a basic lemma on the total variation of time-integrals of time-dependent signed measures.


Download Citation

James Norris. "A consistency estimate for Kac’s model of elastic collisions in a dilute gas." Ann. Appl. Probab. 26 (2) 1029 - 1081, April 2016.


Received: 1 May 2014; Revised: 1 March 2015; Published: April 2016
First available in Project Euclid: 22 March 2016

zbMATH: 1339.60140
MathSciNet: MR3476632
Digital Object Identifier: 10.1214/15-AAP1111

Primary: 60J25
Secondary: 35Q20

Keywords: Boltzmann equation , Kac process , Law of Large Numbers , Wasserstein distance

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.26 • No. 2 • April 2016
Back to Top