TY - JOUR
T1 - Uniform Propagation of Chaos for Kac’s 1D Particle System
AU - Cortez, Roberto
N1 - Funding Information:
The author thanks Joaquin Fontbona and Jean-François Jabir for very useful suggestions and corrections of earlier versions of this manuscript. This work was supported by Fondecyt Postdoctoral Project 3160250.
Publisher Copyright:
© 2016, Springer Science+Business Media New York.
PY - 2016/12/1
Y1 - 2016/12/1
N2 - In this paper we study Kac’s 1D particle system, consisting of the velocities of N particles colliding at constant rate and randomly exchanging energies. We prove uniform (in time) propagation of chaos in Wasserstein distance with explicit polynomial rates in N, for both the squared (i.e., the energy) and non-squared particle system. These rates are of order N- 1 / 3 (almost, in the non-squared case), assuming that the initial distribution of the limit nonlinear equation has finite moments of sufficiently high order (4 + ϵ is enough when using the 2-Wasserstein distance). The proof relies on a convenient parametrization of the collision recently introduced by Hauray, as well as on a coupling technique developed by Cortez and Fontbona.
AB - In this paper we study Kac’s 1D particle system, consisting of the velocities of N particles colliding at constant rate and randomly exchanging energies. We prove uniform (in time) propagation of chaos in Wasserstein distance with explicit polynomial rates in N, for both the squared (i.e., the energy) and non-squared particle system. These rates are of order N- 1 / 3 (almost, in the non-squared case), assuming that the initial distribution of the limit nonlinear equation has finite moments of sufficiently high order (4 + ϵ is enough when using the 2-Wasserstein distance). The proof relies on a convenient parametrization of the collision recently introduced by Hauray, as well as on a coupling technique developed by Cortez and Fontbona.
KW - Kac particle system
KW - Kinetic theory
KW - Propagation of chaos
UR - http://www.scopus.com/inward/record.url?scp=85028256773&partnerID=8YFLogxK
U2 - 10.1007/s10955-016-1674-x
DO - 10.1007/s10955-016-1674-x
M3 - Article
AN - SCOPUS:85028256773
SN - 0022-4715
VL - 165
SP - 1102
EP - 1113
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 6
ER -