Uniform Propagation of Chaos for Kac’s 1D Particle System

Abstract

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 + ǫ 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.

Uniform propagation of chaos for the thermostated Kac model

“…Chaoticity, and thus propagation of chaos, can be made quantitative. For Kac's model this was done in [6] using Wasserstein distances, defined below, and providing explicit convergence rates in N which are uniform in time. Similar quantitative results for the spatially homogeneous Boltzmann equation can be found for instance in [8,14].…”

“…The proof of Theorem 2 is based on a coupling argument developed in [7] and later used in [6]. This argument makes use of a probabilistic object called the Boltzmann process, which is a stochastic proces (Z t ) t≥0 satisfying Law(Z t ) = f t for all t ≥ 0.…”

“…However, notice that Z i t and Z j t have a simultaneous jump whenever V i t and V j t undergo a Kac collision, which implies that Z i t and Z j t are not independent. In order for this construction to be useful, one needs to prove that these Boltzmann processes become asymptotically independent as N → ∞, as is done in [6,7]. This is the content of the following lemma, which moreover provides explicit rates in N , uniformly on time:…”

“…which is a linearized version of (6). This equation has a unique solution in the space C([0, ∞), M) because the mapping ν → B[ν, f t ] is non-expanding in total variation for all t. Since f t is a solution of this linearized version, we must have that ℓ t = f t .…”

“…We now specify the coupling construction that will allow us to prove our main result. We closely follow [7], see also [6]. The key idea is to define a system Z t = (Z 1 t , .…”