We study a class of one-dimensional particle systems with true (Bird type) binary interactions, which includes Kac's model of the Boltzmann equation and nonlinear equations for the evolution of wealth distribution arising in kinetic economic models. We obtain explicit rates of convergence for the Wasserstein distance between the law of the particles and their limiting law, which are linear in time and depend in a mild polynomial manner on the number of particles. The proof is based on a novel coupling between the particle system and a suitable system of nonindependent nonlinear processes, as well as on recent sharp estimates for empirical measures.Comment: Published at http://dx.doi.org/10.1214/15-AAP1107 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We prove propagation of chaos at explicit polynomial rates in Wasserstein distance W 2 for Kac's N -particle system associated with the spatially homogeneous Boltzmann equation for Maxwell molecules. Our approach is mainly based on novel probabilistic coupling techniques. Combining them with recent stabilization results for the particle system we obtain, under suitable moments assumptions on the initial distribution, a uniform-in-time estimate of order almost N −1/3 for W 2 2 .

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.

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We study a stochastic N -particle system representing economic agents in a population randomly exchanging their money, which is associated to a class of onedimensional kinetic equations modelling the evolution of the distribution of wealth in a simple market economy, introduced by Matthes and Toscani [19]. We show that, unless the economic exchanges satisfy some exact conservation condition, the p-moments of the particles diverge with time for all p > 1, and converge to 0 for 0 < p < 1. This establishes a qualitative difference with the kinetic equation, whose solution is known to have bounded p-moments, for all p smaller than the Pareto index of the equilibrium distribution. On the other hand, the case of strictly conservative economies is fully treated: using probabilistic coupling techniques, we obtain stability results for the particle system, such as propagation of moments, exponential equilibration, and uniform (in time) propagation of chaos with explicit rate of order N −1/3 .