We consider a stochastic $N$-particle model for the spatially homogeneousBoltzmann evolution and prove its convergence to the associated Boltzmannequation when $N\to \infty$. For any time $T>0$ we bound the distance betweenthe empirical measure of the particle system and the measure given by theBoltzmann evolution in some homogeneous negative Sobolev space. The control weget is Gaussian, i.e. we prove that the distance is bigger than $x N^{-1/2}$with a probability of type $O(e^{-x^2})$. The two main ingredients are first acontrol of fluctuations due to the discrete nature of collisions, secondly aLipschitz continuity for the Boltzmann collision kernel. The latter condition,in our present setting, is only satisfied for Maxwellian models. Numericalcomputations tend to show that our results are useful in practice.
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机译：我们考虑空间均匀Boltzmann演化的随机$ N $粒子模型，并证明当$ N \ to \ infty $时其收敛到相关的玻耳兹曼方程。在任何时间$ T> 0 $内，我们都限制了质点系统的经验测度与由玻尔兹曼演化给出的测度在某个齐次负Sobolev空间中的距离。我们得到的控制是高斯控制，即我们证明距离大于$ x N ^ {-1/2} $，概率为$ O（e ^ {-x ^ 2}）$。这两个主要因素是：首先是由于碰撞的离散性而导致的波动控制；其次是玻尔兹曼碰撞核的Lipschitz连续性。在我们目前的情况下，后一种条件仅适用于麦克斯韦模型。数值计算倾向于表明我们的结果在实践中是有用的。
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