In this paper, we consider the stochastic dynamics of a finite system of particles in a finite volume (Kac-like particle system) which annihilate with probability \(\alpha \in (0,1)\) or collide elastically with probability \(1-\alpha \). We first establish the well-posedness of the particle system which exhibits no conserved quantities. We rigorously prove that, in some mean-field limit, a suitable hierarchy of kinetic equations is recovered for which tensorized solution to the homogenous Boltzmann with annihilation is a solution. For bounded collision kernels, this shows in particular that propagation of chaos holds true. Furthermore, we make conjectures about the limit behaviour of the particle system when hard-sphere interactions are taken into account.

中文翻译：

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动力学An灭的Kac模型

在本文中，我们考虑了有限体积中的有限粒子系统的随机动力学（类Kac粒子系统），该系统以概率\（\ alpha \ in（0,1）\） an灭或以概率\（ 1- \ alpha \）。我们首先建立没有保留量的粒子系统的适定性。我们严格证明，在某些平均场极限下，可以找到合适的动力学方程式层次结构，对于该方程式，齐次玻尔兹曼的an态张量解是一个解。对于有界碰撞核，这特别表明混沌传播是正确的。此外，当考虑到硬球相互作用时，我们可以推测粒子系统的极限行为。