论文标题
从广义KAC模型中具有高阶碰撞的Boltzmann方程的推导
Derivation of a Boltzmann equation with higher-order collisions from a generalized Kac model
论文作者
论文摘要
在这项工作中，我们概括了M. KAC的原始多粒子二元随机模型，以得出一个均匀的玻尔兹曼方程，其中包括高阶碰撞项的线性组合。首先，我们证明了一个关于从有限层次结构到无限耦合方程层次结构的收敛的抽象定理。我们将此收敛定理应用于与上述广义KAC模型相对应的边际的层次结构。作为推论，我们证明了与广义KAC模型相关的边缘的混乱传播。特别是，第一个边际收敛于玻尔兹曼方程的解，其中包括有限顺序的相互作用，并且其碰撞内核是Maxwell-type，并具有截止性。
In this work, we generalize M. Kac's original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges towards the solution of a Boltzmann equation including interactions up to a finite order, and whose collision kernel is of Maxwell-type with cut-off.
