$N$ -保守Kac的统一定量Tanaka定理$N$ -具有Maxwell分子的粒子系统
A $N$-uniform quantitative Tanaka's theorem for the conservative Kac's $N$-particle system with Maxwell molecules
摘要：
本文研究具有任意角分布的麦克斯韦分子的空间齐次玻尔兹曼方程。紧接着Kac的程序，重点讨论了与之相关的守恒Kac随机$N$ -粒子系统，一个具有能量守恒和总动量守恒的二元碰撞的马尔可夫过程.一个明确的马尔可夫耦合（一个概率，马尔可夫耦合的两个副本的过程）的构建，同时使用碰撞，并平行耦合的每个二进制随机碰撞的碰撞方向的球体上。两个耦合系统之间的欧几里德距离几乎肯定随时间减小，并且明确计算相关的二次耦合创建（平均平方耦合距离的时间变化）。然后，一族（索引为$δ> 0$）的$N$ -一致的“弱"耦合/耦合创建不等式被证明，导致$N$ -一致的幂律趋于平衡的顺序${\sim}_{ t \to + \infty} t^{-δ} $，常数取决于时刻的速度分布严格大于$2(1 + δ)$。对$4$阶矩的情况进行了显式处理，实现了Kac的程序，而不需要任何混沌传播分析。最后，两个反例表明，该方法：（i）需要依赖于$>2$ -时刻，（ii）不能提供收缩在二次Wasserstein距离在任何情况下。
英文摘要：
This paper considers the space homogenous Boltzmann equation with Maxwell molecules and arbitrary angular distribution. Following Kac's program, emphasis is laid on the the associated conservative Kac's stochastic $N$-particle system, a Markov process with binary collisions conserving energy and total momentum. An explicit Markov coupling (a probabilistic, Markovian coupling of two copies of the process) is constructed, using simultaneous collisions, and parallel coupling of each binary random collision on the sphere of collisional directions. The euclidean distance between the two coupled systems is almost surely decreasing with respect to time, and the associated quadratic coupling creation (the time variation of the averaged squared coupling distance) is computed explicitly. Then, a family (indexed by $δ> 0$) of $N$-uniform ''weak'' coupling / coupling creation inequalities are proven, that leads to a $N$-uniform power law trend to equilibrium of order ${\sim}_{ t \to + \infty} t^{-δ} $, with constants depending on moments of the velocity distributions strictly greater than $2(1 + δ)$. The case of order $4$ moment is treated explicitly, achieving Kac's program without any chaos propagation analysis. Finally, two counter-examples are suggested indicating that the method: (i) requires the dependance on $>2$-moments, and (ii) cannot provide contractivity in quadratic Wasserstein distance in any case.
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