Let $f(\cdot,t)$ be the probability density function which represents the solution of Kac's equation at time $t$, with initial data $f_0$, and let $g_σ$ be the Gaussian density with zero mean and variance $σ^2$, $σ^2$ being the value of the second moment of $f_0$. This is the first study which proves that the total variation distance between $f(\cdot,t)$ and $g_σ$ goes to zero, as $t\to +\infty$, with an exponential rate equal to -1/4. In the present paper, this fact is proved on the sole assumption that $f_0$ has finite fourth moment and its Fourier transform $φ_0$ satisfies $|φ_0(ξ)|=o(|ξ|^{-p})$ as $|ξ|\to+\infty$, for some $p>0$. These hypotheses are definitely weaker than those considered so far in the state-of-the-art literature, which in any case, obtains less precise rates.
Large Deviations of Kac's Conservative Particle System and Energy Non-Conserving Solutions to the Boltzmann Equation: A Counterexample to the Predicted Rate Function
本文研究了不可压缩粒子系统在碰撞过程中的混沌传播和平均场极限。我们将考虑的主要例子是产生玻尔兹曼方程的卡茨和麦基恩\cite{Kac1956,McKean1967}的多粒子跳跃过程。我们解决了卡茨提出的猜想\cite{Kac1956}，激发他的计划，在严格的连接之间的长期行为的碰撞多粒子系统和它的平均场极限之一，有界以及无界碰撞率。受Grünbaum \cite{Grunbaum}的启发，我们开发了一种抽象方法，将混沌传播问题简化为证明生成算子的纯函数估计（{ \em一致性估计}），以及非线性极限方程流的可微性估计（{ \em稳定性估计}）。这使我们能够在平均场极限方程的水平上而不是在粒子系统的水平上（如卡茨所提出的）利用耗散性。利用这种方法，我们证明了：（1）对一族态的混沌性的定量估计，即在时间上一致的估计。(2)如\cite{CCLLV}中定义的{ \it熵无序性}的传播。(3)估计松弛至平衡的时间，即\emph{independent of the number of particles in the system}。我们的结果涵盖了碰撞率无界的两个主要玻尔兹曼物理碰撞过程：硬球和\emph{true}麦克斯韦分子相互作用。这些模型的\emph{stability estimates}的证明需要大量的分析工作和新的估计。
