动量守恒和能量守恒碰撞Kac模型谱隙的确定
Determination of the spectral gap in the Kac model for physical momentum and energy conserving collisions
摘要：
Kac模型描述了具有三维速度的$N$粒子气体通过随机游走的局部演化，其中步骤对应于守恒动量以及能量的二元碰撞。这个行走的状态空间是一个维度为$3N - 4$的球体。卡茨猜想涉及的一步过渡算子$Q$的谱间隙为这种行走。在本文中，我们计算的精确谱间隙。正如在以前的工作中由卡伦，卡瓦略和损失的频谱间隙的下限被证明，我们使用的方法，涉及的频谱特性的$Q$的频谱特性的一个更简单的运营商$P$，这只是一个平均值的某些非交换投影。新的特点是，我们展示了如何使用知识的某些本征函数和本征值的$P$，以确定$Q$的光谱特性，而不是简单地使用$P$的频谱间隙绑定的$Q$的频谱间隙，归纳在$N$，在以前的工作。这里开发的方法可以应用到许多其他的高维随机过程，我们将解释。我们还使用一些深的结果Jacobi多项式获得所需的频谱信息$P$，我们展示了如何通过Jacobi多项式进入我们的问题的身份可以用来获得新的界限Jacobi多项式。
英文摘要：
The Kac model describes the local evolution of a gas of $N$ particles with three dimensional velocities by a random walk in which the steps correspond to binary collisions that conserve momentum as well as energy. The state space of this walk is a sphere of dimension $3N - 4$. The Kac conjecture concerns the spectral gap in the one step transition operator $Q$ for this walk. In this paper, we compute the exact spectral gap. As in previous work by Carlen, Carvalho and Loss where a lower bound on the spectral gap was proved, we use a method that relates the spectral properties of $Q$ to the spectral properties of a simpler operator $P$, which is simply an average of certain non commuting projections. The new feature is that we show how to use a knowledge of certain eigenfunctions and eigenvalues of $P$ to determine spectral properties of $Q$, instead of simply using the spectral gap for $P$ to bound the spectral gap for $Q$, inductively in $N$, as in previous work. The methods developed here can be applied to many other high--dimensional stochastic process, as we shall explain. We also use some deep results on Jacobi polynomials to obtain the required spectral information on $P$, and we show how the identity through which Jacobi polynomials enter our problem may be used to obtain new bounds on Jacobi polynomials.
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