Determination of the Spectral Gap in the Kac Model for Physical Momentum and Energy-Conserving Collisions
Determination of the Spectral Gap in the Kac Model for Physical Momentum and Energy-Conserving Collisions
复制标题
物理动量和节能碰撞 Kac 模型中谱间隙的确定
DOI：
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发表时间：
2007
影响因子：
2
通讯作者：
M. Loss
中科院分区：
文献类型：
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作者：
E. Carlen;J. Geronimo;M. Loss
Kac模型描述了具有三维速度的N个粒子的气体通过随机行走的局部演化，其中步骤对应于守恒动量和能量的二元碰撞。这个行走的状态空间是一个维度为3 N- 4$的球体。卡茨猜想关注的是这种行走的一步转移算子Q中的谱间隙。在本文中，我们计算的精确谱间隙。在以前的工作中，由卡伦，卡瓦略，和损失，其中一个下界的谱间隙被证明，我们使用的方法，涉及的频谱性质的Q的频谱性质的一个更简单的运营商P，这只是一个平均的某些noncommuting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，以确定频谱特性的Q，而不是简单地使用频谱间隙的P约束的频谱间隙的Q，归纳在N，如在以前的工作。这里开发的方法可以应用于许多其他高…
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 noncommuting 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 hig...