Sinai Billiards Under Small External Forces
Sinai Billiards Under Small External Forces
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小外力下的西奈台球
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
N. Chernov
中科院分区：
文献类型：
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作者：
N. Chernov
抽象的。考虑一个粒子在环面上自由运动，并与一些固定的凸体弹性碰撞。这种模型被称为周期洛伦兹气体，或西奈台球。它是一个具有光滑不变测度的哈密顿系统，其遍历和统计性质已经得到了很好的研究。现在让粒子受到一个小的外力。这个新系统不太可能有一个平滑的不变度量。然后，西奈-鲁埃尔-鲍恩(SRB)度量描述了典型相轨迹的演变。我们发现关于碰撞映射的SRB测度存在的外力的一般充分条件是唯一的，并且具有良好的遍历性和统计性质，包括伯努利性质和关联的指数衰减。
Abstract. Consider a particle moving freely on the torus and colliding elastically with some fixed convex bodies. This model is called a periodic Lorentz gas, or a Sinai billiard. It is a Hamiltonian system with a smooth invariant measure, whose ergodic and statistical properties have been well investigated. Now let the particle be subjected to a small external force. This new system is not likely to have a smooth invariant measure. Then a Sinai-Ruelle-Bowen (SRB) measure describes the evolution of typical phase trajectories. We find general sufficient conditions on the external force under which the SRB measure for the collision map exists, is unique, and enjoys good ergodic and statistical properties, including Bernoulliness and an exponential decay of correlations.