具有缓变相空间结构的Hamilton系统的输运
Transport in Hamiltonian systems with slowly changing phase space structure
摘要：
具有弱混沌扰动的哈密顿系统的输运问题在过去已经得到了大量的研究。本文介绍了一类新的问题：具有缓变相空间结构的哈密顿系统中的输运问题，这些问题不是给定哈密顿量的一阶扰动。这类问题在许多应用中是非常重要的，例如在天体力学中。作为一个例子，我们研究了一类一维哈密顿量，明确依赖于时间和随机外部参数。外部参数的变化导致相空间结构的畸变：混沌、弱混沌和规则集随时间变化。我们表明，理论预测的传输率可以在限制的随机参数的变化是非常缓慢的相比，哈密顿动力学。在哈密顿动力学对固定的参数值可积的经典情形下，可以得到精确的渐近结果。对于更有趣的混沌哈密顿动力学的情况下，我们表明，两种机制有助于运输。对于一定范围的参数变化，一个机制-称为迁移与混合区一起传输-是占主导地位的。然后，我们能够模拟运输相空间的马尔可夫模型，局部扩散模型，并给出合理良好的运输估计。
英文摘要：
Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are not order one perturbations of a given Hamiltonian. This class of problems is very important for many applications, for instance in celestial mechanics. As an example, we study a class of one-dimensional Hamiltonians that depend explicitly on time and on stochastic external parameters. The variations of the external parameters are responsible for a distortion of the phase space structures: chaotic, weakly chaotic and regular sets change with time. We show that theoretical predictions of transport rates can be made in the limit where the variations of the stochastic parameters are very slow compared to the Hamiltonian dynamics. Exact asymptotic results can be obtained in the classical case where the Hamiltonian dynamics is integrable for fixed values of the parameters. For the more interesting chaotic Hamiltonian dynamics case, we show that two mechanisms contribute to the transport. For some range of the parameter variations, one mechanism -called transport by migration together with the mixing regions - is dominant. We are then able to model transport in phase space by a Markov model, the local diffusion model, and to give reasonably good transport estimates.
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