Ergodicity and the numerical simulation of Hamiltonian systems
Ergodicity and the numerical simulation of Hamiltonian systems
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DOI：
10.1137/040603802
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发表时间：
2005-01-01
影响因子：
2.1
通讯作者：
Tupper, PF
中科院分区：
文献类型：
--
作者：
Tupper, PF
我们讨论常微分方程哈密顿系统的长期数值模拟。我们的目标是解释辛积分方案（例如 Stormer-Verlet 方法）在分子动力学背景下计算这些系统的准确长期平均值的能力。本文介绍了遍历性的弱化版本，使我们能够研究这个问题。首先，我们通过证明弱遍历性定义是哈密顿系统对扰动具有鲁棒性的属性来证明弱遍历性定义的实用性。其次，我们研究哈密顿系统的遍历性减弱对该系统的数值模拟意味着什么。在数值方法是体积守恒且近似能量守恒的情况下，我们表明对于足够小的步长，长期平均值可以很好地近似。
We discuss the long- time numerical simulation of Hamiltonian systems of ordinary differential equations. Our goal is to explain the ability of symplectic integration schemes such as the Stormer-Verlet method to compute accurate long- time averages for these systems in the context of molecular dynamics. This paper introduces a weakened version of ergodicity that allows us to study this problem. First, we demonstrate the utility of the weakened ergodicity definition by showing that it is a property of Hamiltonian systems robust to perturbations. Second, we study what the weakened ergodicity of a Hamiltonian system implies about numerical simulations of the system. In the case where a numerical method is volume- conserving and approximately energy- conserving, we show that long- time averages are approximated well for sufficiently small step lengths.