标题： 经典求和规则中的涨落 显示英文标题

标题： Fluctuations in classical sum rules

摘要： 经典求和规则出现在各种物理情境中。 对于许多这些求和规则，在轨道周期较长（或作用量较大）的极限情况下已经得出了渐近表达式。 尽管从全局角度来看，混沌系统的求和规则收敛可能非常迅速地指数级加快，但求和中的各个贡献可能会随时间而波动，其宽度会发散。 我们的兴趣在于求和规则的全局收敛性以及它们的局部波动。 结果表明，一个简单的懒惰面包师映射提供了一个理想的系统，其中可以解析地推导出经典求和规则、它们的修正项以及它们的波动。 这一点在汉恩-奥齐奥里奥求和规则中得到了详细推导。 在这一特定情况下，发现求和规则的收敛速率由波利科特-鲁埃勒共振所支配，并给出了求和规则可能收敛的局部和全局边界。 此外，还考虑了波动的宽度并进行了解析推导，结果表明其与求和规则应用区域的位置有有趣的依赖关系。 还发现，随着应用区域的大小减小，波动会增大。 这表明，通过考虑一个随时间变化的相空间体积，可以控制波动的长度尺度，而对于懒惰面包师映射来说，这个相空间体积随时间呈指数级快速减小。 显示英文摘要

摘要： Classical sum rules arise in a wide variety of physical contexts. Asymptotic expressions have been derived for many of these sum rules in the limit of long orbital period (or large action). Although sum rule convergence may well be exponentially rapid for chaotic systems in a global sense with time, individual contributions to the sums may fluctuate with a width which diverges in time. Our interest is in the global convergence of sum rules as well as their local fluctuations. It turns out that a simple version of a lazy baker map gives an ideal system in which classical sum rules, their corrections, and their fluctuations can be worked out analytically. This is worked out in detail for the Hannay-Ozorio sum rule. In this particular case the rate of convergence of the sum rule is found to be governed by the Pollicott-Ruelle resonances, and both local and global boundaries for which the sum rule may converge are given. In addition, the width of the fluctuations is considered and worked out analytically, and it is shown to have an interesting dependence on the location of the region over which the sum rule is applied. It is also found that as the region of application is decreased in size the fluctuations grow. This suggests a way of controlling the length scale of the fluctuations by considering a time dependent phase space volume, which for the lazy baker map decreases exponentially rapidly with time.