Fluctuations in classical sum rules

TL;DR

This paper analytically investigates the convergence and fluctuations of classical sum rules in chaotic systems, using a lazy baker map to understand how local and global behaviors are influenced by system parameters and phase space regions.

Contribution

It provides an analytical study of sum rule fluctuations and convergence rates in a simple chaotic map, linking these to Pollicott-Ruelle resonances and phase space region size.

Findings

Sum rule convergence is governed by Pollicott-Ruelle resonances.

Fluctuation width depends on the size of the phase space region.

Decreasing the region size increases fluctuation magnitude.

Abstract

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.