# Fluctuations in classical sum rules - Nonlinear Sciences > Chaotic Dynamics

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Abstract: Classical sum rules arise in a wide variety of physical contexts. Asymptoticexpressions have been derived for many of these sum rules in the limit of longorbital period or large action. Although sum rule convergence may well beexponentially rapid for chaotic systems in a global sense with time, individualcontributions 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 localfluctuations. It turns out that a simple version of a lazy baker map gives anideal system in which classical sum rules, their corrections, and theirfluctuations can be worked out analytically. This is worked out in detail forthe Hannay-Ozorio sum rule. In this particular case the rate of convergence ofthe sum rule is found to be governed by the Pollicott-Ruelle resonances, andboth local and global boundaries for which the sum rule may converge are given.In addition, the width of the fluctuations is considered and worked outanalytically, and it is shown to have an interesting dependence on the locationof the region over which the sum rule is applied. It is also found that as theregion of application is decreased in size the fluctuations grow. This suggestsa way of controlling the length scale of the fluctuations by considering a timedependent phase space volume, which for the lazy baker map decreasesexponentially rapidly with time.

Author: ** John R. Elton, Arul Lakshminarayan, Steven Tomsovic**

Source: https://arxiv.org/