Fluctuations in classical sum rules

John R Elton; Arul Lakshminarayan; Steven Tomsovic

doi:10.1103/PhysRevE.82.046223

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Fluctuations in classical sum rules

John R Elton, Arul Lakshminarayan and Steven Tomsovic

Physical review. E, Statistical, nonlinear, and soft matter physics, Vol.82(4), pp.046223-046223

05/21/2009

DOI: https://doi.org/10.1103/PhysRevE.82.046223

Handle:

https://hdl.handle.net/2376/114975

PMID: 21230381

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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.

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Title: Fluctuations in classical sum rules
Creators: John R Elton
Arul Lakshminarayan
Steven Tomsovic
Publication Details: Physical review. E, Statistical, nonlinear, and soft matter physics, Vol.82(4), pp.046223-046223
Academic Unit: Physics and Astronomy, Department of
Identifiers: 99900547778801842
Language: English
Resource Type: Journal article