Fluctuations in classical sum rules
| Autor: | Steven Tomsovic, John R. Elton, Arul Lakshminarayan |
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| Rok vydání: | 2010 |
| Předmět: |
Length scale
Sum rule in integration FOS: Physical sciences Dynamical Systems (math.DS) Global convergence 01 natural sciences Phase-space volume 010305 fluids & plasmas Time dependent Ideal systems Simple (abstract algebra) Mesoscale and Nanoscale Physics (cond-mat.mes-hall) 0103 physical sciences Convergence (routing) FOS: Mathematics Phase spaces Limit (mathematics) Statistical physics Mathematics - Dynamical Systems High energy physics 010306 general physics Mathematics Discrete mathematics Asymptotic expressions Condensed Matter - Mesoscale and Nanoscale Physics Local fluctuations Phase space methods Approximation theory Chaotic systems Rate of convergence Nonlinear Sciences - Chaotic Dynamics Orbital periods Action (physics) Sum rule in quantum mechanics Chaotic Dynamics (nlin.CD) Sum rule |
| Zdroj: | Physical Review E. 82 |
| ISSN: | 1550-2376 1539-3755 |
| Popis: | 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. Comment: 15 pages, 2 figures, submitted to Phys. Rev. E |
| Databáze: | OpenAIRE |
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