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Fluctuations in classical sum rules.

John R Elton¹, Arul Lakshminarayan, Steven Tomsovic

¹Max-Planck-Institut für Physik Komplexer Systeme, D-01187 Dresden, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|January 15, 2011

Summary

This summary is machine-generated.

Classical sum rules are analyzed for global convergence and local fluctuations. A lazy baker map model reveals sum rule convergence is governed by Pollicott-Ruelle resonances, with fluctuations controlled by phase-space volume.

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Area of Science:

Classical mechanics
Dynamical systems theory
Statistical physics

Background:

Classical sum rules appear in diverse physical systems, with asymptotic expressions often derived for large orbital periods.
While chaotic systems may exhibit rapid global sum-rule convergence, individual contributions can fluctuate with time-dependent widths.

Purpose of the Study:

To investigate the global convergence and local fluctuations of classical sum rules.
To analytically explore these properties using an idealized dynamical system.

Main Methods:

Utilizing a simplified lazy baker map as an analytical model for classical sum rules.
Applying the model to the Hannay-Ozorio sum rule to derive corrections and fluctuation characteristics.

Main Results:

The convergence rate of the Hannay-Ozorio sum rule is determined by Pollicott-Ruelle resonances.
Analytical expressions for local and global convergence boundaries are established.
The width of fluctuations depends on the region of application, growing as the region shrinks.

Conclusions:

The lazy baker map provides an effective framework for studying classical sum rules, their convergence, and fluctuations.
Fluctuations can be controlled by adjusting the time-dependent phase-space volume, which decays exponentially in this model.