Abstract
For a class of idealized chaotic systems (hyperbolic systems) correlations decay exponentially in time. This result is asymptotic and rigorous. The decay rate is related to the Ruelle-Pollicott resonances. Nearly all chaotic model systems, that are studied by physicists, are not hyperbolic. For many such systems it is known that exponential decay takes place for a long time. It may not be asymptotic, but it may persist for a very long time, longer than any time of experimental relevance. In this review a heuristic method for calculation of this exponential decay of correlations in time is presented. It can be applied to model systems, where there are no rigorous results concerning this exponential decay. It was tested for several realistic systems (kicked rotor and kicked top) in addition to idealized systems (baker map and perturbed cat map). The method consists of truncation of the evolution operator (Frobenius-Perron operator), and performing all calculations with the resulting finite dimensional matrix. This finite dimensional approximation can be considered as coarse graining, and is equivalent to the effect of noise. The exponential decay rate of the chaotic system is obtained when the dimensionality of the approximate evolution operator is taken to infinity, resulting in infinitely fine resolution, that is equivalent to vanishing noise. The corresponding Ruelle-Pollicott resonances can be calculated for many systems that are beyond the validity of the Ruelle-Pollicott theorem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
E. Ott, Chaos in Dynamical Systems, (Cambridge University Press, Cambridge, 1997).
V.I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (Addison-Wesley NY, 1989).
F. Haake, Quantum Signatures of Chaos, (Springer-Verlag, Berlin, 1991).
B.G Schuster, Deterministic Chaos, An Introduction, (Physik-Verlag, Weinheim, 1984).
P. Gaspard, Chaos, Scattering and Statistical Mechanics, (Cambridge Press, Cambridge 1998).
J.R. Dorfman, An Introduction to Chaos in Non-Equilibrium Statistical Mechanics, (Cambridge Press, Cambridge 1999).
A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer, NY 1983).
F. L. Moore, J. C. Robinson, C. F. Bharucha, Bala Sundaram and M. G. Raizen, Phys. Rev. Lett. 75, 4598 (1995); C. F. Bharucha, J. C. Robinson, F. L. Moore, Qian Niu, Bala Sundaram and M. G. Raizen, Phys. Rev. E 60, 3881 (1999).
B. Fischer, A. Rosen, A. Bekker and S. Fishman, Phys. Rev. E, R4694 (2000).
**J. Weber, F. Haake, P.A. Braun, C. Manderfeld and P. Å eba, J. Phys. A34, 7195 (2001).
M. V. Berry, in New trends in Nuclear Collective Dynamics, eds: Y. Abe, H. Horiuchi, K. Matsuyanagi, Springer proceedings in Physics. vol 58, 183–186 (1992).
D. Ruelle, Phys. Rev. Lett. 56, 405 (1986).
**H.H. Hasegawa and W.C. Saphir, Phys. Rev. A46, 7401 (1992).
S. Fishman, in Supersymmetry and Trace Formulae, Chaos and Disorder, edited by I.V. Lerner, J.P. Keating, D.E. Khmelnitskii (Kluwer Academic / Plenum Publishers, New York, 1999).
A. Jordan and M. Srednicki, The approach to Ergodicity in the Quantum Baker’s Map, nlin.CD/0108024 (2001), This paper contains also some results on the classical baker’s map, that are relevant for the present review.
G. Blum and O. Agam, Phys. Rev. E62, 1977 (2000).
J. Weber, F. Haake and P. Å eba, Phys. Rev. Lett. 85, 3620 (2000).
C. Manderfeld, J. Weber and F. Haake, J. Phys. A34, 9893 (2001).
**M. Khodas, S. Fishman and O. Agam, Phys. Rev. E62, 4769 (2000).
M. Khodas and S. Fishman, Phys. Rev. Lett. 84, 2837 (2000); Erratum 84, 5918 (2000).
We thank E. Bogomolny, M. Saraceno, M. Srednicki and A. Jordan for illuminating comments.
M. Basilio de Matos and A.M. Ozorio de Almeida, Ann. Phys. (N.Y.) 237, 46 (1995).
R. Balescu, Statistical Dynamics, Matter out of Equilibrium, (Imperial College Press, Singapore, 1983).
A.B. Rechester and R.B. White, Phys. Rev. Lett. 44, 1586 (1980); A.B. Rechester M.N. Rosenbluth and R.B. White, Phys. Rev. A23, 2664 (1981); E. Doron and S. Fishman, Phys. Rev. A37, 2144 (1988).
B. Sundaram and G.M. Zaslavsky, Phys. Rev. E59, 7231 (1999); G.M. Zaslavsky, M. Edelman and B.A. Niyazov, Chaos 7, 159 (1997).
P. Gaspard, G. Nicolis, A. Provata and S. Tasaki, Phys. Rev. E 51, 74 (1995).
A. Ostruszka and K. Życzkowski, Phys. Lett. A 289, 306 (2001).
K. Knopp, Infinite Sequences and Series, (Dover publ. NY, 1956).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fishman, S., Rahav, S. (2002). Relaxation and Noise in Chaotic Systems. In: Garbaczewski, P., Olkiewicz, R. (eds) Dynamics of Dissipation. Lecture Notes in Physics, vol 597. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46122-1_7
Download citation
DOI: https://doi.org/10.1007/3-540-46122-1_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44111-3
Online ISBN: 978-3-540-46122-7
eBook Packages: Springer Book Archive