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Self-Organized Regularity in Long-Range Systems

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Nonlinear Dynamics New Directions

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 12))

Abstract

Dynamics of many-body long-range interacting systems are investigated. Using the XY-Hamiltonian mean-field model as a case study, we show that regular trajectories, associated with invariant tori of the single-particle dynamics emerge as the number of particles is increased. Moreover, the construction of stationary solutions as well as studies of the maximal Lyapunov exponent of the systems show the same trend towards integrability. This feature provides a dynamical interpretation of the emergence of long-lasting out-of-equilibrium regimes observed generically in long-range systems. Extensions beyond the mean-field system are considered and display similar features.

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Notes

  1. 1.

    We do not differentiate between the observable M and its statistical average \(\langle M\rangle\), for simplicity in notations, except when needed

  2. 2.

    Note that even if one accounts for the linear stability criteria, there are still large sets of distributions which shall generate quasi-stationary dynamics.

  3. 3.

    Actually due to numerical error we renormalize to \((N+\sqrt{N}/4)\tilde{\rho}\), which is in range with expected typical fluctuations.

  4. 4.

    A comparison with higher scheme has been made and gave identical results.

  5. 5.

    Note that \(1/\log(N)\) scalings are also good, and can be explained by the fact that the typical time scale in the separatrix can be estimated from the period of regular motion of the last regular trajectory \(T\sim\log(\delta E)\sim\log(\delta M)\sim\log(N)\).

  6. 6.

    This approximation can be rigorously justified via a detailed expansion see [36] for details.

  7. 7.

    Actually only the ones with \(\kappa<\kappa_c\).

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Acknowledgements

I am grateful to Romain Bachelard for a careful reading of the manuscript, suggesting corrections, references, and extensions to the original text. I am also thankful to him for preparing Figs. 5 and 6. His input was much appreciated. Most of the work presented in this chapter is shared with my coauthors: Duccio Fanelli, Tineke Van den Berg, Stefano Ruffo, Cristel Chandre. I use this occasion to thank them again. Discussions about the possible implications of self-organized regularity first occurred between the author and George M. Zaslavsky, in November 2007, and I take this opportunity to express my gratitude.

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  1. Centre de Physique Théorique, Aix-Marseille Université, Campus de Luminy, Case 907 - 13288, Marseille cedex 9, France

    Xavier Leoncini

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  1. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Hernán González-Aguilar

  2. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Edgardo Ugalde

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Leoncini, X. (2015). Self-Organized Regularity in Long-Range Systems. In: González-Aguilar, H., Ugalde, E. (eds) Nonlinear Dynamics New Directions. Nonlinear Systems and Complexity, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-09864-7_4

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  • DOI: https://doi.org/10.1007/978-3-319-09864-7_4

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