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Nonlinear Stability Criteria for the HMF Model

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References

  1. Antoni M., Ruffo S.: Clustering and relaxation in Hamiltonian long-range dynamics. Phys. Rev. E 52, 2361 (1995)

    Article  ADS  Google Scholar 

  2. Antoniazzi A., Fanelli D., Ruffo S., Yamaguchi Y.Y.: Nonequilibrium tricritical point in a system with long-range interactions. Phys. Rev. Lett. 99, 040601 (2007)

    Article  ADS  Google Scholar 

  3. Barré J., Bouchet F., Dauxois T., Ruffo S., Yamaguchi Y.Y.: The Vlasov equation and the Hamiltonian mean-field model. Phys. A 365, 177 (2006)

    Article  Google Scholar 

  4. Barré, J., Olivetti, A., Yamaguchi, Y.Y.: Dynamics of perturbations around inhomogeneous backgrounds in the HMF model. J. Stat. Mech. 2010, P08002 (2010)

  5. Barré J., Olivetti A., Yamaguchi Y.Y.: Algebraic damping in the one-dimensional Vlasov equation. J. Phys. A Math. Gen. 44, 405502 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barré J., Yamaguchi Y.Y.: Small traveling clusters in attractive and repulsive Hamiltonian mean-field models. Phys. Rev. E 79, 036208 (2009)

    Article  ADS  Google Scholar 

  7. Barré J., Yamaguchi Y.Y.: On the neighborhood of an inhomogeneous stable stationary solution of the Vlasov equation—case of an attractive cosine potential. J. Math. Phys. 56, 081502 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Caglioti E., Rousset F.: Long time estimates in the mean field limit. Arch. Ration. Mech. Anal. 190(3), 517–547 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caglioti E., Rousset F.: Quasi-stationary states for particle systems in the mean-field limit. J. Stat. Phys. 129(2), 241–263 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Chavanis P.-H.: Lynden-Bell and Tsallis distributions for the HMF model. Eur. Phys. J. B 53, 487 (2006)

    Article  ADS  Google Scholar 

  11. Chavanis P.-H., Campa A.: Inhomogeneous Tsallis distributions in the HMF model. Eur. Phys. J. B 76, 581 (2010)

    Article  ADS  MATH  Google Scholar 

  12. Chavanis P.-H., Vatteville J., Bouchet F.: Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model. Eur. Phys. J. B 46, 61 (2005)

    Article  ADS  Google Scholar 

  13. Faou E., Rousset F.: Landau damping in Sobolev spaces for the Vlasov-HMF model. Arch. Ration. Mech. Anal. 219, 887–902 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lemou M.: Extended rearrangement inequalities and applications to some quantitative stability results. Commun. Math. Phys. 348(2), 695–727 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Lemou M., Méhats F., Raphaël P.: Orbital stability of spherical galactic models. Invent. Math. 187, 145–194 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Lemou M., Méhats F., Raphaël P.: A new variational approach to the stability of gravitational systems. Comm. Math. Phys. 302(1), 161–224 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Lieb, E.H., Loss, M.: Analysis. Second Edition. Graduate Studies in Mathematics, vol. 14. American Mathematical Society, Providence, 2001

  18. Messer J., Spohn H.: Statistical mechanics of the isothermal Lane–Emden equation. J. Stat. Phys. 29, 561 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  19. Ogawa S.: Spectral and formal stability criteria of spatially inhomogeneous solutions to the Vlasov equation for the Hamiltonian mean-field model. Phys. Rev. E 87, 062107 (2013)

    Article  ADS  Google Scholar 

  20. Ogawa S., Yamaguchi Y.Y.: Precise determination of the nonequilibrium tricritical point based on Lynden-Bell theory in the Hamiltonian mean-field model. Phys. Rev. E 84, 061140 (2011)

    Article  ADS  Google Scholar 

  21. Rakotoson, J.-M.: Réarrangement relatif, Un instrument d’estimations dans les problèmes aux limites. Mathématiques & Applications, vol. 64. Springer, Berlin, 2008

  22. Staniscia, F., Chavanis, P.H., De Ninno, G.: Out-of-equilibrium phase transitions in the HMF model: a closer look. Phys. Rev. E 83, 051111, 2011

  23. Yamaguchi Y.Y.: Construction of traveling clusters in the Hamiltonian mean-field model by nonequilibrium statistical mechanics and Bernstein–Greene–Kruskal waves. Phys. Rev. E 84, 016211 (2011)

    Article  ADS  Google Scholar 

  24. Yamaguchi Y.Y., Barré J., Bouchet F., Dauxois T., Ruffo S.: Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model. Phys. A 337, 36 (2004)

    Article  Google Scholar 

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Correspondence to Mohammed Lemou.

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Communicated by L. Saint-Raymond

The authors acknowledge support by the ANR project Moonrise ANR-14-CE23-0007-01. The work of A. M. Luz was supported by the Brazilian National Council for Scientific and Technological Development (CNPq) under the program “Science without Borders” 249279/2013-4.

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Lemou, M., Luz, A.M. & Méhats, F. Nonlinear Stability Criteria for the HMF Model. Arch Rational Mech Anal 224, 353–380 (2017). https://doi.org/10.1007/s00205-017-1077-4

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  • DOI: https://doi.org/10.1007/s00205-017-1077-4

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