Advertisement

Importance sampling of rare events in chaotic systems

  • Jorge C. Leitão
  • João M. Viana Parente Lopes
  • Eduardo G. Altmann
Open Access
Colloquium

Abstract

Finding and sampling rare trajectories in dynamical systems is a difficult computational task underlying numerous problems and applications. In this paper we show how to construct Metropolis-Hastings Monte-Carlo methods that can efficiently sample rare trajectories in the (extremely rough) phase space of chaotic systems. As examples of our general framework we compute the distribution of finite-time Lyapunov exponents (in different chaotic maps) and the distribution of escape times (in transient-chaos problems). Our methods sample exponentially rare states in polynomial number of samples (in both low- and high-dimensional systems). An open-source software that implements our algorithms and reproduces our results can be found in reference [J. Leitao, A library to sample chaotic systems, 2017, https://github.com/jorgecarleitao/chaospp].

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    S. Albeverio, V. Jentsch, H. Kantz, Extreme Events in Nature and Society (Springer, 2006), ISBN 978-3-540-28610-3Google Scholar
  2. 2.
    P. Holmes, Phys. Rep. 193, 137 (1990)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    A.E. Motter, D.K. Campbell, Phys. Today 66, 27 (2013)ADSCrossRefGoogle Scholar
  4. 4.
    E. Ott, Chaos in Dynamical Systems, 2nd edn. (Cambridge University Press, Cambridge, 1993), ISBN 0 521 43215 4Google Scholar
  5. 5.
    E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963)ADSCrossRefGoogle Scholar
  6. 6.
    M.E.J. Newman, G.T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press, New York, USA, 2002), ISBN 0198517971Google Scholar
  7. 7.
    C.P. Robert, G. Casella, Monte Carlo statistical methods, Springer texts in statistics, 2nd edn. (Springer, Berlin, 2005), ISBN 0387212396Google Scholar
  8. 8.
    J.A. Bucklew, Introduction to Rare Event Simulation, Springer Series in Statistics (Springer New York, New York, NY, 2004), ISBN 978-1-4419-1893-2, http://link.springer.com/10.1007/978-1-4757-4078-3
  9. 9.
    V. Lucarini, D. Faranda, A.C.M. Freitas, J.M. Freitas, T. Kuna, M. Holland, M. Nicol, M. Todd, S. Vaienti, arXiv:1605.07006, 2016
  10. 10.
    Y.C. Lai, T. Tél, Transient chaos: complex dynamics in finite time scales, 1st edn. (Springer, New York, 2010), Vol. 173Google Scholar
  11. 11.
    D. Sweet, H.E. Nusse, J.A. Yorke, Phys. Rev. Lett. 86, 2261 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    E.M. Bollt, Int. J. Bifurc. Chaos 15, 1615 (2005)CrossRefGoogle Scholar
  13. 13.
    C. Dellago, P.G. Bolhuis, P.L. Geissler, Adv. Chem. Phys. 123, 1 (2002)Google Scholar
  14. 14.
    P.G. Bolhuis, D. Chandler, C. Dellago, P.L. Geissler, Annu. Rev. Chem. 53, 291 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    C. Giardinà, J. Kurchan, L. Peliti, Phys. Rev. Lett. 96, 120603 (2006)ADSCrossRefGoogle Scholar
  16. 16.
    J. Tailleur, J. Kurchan, Nat. Phys. 3, 203 (2007)CrossRefGoogle Scholar
  17. 17.
    T. Yanagita, Y. Iba, J. Stat. Mech.: Theory Exp. 2009, P02043 (2009)CrossRefGoogle Scholar
  18. 18.
    A. Kitajima, Y. Iba, Comput. Phys. Commun. 182, 251 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    T. Laffargue, K.D.N.T. Lam, J. Kurchan, J. Tailleur, J. Phys. A: Math. Theor. 46, 254002 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    J. Wouters, F. Bouchet, arXiv:1511.02703, 2015, pp. 1–29
  21. 21.
    P. Geiger, C. Dellago, Chem. Phys. 375, 309 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    S.I. Sasa, K. Hayashi, Europhys. Lett. 74, 156 (2006)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    M. Grünwald, C. Dellago, P.L. Geissler, J. Chem. Phys. 129, 194101 (2008)ADSCrossRefGoogle Scholar
  24. 24.
    J.C. Leitão, J.M.V.P. Lopes, E.G. Altmann, Phys. Rev. Lett. 110, 220601 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    J.C. Leitão, J.M.V.P. Lopes, E.G. Altmann, Phys. Rev. E 90, 052916 (2014)ADSCrossRefGoogle Scholar
  26. 26.
    P. Cvitanovic, R. Artuso, R. Mainieri, G. Tanner, G. Vattay, Chaos book (Niels Bohr Institute, Copenhagen, 2016)Google Scholar
  27. 27.
    P. Grassberger, H. Kantz, Phys. Lett. A 113, 167 (1985)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    P. Grassberger, R. Badii, A. Politi, J. Stat. Phys. 51, 135 (1988)ADSCrossRefGoogle Scholar
  29. 29.
    M. Sepúlveda, R. Badii, E. Pollak, Phys. Rev. Lett. 63, 1226 (1989)ADSCrossRefGoogle Scholar
  30. 30.
    D. Beigie, A. Leonard, S. Wiggins, Phys. Rev. Lett. 70, 275 (1993)ADSCrossRefGoogle Scholar
  31. 31.
    C. Amitrano, R. Berry, Phys. Rev. Lett. 68, 729 (1992)ADSCrossRefGoogle Scholar
  32. 32.
    S. Olmi, Chaos (Woodbury, NY) 25, 123125 (2015)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    G.M. Zaslavsky, Phys. Rep. Rev. Sect. Phys. Lett. 371, 461 (2002)Google Scholar
  34. 34.
    J. Szezech, S. Lopes, R. Viana, Phys. Lett. A 335, 394 (2005)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    R. Artuso, C. Manchein, Phys. Rev. E 80, 036210 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    C. Manchein, M.W. Beims, J.M. Rost, Chaos (Woodbury, NY) 22, 033137 (2012)ADSCrossRefGoogle Scholar
  37. 37.
    H.D.I. Abarbanel, R. Brown, M.B. Kennel, J. Nonlinear Sci. 1, 175 (1991)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    H.D.I. Abarbanel, R. Brown, M.B. Kennel, J. Nonlinear Sci. 2, 343 (1992)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    C. Beck, F. Schögl, Thermodynamics of chaotic systems: an introduction (Cambridge University Press, 1995), ISBN 9780521433679Google Scholar
  40. 40.
    A. Prasad, R. Ramaswamy, Phys. Rev. E 60, 2761 (1999)ADSCrossRefGoogle Scholar
  41. 41.
    H. Richter, Int. J. Parallel Emerg. Distrib. Syst. 1 (2017)Google Scholar
  42. 42.
    U.H.E. Hansmann, Y. Okamoto, J. Comput. Chem. 14, 1333 (1993)CrossRefGoogle Scholar
  43. 43.
    T. Tél, Y.C. Lai, Phys. Rep. 460, 245 (2008)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    E.G. Altmann, J.S.E. Portela, T. Tél, Rev. Mod. Phys. 85, 869 (2013)ADSCrossRefGoogle Scholar
  45. 45.
    M. Sala, J.C. Leitão, E.G. Altmann, Chaos: Interdiscip. J. Nonlinear Sci. 26, 123124 (2016)CrossRefGoogle Scholar
  46. 46.
    M. Dhamala, Y.C. Lai, E.J. Kostelich, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 64, 056207 (2001)CrossRefGoogle Scholar
  47. 47.
    G. Torrie, J. Valleau, J. Comput. Phys. 23, 187 (1977)ADSCrossRefGoogle Scholar
  48. 48.
    R.H. Swendsen, J.S. Wang, Phys. Rev. Lett. 57, 2607 (1986)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    E. Marinari, G. Parisi, Europhys. Lett. 19, 451 (1992)ADSCrossRefGoogle Scholar
  50. 50.
    Y. Sugita, Y. Okamoto, Chem. Phys. Lett. 314, 141 (1999)ADSCrossRefGoogle Scholar
  51. 51.
    J. Lee, Phys. Rev. Lett. 71, 211 (1993)ADSCrossRefGoogle Scholar
  52. 52.
    B.A. Berg, T. Neuhaus, Phys. Lett. B 267, 249 (1991)ADSCrossRefGoogle Scholar
  53. 53.
    F. Wang, D.P. Landau, Phys. Rev. Lett. 86, 2050 (2001)ADSCrossRefGoogle Scholar
  54. 54.
    J. Viana Lopes, M. Costa, J. Lopes dos Santos, R. Toral, Phys. Rev. E 74, 046702 (2006)ADSCrossRefGoogle Scholar
  55. 55.
    C.G. Zhou, R.N. Bhatt, Phys. Rev. E 72, 025701 (2005)ADSCrossRefGoogle Scholar
  56. 56.
    R.E. Belardinelli, S. Manzi, V.D. Pereyra, Phys. Rev. E 78, 067701 (2008)ADSCrossRefGoogle Scholar
  57. 57.
    S. Trebst, D.A. Huse, M. Troyer, Phys. Rev. E 70, 046701 (2004)ADSCrossRefGoogle Scholar
  58. 58.
    P. Dayal, S. Trebst, S. Wessel, D. Wurtz, M. Troyer, S. Sabhapandit, S.N. Coppersmith, Phys. Rev. Lett. 92, 097201 (2004)ADSCrossRefGoogle Scholar
  59. 59.
    J. Viana Lopes, Ph.D. thesis, Universidade do Porto, 2006Google Scholar
  60. 60.
    M.D. Costa, J.V. Lopes, J.M.B.L. dos Santos, Europhys. Lett. 72, 802 (2005)ADSCrossRefGoogle Scholar
  61. 61.
    G.O. Roberts, A. Gelman, W.R. Gilks, Ann. Appl. Prob. 7, 110 (1997)CrossRefGoogle Scholar
  62. 62.
    R. Fischer, J.C. Leitão, T.P. Peixoto, E.G. Altmann, Phys. Rev. Lett. 115, 188701 (2015)ADSCrossRefGoogle Scholar
  63. 63.
    P. Grassberger, Phys. Rev. E 56, 3682 (1997)ADSCrossRefGoogle Scholar
  64. 64.
    C. Grebogi, S. Hammel, J. Yorke, T. Sauer, Phys. Rev. Lett. 65, 1527 (1990)ADSMathSciNetCrossRefGoogle Scholar
  65. 65.
    Y. Iba, N. Saito, A. Kitajima, Ann. Inst. Stat. Math. 66, 611 (2014)CrossRefGoogle Scholar
  66. 66.
    J. Leitao, A library to sample chaotic systems, 2017, https://github.com/jorgecarleitao/chaospp
  67. 67.
    M. Grünwald, Ph.D. thesis, University of Wien, 2009Google Scholar
  68. 68.
    C.N. Rowley, T.K. Woo, J. Chem. Phys. 131, 234102 (2009)ADSCrossRefGoogle Scholar
  69. 69.
    N. Eidelson, B. Peters, J. Chem. Phys. 137, 094106 (2012)ADSCrossRefGoogle Scholar
  70. 70.
    B. Schaefer, S. Mohr, M. Amsler, S. Goedecker, arXiv:1401.8081, 2014, p. 15
  71. 71.
    J.C. Leitão, Ph.D., Technical University of Dresden, 2016, http://www.qucosa.de/fileadmin/data/qucosa/documents/20901/main.pdf
  72. 72.
    S. Gupta, J.C. Leitao, E.G. Altmann, Phys. Rev. E 96, 012201 (2017)ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Max Planck Institute for the Physics of Complex SystemsDresdenGermany
  2. 2.DTU Compute, Technical University of DenmarkKgs. LyngbyDenmark
  3. 3.Department of Physics and Center of PhysicsUniversity of MinhoBragaPortugal
  4. 4.Physics Engineering Department, Engineering Faculty of the University of PortoPortoPortugal
  5. 5.School of Mathematics and Statistics, University of SydneySydneyAustralia

Personalised recommendations