Skip to main content
SpringerLink
Log in

Pseudo generators for under-resolved molecular dynamics

  • Regular Article
  • B. Bridging of Time Scales and Methods for Rare Events
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

An Addendum to this article was published on 15 December 2017

This article has been updated

Abstract

Many features of a molecule which are of physical interest (e.g. molecular conformations, reaction rates) are described in terms of its dynamics in configuration space. This article deals with the projection of molecular dynamics in phase space onto configuration space. Specifically, we study the situation that the phase space dynamics is governed by a stochastic Langevin equation and study its relation with the configurational Smoluchowski equation in the three different scaling regimes: Firstly, the Smoluchowski equations in non-Cartesian geometries are derived from the overdamped limit of the Langevin equation. Secondly, transfer operator methods are used to describe the metastable behaviour of the system at hand, and an explicit small-time asymptotics is derived on which the Smoluchowski equation turns out to govern the dynamics of the position coordinate (without any assumptions on the damping). By using an adequate reduction technique, these considerations are then extended to one-dimensional reaction coordinates. Thirdly, we sketch three different approaches to approximate the metastable dynamics based on time-local information only.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

  • 15 December 2017

    We would like to add the following acknowledgement to our article published in Eur. Phys. J. Special Topics, September 2015, Volume 224, Issue 12, pp. 2463–2490.

References

  1. R.B. Best, G. Hummer, Proc. Natl. Acad. Sci. USA 107(3), 1088 (2010)

    Article  ADS  Google Scholar 

  2. A. Bittracher, P. Koltai, O. Junge, Pseudo generators of spatial transfer operators (preprint), (2014) [arXiv:1412.1733]

  3. J.D. Chodera, W.C. Swope, J.W. Pitera, K.A. Dill, Multiscale Model. Simul. 5(4), 1214 (2006)

    Article  MathSciNet  Google Scholar 

  4. A.J. Chorin, O.H. Hald, R. Kupferman, Proc. Natl. Acad. Sci. 97(7), 2968 (2000)

    Article  ADS  Google Scholar 

  5. A.J. Chorin, O.H. Hald, R. Kupferman, Phys. D: Nonlin. Phenom. 166(3), 239 (2002)

    Article  ADS  Google Scholar 

  6. G. Ciccotti, T. Lelièvre, E. Vanden-Eijnden, Comm. Pure Appl. Math. 61(3), 371 (2008)

    Article  MathSciNet  Google Scholar 

  7. M. Dellnitz, A. Hohmann, Numerische Mathematik 75(3), 293 (1997)

    Article  MathSciNet  Google Scholar 

  8. M. Dellnitz, O. Junge, SIAM J. Num. Anal. 36(2) (1999)

  9. P. Deuflhard, M. Dellnitz, O. Junge, C. Schütte, Computational molecular dynamics: challenges, methods, ideas (Springer, 1999), p. 98

  10. P. Deuflhard, M. Weber, Linear Algebra Appl. 398, 161 (2004), Special Issue on Matrices and Mathematical Biology

    Article  Google Scholar 

  11. A.R. Dinner, A. Ali, L.J. Smith, C.M. Dobson, M. Karplus, Trends Biochem. Sci. 25(7), 331 (2000)

    Article  Google Scholar 

  12. W. E, E. Vanden-Eijnden, Multiscale modelling and simulation(Springer, 2004), p. 35

  13. W. E, E. Vanden-Eijnden, J. Stat. Phys. 123(3), 503 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  14. D.J. Evans, G.P. Morriss, Statistical mechanics of nonequilibrium liquids (Anu E Press, 2007)

  15. H. Federer, Geometric measure theory, Vol. 1996 (Springer New York, 1969)

  16. G. Froyland, O. Junge, P. Koltai, SIAM J. Numer. Anal. 51(1), 223 (2013)

    Article  MathSciNet  Google Scholar 

  17. C. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences (Springer, 2009)

  18. D. Givon, R. Kupferman, O.H. Hald, Israel J. Math. 145, 221 (2004)

    Article  Google Scholar 

  19. P. Hänggi, F. Marchesoni, Chaos 15(2), 026101 (2005)

    Article  ADS  Google Scholar 

  20. P. Hänggi, P. Talkner, M. Borkovec, Rev. Mod. Phys. 62, 251 (1990)

    Article  ADS  Google Scholar 

  21. C. Hartmann, Model Reduction in Classical Molecular Dynamics, Ph.D. thesis, Freie Universität Berlin , 2007

  22. C. Hartmann, J. Stat. Phys. 130(4), 687 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  23. C. Hartmann, T. Yanao, Mol. Phys. 111(22-23), 3534 (2013)

    Article  ADS  Google Scholar 

  24. F. Hérault, F. Nier, Arch. Rational Mech. Anal. 171(2), 151 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  25. W. Huisinga, B. Schmidt, Metastability and dominant eigenvalues of transfer operators, edited by B. Leimkuhler, C. Chipot, R. Elber, A. Laaksonen, A. Mark, T. Schlick, C. Schütte, R. Skeel New Algorithms for Macromolecular Simulation, Vol. 49, Lecture Notes in Computational Science and Engineering (Springer Berlin Heidelberg, 2006), p. 167

  26. I. Karatzas, Brownian motion and stochastic calculus, Vol. 113 (Springer, 1991)

  27. H.A. Kramers, Physica 7(4), 284 (1940)

    Article  ADS  MathSciNet  Google Scholar 

  28. A. Lasota, M.C. Mackey, Chaos, fractals, and noise: stochastic aspects of dynamics, Vol. 97 (Springer, 1994)

  29. F. Legoll, T. Lelièvre, Nonlinearity 23(9), 2131 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  30. T. Lelièvre, M. Rousset, G. Stoltz, Free energy computations: a mathematical perspective (Imperial College Press, 2010)

  31. J. Lu, E. Vanden-Eijnden, J. Chem. Phys. 141 (044109) (2014)

    ADS  Google Scholar 

  32. S. Maire, E. Tanré, Stochastic spectral formulations for elliptic problems, edited by P. L’ Ecuyer, A.B. Owen, Monte Carlo and Quasi-Monte Carlo Methods 2008 (Springer Berlin Heidelberg, 2009), p. 513

  33. J.C. Mattingly, A.M. Stuart, Markov Process. Related Fields 8(2), 199 (2002)

    MathSciNet  Google Scholar 

  34. J.C. Mattingly, A.M. Stuart, D.J. Higham, Stoch. Proc. Appl. 101(2), 185 (2002)

    Article  Google Scholar 

  35. E. Nelson, Dynamical theories of Brownian motion, Vol. 17 (Princeton University Press Princeton, 1967)

  36. B. Øksendal, Stochastic differential equations (Springer, 2003)

  37. G. Papanicolaou, Rocky Mountain J. Math. 6(4), 653 (1976)

    Article  MathSciNet  Google Scholar 

  38. E. Pardoux, A.Y. Veretennikov, Ann. Probab. 33(3), 1111 (2005)

    Article  MathSciNet  Google Scholar 

  39. G. Pavliotis, A. Stuart, Multiscale methods: averaging and homogenization, Vol. 53 (Springer, 2008)

  40. A. Pazy, Semigroups of linear operators and applications to partial differential equations (Springer-Verlag, New York, 1983)

  41. M.A. Peletier, G. Savarè, M. Veneroni, SIAM J. Math. Analysis 42(4), 1805 (2010)

    Article  MathSciNet  Google Scholar 

  42. C. Schütte, Conformational dynamics: Modelling, theory, algorithm, and application to biomolecules, 1999, Habilitation Thesis

  43. C. Schütte, M. Sarich, Metastability and Markov State Models in Molecular Dynamics, Courant Lecture Notes in Mathematics (2013)

  44. C. Schütte, M. Sarich, Eur. Phys. J. Special Topics 224 (2015), this volume

  45. J.L. Skinner, P.G. Wolynes, Physica A 96(3), 561 (1979)

    Article  ADS  Google Scholar 

  46. M.V. Smoluchowski, Z. Physik 17, 557 (1916)

    ADS  Google Scholar 

  47. W.C. Swope, J.W. Pitera, F. Suits, J. Phys. Chem. B 108(21), 6571 (2004)

    Article  Google Scholar 

  48. L.N. Trefethen, M. Embree, Spectra and pseudospectra: the behavior of nonnormal matrices and operators (Princeton University Press, 2005)

  49. E. Vanden-Eijnden, Ann. Rev. Phys. Chem., 61, 391 (2010)

    Article  Google Scholar 

  50. M. Weber, A subspace approach to molecular Markov state models via a new infinitesimal generator (2012) Habilitation thesis

  51. E. Zeidler, Applied functional analysis, Vol. 108 (Springer, 1995)

  52. R. Zwanzig, J. Stat. Phys. 9(3), 215 (1973)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität München, 85748, Garching, Germany

    A. Bittracher & O. Junge

  2. Institut für Mathematik, Freie Universität Berlin, 14195, Berlin, Germany

    C. Hartmann & P. Koltai

Authors
  1. A. Bittracher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Hartmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. O. Junge
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. Koltai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Bittracher.

Additional information

An erratum to this article is available at https://doi.org/10.1140/epjst/e2017-70091-7.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bittracher, A., Hartmann, C., Junge, O. et al. Pseudo generators for under-resolved molecular dynamics. Eur. Phys. J. Spec. Top. 224, 2463–2490 (2015). https://doi.org/10.1140/epjst/e2015-02422-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjst/e2015-02422-y

Search

Navigation