Advertisement

Numerical Methods for Stochastic Molecular Dynamics

  • Ben Leimkuhler
  • Charles Matthews
Chapter
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 39)

Abstract

In this chapter, we discuss principles for the design of algorithms for canonical sampling, based on the numerical discretization of stochastic dynamics models (such as Langevin dynamics) introduced in the previous chapter. Before we begin our discussion, let us consider the motivation for computing stationary averages using molecular dynamics.

Keywords

Stochastic Differential Equation Potential Energy Function Invariant Distribution Langevin Dynamic Canonical Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 6.
    Allen, M., Quigley, D.: Some comments on Monte Carlo and molecular dynamics methods. Mol. Phys. 111, 3442–3447 (2013). doi:10.1080/00268976.2013.817623 CrossRefGoogle Scholar
  2. 22.
    Barash, D., Yang, L., Qian, X., Schlick, T.: Inherent speedup limitations in multiple time step/particle mesh Ewald algorithms. J. Comput. Chem. 24, 77–88 (2003). doi:10.1002/jcc.10196 CrossRefGoogle Scholar
  3. 29.
    Batcho, P.F., Case, D., Schlick, T.: Optimized particle-mesh Ewald/multiple-time step integration for molecular dynamics simulations. J. Chem. Phys. 115, 4003–4018 (2001). doi:10.1063/1.1389854 CrossRefGoogle Scholar
  4. 44.
    Bou-Rabee, N.: Time integrators for molecular dynamics. Entropy 16(1), 138–162 (2014). doi:10.3390/e16010138 CrossRefGoogle Scholar
  5. 46.
    Bou-Rabee, N., Owhadi, H.: Long-run accuracy of variational integrators in the stochastic context. SIAM J. Numer. Anal. 48, 278–297 (2010). doi:10.1137/090758842 CrossRefzbMATHMathSciNetGoogle Scholar
  6. 47.
    Bou-Rabee, N., Vanden-Eijnden, E.: Pathwise accuracy and ergodicity of metropolized integrators for SDEs. Commun. Pure Appl. Math. 63, 655–696 (2010). doi:10.1002/cpa.20306 zbMATHMathSciNetGoogle Scholar
  7. 48.
    Bou-Rabee, N., Vanden-Eijnden, E.: A patch that imparts unconditional stability to explicit integrators for Langevin-like equations. J. Comput. Phys. 231, 2565–2580 (2012). doi:10.1016/j.jcp.2011.12.007 CrossRefzbMATHMathSciNetGoogle Scholar
  8. 55.
    Brünger, A., Brooks III, C., Karplus, M.: Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett. 105, 495–500 (1984). doi:10.1016/0009-2614(84)80098-6
  9. 59.
    Burrage, K., Lythe, G.: Accurate stationary densities with partitioned numerical methods for stochastic differential equations. SIAM J. Numer. Anal. 47, 1601–1618 (2009). doi:10.1137/060677148 CrossRefzbMATHMathSciNetGoogle Scholar
  10. 62.
    Bussi, G., Parrinello, M.: Accurate sampling using Langevin dynamics. Phys. Rev. E 75, 056,707 (2007). doi:10.1103/PhysRevE.75.056707
  11. 81.
    Ciccotti, G., Kalibaeva, G.: Deterministic and stochastic algorithms for mechanical systems under constraints. Philos. Trans. R. Soc. Lond. Series A 362, 1583–1594 (2004). doi:10.1098/rsta.2004.1400 CrossRefzbMATHMathSciNetGoogle Scholar
  12. 96.
    De Fabritiis, G., Serrano, M., Español, P., Coveney, P.: Efficient numerical integrators for stochastic models. Physica A 361(2), 429–440 (2006). doi:10.1016/j.physa.2005.06.090 CrossRefGoogle Scholar
  13. 99.
    Debussche, A., Faou, E.: Weak backward error analysis for SDEs. SIAM J. Numer. Anal. 50(3), 1735–1752 (2012). doi:10.1137/110831544 CrossRefzbMATHMathSciNetGoogle Scholar
  14. 118.
    Eastman, P., Doniach, S.: Multiple time step diffusive Langevin dynamics for proteins. Proteins 30, 215–227 (1998). doi:10.1002/(SICI)1097-0134(19980215)30:3¡215::AID-PROT1¿3.0.CO;2-JCrossRefGoogle Scholar
  15. 133.
    Feng, K., Shang, Z.: Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71, 451–463 (1995). doi:10.1007/s002110050153 CrossRefzbMATHMathSciNetGoogle Scholar
  16. 164.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31. Springer, New York (2006). ISBN:978-3-540-30666-5Google Scholar
  17. 170.
    Hardy, D.: NAMD-Lite. University of Illinois at Urbana-Champaign, http://www.ks.uiuc.edu/Development/MDTools/namdlite/ (2007)
  18. 178.
    Hoare, M.: Structure and dynamics of simple microclusters. Adv. Chem. Phys. 40, 49–135 (1979). doi:10.1002/9780470142592.ch2 Google Scholar
  19. 193.
    Jepps, O., Ayton, G., Evans, D.: Microscopic expressions for the thermodynamic temperature. Phys. Rev. E 62, 4757–4763 (2000). doi:10.1103/PhysRevE.62.4757 CrossRefGoogle Scholar
  20. 200.
    Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics. Springer, New York (1992). ISBN:978-3540540625CrossRefzbMATHGoogle Scholar
  21. 211.
    Landau, L.D., and Lifshitz, E.M., Statistical Physics (Volume 5, Course of Theoretical Physics), Third Edition, Butterworth-Heinemann (1980), ISBN: 978-0-750-63372-7.Google Scholar
  22. 221.
    Leimkuhler, B., Matthews, C.: Rational construction of stochastic numerical methods for molecular sampling. Appl. Math. Res. Express 1, 4–56 (2013). doi:10.1093/amrx/abs010 Google Scholar
  23. 222.
    Leimkuhler, B., Matthews, C.: Robust and efficient configurational molecular sampling via Langevin dynamics. J. Chem. Phys. 138, 174,102 (2013). doi:10.1063/1.4802990
  24. 223.
    Leimkuhler, B., Matthews, C. and Stoltz G.: The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics. IMA J Numer Anal (2015). doi:10.1093/imanum/dru056Google Scholar
  25. 224.
    Leimkuhler, B., Matthews, C., Tretyakov, M.V.: On the long-time integration of stochastic gradient systems. Proc. R. Soc. A 470(2170) (2014). doi:10.1098/rspa.2014.0120
  26. 232.
    Lelièvre, T., Rousset, M., Stoltz, G.: Langevin dynamics with constraints and computation of free energy differences. Math. Comput. 81, 2071 (2012). doi:10.1090/S0025-5718-2012-02594-4 CrossRefzbMATHGoogle Scholar
  27. 233.
    Lelièvre, T., Stoltz, G., Rousset, M.: Free Energy Computations: A Mathematical Perspective. World Scientific, Singapore (2010)CrossRefGoogle Scholar
  28. 246.
    MacKerell Jr., A., Brooks III, C., Nilsson, L., Roux, B., Won, Y., Karplus, M.: CHARMM: The Energy Function and Its Parameterization with an Overview of the Program. The Encyclopedia of Computational Chemistry, vol. 1, pp. 271–277. Wiley, Chichester (1998). http://www.charmm.org
  29. 261.
    McLachlan, R., Quispel, G.: Geometric integration of conservative polynomial ODEs. Appl. Numer. Math. 45, 411–418 (2003). doi:10.1016/S0168-9274(03)00022-9
  30. 265.
    Melchionna, S.: Design of quasisymplectic propagators for Langevin dynamics. J. Chem. Phys. 127(4), 044108 (2007). doi:10.1063/1.2753496 CrossRefGoogle Scholar
  31. 270.
    Milstein, G., Tretyakov, M.: Stochastic Numerics for Mathematical Physics. Springer, New York (2004). doi:10.1007/978-3-662-10063-9 Google Scholar
  32. 274.
    Miyamoto, S., Kollman, P.: Settle: An analytical version of the SHAKE and RATTLE algorithm for rigid water models. J. Comput. Chem. 13, 952–962 (1992). doi:10.1002/jcc.540130805 CrossRefGoogle Scholar
  33. 278.
    Neal, R.: MCMC using Hamiltonian dynamics. In: Handbook of Markov Chain Monte Carlo, pp. 113–162. Chapman and Hall, Boca Raton (2011)Google Scholar
  34. 302.
    Phillips, J., Braun, R., Wang, W., Gumbart, J., Tajkhorshid, E., Villa, E., Chipot, C., Skeel, R., Kalé, L., Schulten, K.: Scalable molecular dynamics with NAMD. J. Comput. Chem. 26(16), 1781–1802 (2005). doi:10.1002/jcc.20289. http://www.ks.uiuc.edu/Research/namd/
  35. 319.
    Rugh, H.: Dynamical approach to temperature. Phys. Rev. Lett. 78, 772–774 (1997). doi:10.1103/PhysRevLett.78.772 CrossRefGoogle Scholar
  36. 322.
    Ryckaert, J., Ciccotti, G., Berendsen, H.: Numerical integration of the Cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes. J. Comput. Phys. 23, 327–341 (1977). doi:10.1016/0021-9991(77)90098-5
  37. 356.
    Talay, D.: Simulation and Numerical Analysis of Stochastic Differential Systems: A Review. Rapports de recherche. Institut National de Recherche en Informatique et en Automatique (1990)Google Scholar
  38. 357.
    Talay, D.: Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Process. Relat. Fields 8, 163–198 (2002)zbMATHMathSciNetGoogle Scholar
  39. 358.
    Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8, 483–509 (1990). doi:10.1080/07362999008809220 CrossRefzbMATHMathSciNetGoogle Scholar
  40. 363.
    Thalmann, F., Farago, J.: Trotter derivation of algorithms for Brownian and dissipative particle dynamics. J. Chem. Phys. 127, 124,109 (2007). doi:10.1063/1.2764481
  41. 380.
    Tupper, P.: A non-existence result for Hamiltonian integrators (2006). http://arxiv.org/abs/math/0607641
  42. 383.
    Vanden-Eijnden, E., Ciccotti, G.: Second-order integrators for Langevin equations with holonomic constraints. Chem. Phys. Lett. 429, 310–316 (2006). doi:10.1016/j.cplett.2006.07.086 CrossRefGoogle Scholar
  43. 396.
    White, T., Ciccotti, G., Hansen, J.P.: Brownian dynamics with constraints. Mol. Phys. 99(24), 2023–2036 (2001). doi:10.1080/00268970110090854 CrossRefGoogle Scholar
  44. 400.
    Zhong, G., Marsden, J.E.: Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133, 134–139 (1988). doi:10.1016/0375-9601(88)90773-6
  45. 401.
    Zuckerman, D.M.: Equilibrium sampling in biomolecular simulations. Ann. Rev. Biophys. 40(1), 41–62 (2011). doi:10.1146/annurev-biophys-042910-155255 CrossRefMathSciNetGoogle Scholar
  46. 403.
    Zygalakis, K.: On the existence and the applications of modified equations for stochastic differential equations. SIAM J. Sci. Comput. 33, 102–130 (2011). doi:10.1137/090762336 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ben Leimkuhler
    • 1
  • Charles Matthews
    • 2
  1. 1.School of MathematicsUniversity of EdinburghEdinburghUK
  2. 2.Gordon Center for Integrative ScienceUniversity of ChicagoChicagoUSA

Personalised recommendations