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Numerical Methods for Stochastic Molecular Dynamics

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Molecular Dynamics

Part of the book series: Interdisciplinary Applied Mathematics ((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.

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Notes

  1. 1.

    In practice all the methods presented here are convergent in the strong sense, but their strong convergence orders may be low, typically r = 1∕2.

References

  1. 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

    Article  Google Scholar 

  2. 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

    Article  Google Scholar 

  3. 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

    Article  Google Scholar 

  4. Bou-Rabee, N.: Time integrators for molecular dynamics. Entropy 16(1), 138–162 (2014). doi:10.3390/e16010138

    Article  Google Scholar 

  5. 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

    Article  MATH  MathSciNet  Google Scholar 

  6. 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

    MATH  MathSciNet  Google Scholar 

  7. 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

    Article  MATH  MathSciNet  Google Scholar 

  8. 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. 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

    Article  MATH  MathSciNet  Google Scholar 

  10. Bussi, G., Parrinello, M.: Accurate sampling using Langevin dynamics. Phys. Rev. E 75, 056,707 (2007). doi:10.1103/PhysRevE.75.056707

  11. 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

    Article  MATH  MathSciNet  Google Scholar 

  12. 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

    Article  Google Scholar 

  13. Debussche, A., Faou, E.: Weak backward error analysis for SDEs. SIAM J. Numer. Anal. 50(3), 1735–1752 (2012). doi:10.1137/110831544

    Article  MATH  MathSciNet  Google Scholar 

  14. 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-J

    Article  Google Scholar 

  15. Feng, K., Shang, Z.: Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71, 451–463 (1995). doi:10.1007/s002110050153

    Article  MATH  MathSciNet  Google Scholar 

  16. 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:978-3-540-30666-5

    Google Scholar 

  17. Hardy, D.: NAMD-Lite. University of Illinois at Urbana-Champaign, http://www.ks.uiuc.edu/Development/MDTools/namdlite/ (2007)

  18. Hoare, M.: Structure and dynamics of simple microclusters. Adv. Chem. Phys. 40, 49–135 (1979). doi:10.1002/9780470142592.ch2

    Google Scholar 

  19. 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

    Article  Google Scholar 

  20. Kloeden, P., Platen, E.: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics. Springer, New York (1992). ISBN:978-3540540625

    Book  MATH  Google Scholar 

  21. 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. 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. 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. 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/dru056

    Google Scholar 

  25. 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. 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

    Article  MATH  Google Scholar 

  27. Lelièvre, T., Stoltz, G., Rousset, M.: Free Energy Computations: A Mathematical Perspective. World Scientific, Singapore (2010)

    Book  Google Scholar 

  28. 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. 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. Melchionna, S.: Design of quasisymplectic propagators for Langevin dynamics. J. Chem. Phys. 127(4), 044108 (2007). doi:10.1063/1.2753496

    Article  Google Scholar 

  31. Milstein, G., Tretyakov, M.: Stochastic Numerics for Mathematical Physics. Springer, New York (2004). doi:10.1007/978-3-662-10063-9

    Google Scholar 

  32. 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

    Article  Google Scholar 

  33. 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. 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. Rugh, H.: Dynamical approach to temperature. Phys. Rev. Lett. 78, 772–774 (1997). doi:10.1103/PhysRevLett.78.772

    Article  Google Scholar 

  36. 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. 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. 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)

    MATH  MathSciNet  Google Scholar 

  39. 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

    Article  MATH  MathSciNet  Google Scholar 

  40. 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. Tupper, P.: A non-existence result for Hamiltonian integrators (2006). http://arxiv.org/abs/math/0607641

  42. 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

    Article  Google Scholar 

  43. White, T., Ciccotti, G., Hansen, J.P.: Brownian dynamics with constraints. Mol. Phys. 99(24), 2023–2036 (2001). doi:10.1080/00268970110090854

    Article  Google Scholar 

  44. 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. Zuckerman, D.M.: Equilibrium sampling in biomolecular simulations. Ann. Rev. Biophys. 40(1), 41–62 (2011). doi:10.1146/annurev-biophys-042910-155255

    Article  MathSciNet  Google Scholar 

  46. 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

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, UK

    Ben Leimkuhler

  2. Gordon Center for Integrative Science, University of Chicago, Chicago, IL, USA

    Charles Matthews

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  1. Ben Leimkuhler
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  2. Charles Matthews
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Leimkuhler, B., Matthews, C. (2015). Numerical Methods for Stochastic Molecular Dynamics. In: Molecular Dynamics. Interdisciplinary Applied Mathematics, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-16375-8_7

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