An averaging principle for fast degrees of freedom exhibiting long-term correlations

Multiscale Modeling and Simulation

Volumn 2, Issue 3, 2004, Pages 501-526

  Schütte, Christof ^a   Walter, Jessika ^a   Hartmann, Carsten ^a   Huisinga, Wilhelm ^a  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Conditional averaging; Conformational free energy landscape; Deviations from averaging; Diffusion process; Dominant spectrum; Fixman potential; Fokker Planck generator; Long term correlations; Metastability; Multiscale asymptotics; Transfer operator approach

Indexed keywords

DEGREES OF FREEDOM (MECHANICS); DYNAMICAL SYSTEMS; MARKOV PROCESSES; MOLECULAR DYNAMICS; STOCHASTIC SYSTEMS;

ASYMPTOTICS; CONDITIONAL AVERAGING; CONFORMATIONAL FREE ENERGY; DEVIATIONS FROM AVERAGING; DIFFUSION PROCESS; DOMINANT SPECTRUM; FIXMAN POTENTIAL; FOKKER PLANCK; LONG-TERM CORRELATIONS; METASTABILITIES; TRANSFER OPERATOR;

MECHANICS;

EID: 84970882180     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/030600308     Document Type: Article

References (56)

1. M. P. Allen and D. J. Tildesley, Computer Simulations of Liquids, Clarendon, Oxford, 1987.
2. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, Berlin, 1993.
3. E. Barth, K. Kuczera, B. Leimkuhler, and R. D. Skeel, Algorithms for constrained molecular dynamics, J. Comp. Chem., 16 (1995), pp. 1192-1209.
4. P. G. Bolhuis, D. Chandler, C. Dellago, and P. Geissler, Transition path sampling: Throwing ropes over mountain passes, in the dark, Ann. Rev. Phys. Chem., 59 (2002), pp. 291-318.
5. M. Born and V. Fock, Beweis des Adiabatensatzes, Z. Phys., 51 (1928), pp. 165-180.
6. F. A. Bornemann, Homogenization in Time of Singularly Perturbed Mechanical Systems, Lecture Notes in Math. 1687, Springer-Verlag, 1998.
7. F. A. Bornemann and C. Schütte, Homogenization of Hamiltonian systems with a strong constraining potential, Phys. D, 102 (1997), pp. 57-77.
8. F. A. Bornemann and C. Schütte, On the singular limit of the quantum-classical molecular dynamics model, SIAM J. Appl. Math., 59 (1999), pp. 1208-1224.
9. A. Bovier, V. Gayrard, and M. Klein, Metastability in Reversible Diffusion Processes II: Precise Asymptotics for Small Eigenvalues, WIAS Preprint 768, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany, 2002.
10. E. A. Carter, G. Ciccotti, J. T. Hynes, and R. Kapral, Constrained reaction coordinate dynamics for the simulation of rare events, Chem. Phys. Lett., 156 (1989), pp. 472-477.
11. A. Chorin, A. Kast, and R. Kupferman, Optimal prediction of unresolved dynamics, Proc. Natl. Acad. Sci. USA, 95 (1998), pp. 4094-4098.
12. A. J. Chorin, O. H. Hald, and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), pp. 2968-2973.
13. E. B. Davies, Metastable states of symmetric Markov semigroups II, J. London Math. Soc. (2), 26 (1982), pp. 541-556.
14. M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), pp. 491-515.
15. P. Deuflhard, J. Hermans, B. Leimkuhler, A. E. Mark, S. Reich, and R. D. Skeel, eds., Computational Molecular Dynamics: Challenges, Methods, Ideas, Lect. Notes Comput. Sci. Eng. 4, Springer-Verlag, Berlin, 1999.
16. P. Deuflhard, W. Huisinga, A. Fischer, and C. Schütte, Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Linear Algebra Appl., 315 (2000), pp. 39-59.
17. W. E, W. Ren, and E. Vanden-Eijnden, Probing multi-scale energy landscapes using the string method, Phys. Rev. Lett., submitted.
18. C. Fermanian-Kammerer and C. Lasser, Wigner measures and codimension two crossings, J. Math. Phys., 44 (2003), pp. 507-527.
19. A. Fischer, An Uncoupling-Coupling Method for Markov Chain Monte Carlo Simulations with an Application to Biomolecules, Ph.D. thesis, Free University Berlin, Berlin, Germany, 2003.
20. A. Fischer, C. Schütte, P. Deuflhard, and F. Cordes, Hierarchical uncoupling-coupling of metastable conformations, in Computational Methods for Macromolecules: Challenges and Applications, Lect. Notes Comput. Sci. Eng. 24, T. Schlick and H. H. Gan, eds., Springer, Heidelberg, 2002.
21. M. Fixman, Classical statistical mechanics of constraints: A theorem and application to polymers, Proc. Nat. Acad. Sci. U.S.A., 71 (1974), pp. 3050-3053.
22. M. Freidlin, The averaging principle and theorems on large deviations, Russian Math. Surveys, 33 (1978), pp. 107-160.
23. M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren Math. Wiss. 260, Springer, New York, 1984.
24. A. Gelman and X.-L. Meng, Simulating normalizing constants: From importance sampling to bridge sampling to path sampling, Statist. Sci., 13 (1998), pp. 163-185.
25. C. J. Geyer, Estimating Normalizing Constants and Reweighting Mixtures in Markov Chain Monte Carlo, Technical report 568, University of Minnesota, Minneapolis, MN, 1994.
26. G. Hagedorn, A time dependent Born-Oppenheimer approximation, Comm. Math. Phys., 77 (1980), pp. 1-19.
27. G. A. Hagedorn, Molecular propagation through electron energy level crossings, Mem. Amer. Math. Soc., 111 (536) (1994), pp. 1-130.
28. I. Horenko, B. Schmidt, and C. Schütte, A theoretical model for molecules interacting with intense laser pulses: The Floquet-based quantum-classical Liouville equation, J. Chem. Phys., 115 (2001), pp. 5733-5743.
29. W. Huisinga, Metastability of Markovian Systems: A Transfer Operator Based Approach in Application to Molecular Dynamics, Ph.D. thesis, Free University Berlin, Berlin, Germany, 2001.
30. W. Huisinga, S. Meyn, and C. Schütte, Phase transitions and metastability in Markovian and molecular systems, Ann. Appl. Probab., 14 (2004), pp. 419-458.
31. Y. Kifer, Averaging in dynamical systems and large deviations, Invent. Math., 110 (1992), pp. 337-370.
32. Y. Kifer, Stochastic versions of Anosov's and Neistadt's theorems on averaging, Stoch. Dyn., 1 (2001), pp. 1-21.
33. Y. I. Kifer, L2 diffusion approximation for slow motion in averaging, Stoch. Dyn., 3 (2003), pp. 213-246.
34. K. Kuczera, One- and multidimensional conformational free energy simulations, J. Comp. Chem., 17 (1996), pp. 1726-1749.
35. K. Kuczera, Exploration of peptide free energy surfaces, in Computational Molecular Dynamics: Challenges, Methods, Ideas, Lect. Notes Comput. Sci. Eng. 4, Springer-Verlag, Berlin, 1999, pp. 161-175.
36. R. Kupferman and A. Stuart, Fitting SDE Models to Nonlinear Kac-Zwanzig Heat Bath Models, Warwick Preprint 31/2002, University of Warwick, Coventry, UK, 2002.
37. T. G. Kurtz, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), pp. 55-67.
38. L. D. Landau, Zur Theorie der Energieübertragung bei Stößen, Phys. Z. Sowjetunion, 2 (1932), p. 46.
39. A. J. Majda, I. Timofeyev, and E. Vanden-Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), pp. 891-974.
40. A. J. Majda, I. Timofeyev, and E. Vanden-Eijnden, A priori tests of a stochastic mode reduction strategy, Phys. D, 170 (2002), pp. 206-252.
41. J. Mattingly, A. M. Stuart, and D. J. Higham, Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerated noise, Stochastic Process. Appl., 10 (2002), pp. 185-232.
42. S. Meyn and R. Tweedie, Markov Chains and Stochastic Stability, Springer, Berlin, 1993.
43. H. Mori, Transport collective motion and Brownian motion, Prog. Theoret. Phys. Suppl., 33 (1965), pp. 423-455.
44. I. Pavlyukevich, Stochastic Resonance, Logos, Berlin, 2002.
45. S. Reich, Smoothed Langevin dynamics of highly oscillatory systems, Phys. D, 138 (2000), pp. 210-224.
46. H. Rubin and P. Ungar, Motion under strong constraining force, Comm. Pure Appl. Math., 10 (1957), pp. 65-87.
47. R. Samudrala and J. Moult, An all-atom distance-dependent conditional probability discriminatory function for protein structure prediction, J. Mol. Biol., 275 (1998), pp. 895-916.
48. J. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer, New York, 1985.
49. T. Schneider and E. Stoll, Molecular dynamics study of a three-dimensional one-component model for distortive phase-transitions, Phys. Rev. B, 17 (1978), pp. 1302-1322.
50. C. Schütte, A. Fischer, W. Huisinga, and P. Deuflhard, A direct approach to conformational dynamics based on hybrid Monte Carlo, J. Comput. Phys., 151 (1999), pp. 146-168.
51. C. Schütte and W. Huisinga, Biomolecular conformations can be identified as metastable sets of molecular dynamics, in Handbook of Numerical Analysis, Vol. 10, P. G. Ciarlet, ed., North-Holland, Amsterdam, 2003, pp. 699-744.
52. H. Spohn and S. Teufel, Adiabatic decoupling and time-dependent Born-Oppenheimer theory, Comm. Math. Phys., 224 (2001), pp. 113-132.
53. M. Sprik and G. Ciccotti, Free energy from constrained molecular dynamics, J. Chem. Phys., 109 (1998), pp. 7737-7744.
54. F. Takens, Motion under the influence of a strong constraining force, in Global Theory of Dynamical Systems, Lecture Notes in Math. 819, Z. Niteki and C. Robinson, eds., Springer, Evanston, 1979, pp. 425-445.
55. C. Zener, Nonadiabatic crossing of energy levels, Proc. Roy. Soc London Ser. A, 7 (1932), p. 696.
56. R. Zwanzig, Nonlinear generalized Langevin equations, J. Statist. Phys., 9 (1973), pp. 215-220.