A GEOMETRIC APPROACH TO CONSTRAINED MOLECULAR DYNAMICS AND FREE ENERGY *

Communications in Mathematical Sciences

Volumn 3, Issue 1, 2005, Pages

  Hartmann, Carsten ^a   Schütte, Christof ^a  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

Author keywords

affine connection; blue moon method; free energy; holonomic constraints; hybrid Monte Carlo; mean curvature; mean force; metric entropy; molecular dynamics; normal connection

Indexed keywords

EID: 85129686405     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2005.v3.n1.a1     Document Type: Article

References (40)

1. [1] M. Sprik and G. Ciccotti, Free energy from constrained molecular dynamics, J. Chem. Phys., 109, 18, 7737-7744, 1998.
2. [2] E. Darve, M. A. Wilson and A. Pohorille, Calculating free energies using a scaled-force molecular dynamics algorithm, Mol. Sim., 28, 1-2, 113-144, 2002.
3. [3] C. Chipot and D. A. Pearlman, Free energy calculations, the long and winding gilded road, Mol. Sim., 28, 1-2, 1-12, 2002.
4. [4] B. Roux, The calculation of the potential of mean force using computer simulations, Comp. Phys. Comm., 91, 275-282, 1995.
5. [5] J. VandeVondele and U. Rothlisberger, Efficient multidimensional free energy calculations for ab initio molecular dynamics using classical bias potentials, J. Chem. Phys., 113, 12, 48634868, 2000.
6. [6] J. G. Kirkwood, Statistical mechanics of fluid mixtures, J. Chem. Phys., 3, 300-313, 1935.
7. [7] T. Mulders, P. Kruger, W. Swegat and L. Schlitter, Free Energy as the potential of mean force, J. Chem. Phys., 104, 12, 4869-4870, 1996.
8. [8] W. K. den Otter and W. J. Briels, The Calculation of free-energy differences by constrained molecular-dynamics simulations, J. Chem. Phys., 109, 11, 4139-4146, 1998.
9. [9] W. E and E. Vanden-Eijnden, Metastability, conformation dynamics, and transition pathways in complex systems, Multiscale Modelling and Simulation, Springer, Berlin, 35-68, 2004.
10. [10] M. J. Ruiz-Montero, D. Frenkel and J. J. Brey, Efficient schemes to compute diffusive barrier crossing rates, Mol. Phys., 90, 6, 925-941, 1997.
11. [11] 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, 5, 472-477, 1989.
12. [12] W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A, 31, 3, 1695-1697, 1985.
13. [13] B. L. Holian and W. G. Hoover, Numerical test of the Liouville equation, Phys. Rev. A, 34, 5, 4229-4239, 1986.
14. [14] G. J. Martyna, Klein and M. E. Tuckerman, Nose-Hoover chains: the canonical ensemble via continuous dynamics, J. Chem. Phys., 97, 4, 2635-2643, 1992.
15. [15] H. C. Andersen, Molecular dynamics simulations at constant temperature and/or pressure, J. Chem. Phys., 71, 4, 2384-2393, 1980.
16. [16] S. D. Bond, B. J. Leimkuhler and B. B. Laird, The Nosee-Poincaree method for constant tem perature molecular dynamics, J. Comp. Phys., 151, 114-134, 1999.
17. [17] A. J. vander Schaft and B. M. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys., 34, 2, 225-233, 1994.
18. [18] R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer, New York, 1988.
19. [19] M. Spivak, Differential Geometry, Publish or Perish, Boston, 3, 1975.
20. [20] H. Rubin and P. Ungar, Motion under a strong constraining force, Comm. Pure Appl. Math., 10, 65-87, 1957.
21. [21] F. Takens, Motion under the influence of a strong constraining force, in Global Theory of Dynamical Systems, Springer, Berlin, 99, 1980.
22. [22] G. Gallavotti, The Elements of Mechanics, Springer, New York, 1983.
23. [23] V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978.
24. [24] R. Herbst and I. Froese, Realizing holonomic constraints in classical and quantum mechanics, Comm. Math. Phys., 220, 489-535, 2001.
25. [25] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer, New York, 1999.
26. [26] D. Frenkel and B Smit, Understanding Molecular Dynamics: From Algorithms to Applications, Academic Press, London, 2002.
27. [27] J. E. Marsden, R. Montgomery and T. S. Ratiu, Reduction, symmetry and phases in mechanics, Memoirs AMS, 88, 436, 1-110, 1990.
28. [28] P. A. M. Dirac, Generalized Hamiltonian dynamics, Can. J. Math, 2, 129-148, 1950.
29. [29] B. Leimkuhler and S. Reich, Symplectic integration in constrained Hamiltonian systems, Math. Comp., 63, 589-605, 1994.
30. [30] R. I. McLachlan and C. Scovel, Equivariant constrained symplectic integration, J. Nonlinear Sci., 5, 233-256, 1995.
31. [31] M. E. Tuckerman, Y. Liu, G. Ciccotti and G. J. Martyna, Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian dynamics, J. Chem. Phys., 115, 4, 1678-1702, 2001.
32. [32] S. Melchionna, Constrained systems and statistical distribution, Phys Rev. E, 61, 6, 6165-6170, 2000.
33. [33] R. A. Horn, Matrix Analysis, Cambridge University Press, 1987.
34. [34] M. Fixman, Classical statistical mechanics of constraints: a theorem and applications to polymers, PNAS, 71, 3050-3053, 1974.
35. [35] J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer, New York, 2001.
36. [36] M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston, 1992.
37. [37] B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry - Methods and Appli cations: Part I. The Geometry of Surfaces, Transformation Groups, and Fields, Springer, Berlin, 1984.
38. [38] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer, Berlin, 2002.
39. [39] P. Bremaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues, Springer, New York, 1999.
40. [40] J. P. Ryckart and A. Bellmanns, Molecular dynamics of liquid n-Butane near its boiling point, Chem. Phys. Lett., 30, 1, 123-125, 1975.