Generating Equilibrium Ensembles Via Molecular Dynamics

  • Mark E. Tuckerman

Abstract

Over the last several decades, molecular dynamics (MD) has become one of the most important and commonly used approaches for studying condensed phase systems. MD calculations generally serve two often complementary purposes. First, an MD simulation can be used to study the dynamics of a system starting from particular initial conditions. Second, MD can be employed as a means of generating a collection of classical microscopic configurations in a particular equilibrium ensemble. The latter of these uses shows that MD is intimately connected with statistical mechanics and can serve as a computational tool for solving statistical mechanical problems. Indeed, even when MD is used to study a system’s dynamics, one never uses just a single trajectory (generated from a single initial condition). Dynamical properties in the linear response regime, computed according to the rules of statistical mechanics from time correlation functions, require an ensemble of trajectories starting from an equilibrium distribution of initial conditions. These points underscore the importance of having efficient and rigorous techniques capable of generating equilibrium distributions. Indeed while the problem of producing classical trajectories from a distribution of initial conditions is relatively straightforward — one simply integrates Hamilton’s equations of motion — the problem of generating the equilibrium distribution for a complex system is an immense challenge for which advanced sampling techniques are often required.

Keywords

Entropy Enthalpy Compressibility 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G.M. Torrie and J.P. Valleau, “Nonphysical sampling distributions in Monte Carlo free energy estimation: umbrella sampling,” J. Comp. Phys., 23, 187, 1977.CrossRefADSGoogle Scholar
  2. [2]
    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, 472, 1989.CrossRefADSGoogle Scholar
  3. [3]
    M. Sprik and G. Ciccotti, “Free energy from constrained molecular dynamics,” J. Chem. Phys., 109, 7737, 1998.CrossRefADSGoogle Scholar
  4. [4]
    Z. Zhu, M.E. Tuckerman, S.O. Samuelson, and G.J. Martyna, “Using novel variable transformations to enhance conformational sampling in molecular dynamics,” Phys. Rev. Lett., 88, 100201, 2002.CrossRefADSGoogle Scholar
  5. [5]
    J.I. Siepmann and D. Frenkel, “Configurational bias Monte Carlo — a new sampling scheme for flexible chains,” Mol. Phys., 75, 59, 1992.CrossRefADSGoogle Scholar
  6. [6]
    S. Duane, A.D. Kennedy, B.J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B, 195, 216, 1987.CrossRefADSGoogle Scholar
  7. [7]
    S. Plimpton, “Fast parallel algorithms for short-range molecular dynamics,” J. Corn-put. Phys., 117, 1, 1995.MATHADSGoogle Scholar
  8. [8]
    G.J. Martyna, M.E. Tuckerman, D.J. Tobias, and M.L. Klein, “Explicit reversible integrators for extended systems dynamics,” Mol. Phys., 87, 1117, 1996.ADSCrossRefGoogle Scholar
  9. [9]
    M.E. Tuckerman, G.J. Martyna, and B.J. Berne, “Reversible multiple time scale molecular dynamics,” J. Chem. Phys., 97, 1990, 1992.CrossRefADSGoogle Scholar
  10. [10]
    H. Andersen, “Molecular dynamics at constant temperature and/or pressure,” J. Chem. Phys., 72, 2384, 1980.CrossRefADSGoogle Scholar
  11. [11]
    S. Nosé, “A unified formulation of the constant temperature molecular dynamics methods,” J. Chem. Phys., 81, 511, 1984.CrossRefADSGoogle Scholar
  12. [12]
    S.D. Bond, B.J. Leimkuhler, and B.B. Laird, “The nosé-poincaré method for con-stant temperature molecular dynamics,” J. Comput. Phys., 151, 114, 1999.MATHCrossRefMathSciNetADSGoogle Scholar
  13. [13]
    G.J. Martyna, M.E. Tuckerman, and M.L. Klein, “Nosé-Hoover chains: the canoni-cal ensemble via continuous dynamics,” J. Chem. Phys., 97, 2635, 1992.CrossRefADSGoogle Scholar
  14. [14]
    Y. Liu and M.E. Tuckerman, “Generalized Gaussian moment thermostatting: a new continuous dynamical approach to the canonical ensemble,” J. Chem. Phys., 112, 1685, 2000.CrossRefADSGoogle Scholar
  15. [15]
    M.E. Tuckerman, C.J. Mundy, and G.J. Martyna, “On the classical statistical mechanics of non-Hamiltonian systems,” Europhys. Lett., 45, 149, 1999.CrossRefADSGoogle Scholar
  16. [16]
    M.E. Tuckerman, Y. Liu, G. Ciccotti, and G.J. Martyna, “Non-Hamiltonian molecu-lar dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems,” J. Chem. Phys., 115, 1678, 2001.CrossRefADSGoogle Scholar
  17. [17]
    W.G. Hoover, “Canonical dynamics — equilibrium phase space distributions,” Phys. Rev. A, 31, 1695, 1985.CrossRefADSGoogle Scholar
  18. [18]
    M.E. Tuckerman, B.J. Berne, G.J. Martyna, and M.L. Klein, “Efficient molecular dynamics and hybrid Monte Carlo algorithms for path integrals,” J. Chem. Phys., 99, 2796, 1993.CrossRefADSGoogle Scholar
  19. [19]
    M.E. Tuckerman and G.J. Martyna, Comment on “Simple reversible molecular dynamics algorithms for Nose-Hoover chain dynamics,” J. Chem. Phys., 110, 3623, 1999.CrossRefADSGoogle Scholar
  20. [20]
    H. Yoshida, “Construction of higher-order symplectic integrators,” Phys. Lett. A, 150, 262, 1990.CrossRefMathSciNetADSGoogle Scholar
  21. [21]
    M. Suzuki, “General-theory of fractal path-integrals with applications to many-body theories and statistical physics,” J. Math. Phys., 32, 400, 1991.MATHCrossRefMathSciNetADSGoogle Scholar
  22. [22]
    G.J. Martyna, D.J. Tobias, and M.L. Klein, “Constant-pressure molecular-dynamics algorithms,” J. Chem. Phys., 101, 4177, 1994.CrossRefADSGoogle Scholar
  23. [23]
    M. Parrinello and A. Rahman, “Crystal-structure and pair potentials — a molecular-dynamics study,” Phys. Rev. Lett., 45, 1196, 1980.CrossRefADSGoogle Scholar
  24. [24]
    J.P. Ryckaert, G. Ciccotti, and H.J.C. Berendsen, “Numerical-integration of carte-sian equations of motion of a system with constraints — molecular-dynamics of n-alkanes,” J. Comput. Phys., 23, 327, 1977.CrossRefADSGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Mark E. Tuckerman
    • 1
  1. 1.Department of Chemistry, Courant Institute of Mathematical ScienceNew York UniversityNew York

Personalised recommendations