Advertisement

Mathematical Foundations of Accelerated Molecular Dynamics Methods

  • Tony LelièvreEmail author
Reference work entry
  • 14 Downloads

Abstract

The objective of this review chapter is to present recent results on the mathematical analysis of the accelerated dynamics algorithms introduced by A.F. Voter in collaboration with D. Perez and M. Sorensen. Using the notion of quasi-stationary distribution, one is able to rigorously justify the fact that the exit event from a metastable state for the Langevin or overdamped Langevin dynamics can be modeled by a kinetic Monte Carlo model. Moreover, under some geometric assumptions, one can prove that this kinetic Monte Carlo model can be parameterized using Eyring-Kramers formulas. These are the building blocks required to analyze the accelerated dynamics algorithms, to understand their efficiency and their accuracy, and to improve and generalize these techniques beyond their original scope.

Notes

Acknowledgments

This work is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement number 614492. Part of this work was completed during the long programs “Large deviation theory in statistical physics: Recent advances and future challenges” at the International Centre for Theoretical Sciences (Bangalore) and “Complex High-Dimensional Energy Landscapes” at the Institute for Pure and Applied Mathematics (UCLA). The author would like to thank ICTS and IPAM for their hospitality.

References

  1. Allen R, Warren P, ten Wolde P (2005) Sampling rare switching events in biochemical networks. Phys Rev Lett 94(1):018104ADSCrossRefGoogle Scholar
  2. Aristoff D, Lelièvre T (2014) Mathematical analysis of temperature accelerated dynamics. SIAM Multiscale Model Simul 12(1):290–317MathSciNetzbMATHCrossRefGoogle Scholar
  3. Aristoff D, Lelièvre T, Simpson G (2014) The parallel replica method for simulating long trajectories of markov chains. AMRX 2:332–352MathSciNetzbMATHGoogle Scholar
  4. Bal K, Neyts E (2015) Merging metadynamics into hyperdynamics: accelerated molecular simulations reaching time scales from microseconds to seconds. J Chem Theory Comput 11(10):4545–4554CrossRefGoogle Scholar
  5. Binder A, Simpson G, Lelièvre T (2015) A generalized parallel replica dynamics. J Comput Phys 284:595–616ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. Bovier A, Eckhoff M, Gayrard V, Klein M (2004) Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J Eur Math Soc (JEMS) 6:399–424zbMATHCrossRefGoogle Scholar
  7. Bovier A, Gayrard V, Klein M (2005) Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J Eur Math Soc (JEMS) 7:69–99zbMATHCrossRefGoogle Scholar
  8. Bowman G, Pande V, Noé F (Eds) (2014) An introduction to Markov state models and their application to long timescale molecular simulation. Springer, DordrechtzbMATHGoogle Scholar
  9. Cameron M (2014) Metastability, spectrum, and eigencurrents of the Lennard-Jones-38 network. J Chem Phys 141(18):184113CrossRefGoogle Scholar
  10. Cérou F, Guyader A, Lelièvre T, Pommier D (2011) A multiple replica approach to simulate reactive trajectories. J Chem Phys 134:054108ADSCrossRefGoogle Scholar
  11. Collet P, Martínez S, San Martín J (2013) Quasi-Stationary Distributions. Springer, Berlin/HeidelbergzbMATHCrossRefGoogle Scholar
  12. Dellago C, Bolhuis P, Chandler D (1999) On the calculation of reaction rate constants in the transition path ensemble. J Chem Phys 110(14):6617–6625ADSCrossRefGoogle Scholar
  13. Di Gesù G, Le Peutrec D, Lelièvre T, Nectoux B (2017) Precise asymptotics of the first exit point density. https://arxiv.org/abs/1706.08728
  14. Dickson B (2017) Overfill protection and hyperdynamics in adaptively biased simulations. J Chem Theory Comput 13(12):5925–5932CrossRefGoogle Scholar
  15. Eckhoff M (2005) Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime. Ann Probab 33(1):244–299MathSciNetzbMATHCrossRefGoogle Scholar
  16. Faradjian A, Elber R (2004) Computing time scales from reaction coordinates by milestoning. J Chem Phys 120(23):10880–10889ADSCrossRefGoogle Scholar
  17. Ferrari P, Maric N (2007) Quasi-stationary distributions and Fleming-Viot processes in countable spaces. Electron J Probab 12(24):684–702MathSciNetzbMATHCrossRefGoogle Scholar
  18. Freidlin M, Wentzell A (1984) Random perturbations of dynamical systems. Springer, New YorkzbMATHCrossRefGoogle Scholar
  19. Gelman A, Rubin D (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4):457–472zbMATHCrossRefGoogle Scholar
  20. Hänggi P, Talkner P, Borkovec M (1990) Reaction-rate theory: fifty years after Kramers. Rev Mod Phys 62(2):251–342ADSMathSciNetCrossRefGoogle Scholar
  21. Helffer B, Nier F (2006) Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary. Mémoire de la société mathématique de France 105:1–89zbMATHCrossRefGoogle Scholar
  22. Helffer B, Sjöstrand J (1984) Multiple wells in the semi-classical limit I. Commun Partial Diff Equ 9(4):337–408zbMATHCrossRefGoogle Scholar
  23. Helffer B, Klein M, Nier F (2004) Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. Mat Contemp 26:41–85MathSciNetzbMATHGoogle Scholar
  24. Hérau F, Hitrik M, Sjöstrand J (2011) Tunnel effect and symmetries for Kramers-Fokker-Planck type operators. J Inst Math Jussieu 10(3):567–634MathSciNetzbMATHCrossRefGoogle Scholar
  25. Kim S, Perez D, Voter AF (2013) Local hyperdynamics. J Chem Phys 139(14):144110ADSCrossRefGoogle Scholar
  26. Kramers H (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4):284–304ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. Kum O, Dickson B, Stuart S, Uberuaga B, Voter AF (2004) Parallel replica dynamics with a heterogeneous distribution of barriers: application to n-hexadecane pyrolysis. J Chem Phys 121:9808–9819ADSCrossRefGoogle Scholar
  28. Le Bris C, Lelièvre T, Luskin M, Perez D (2012) A mathematical formalization of the parallel replica dynamics. Monte Carlo Methods Appl 18(2):119–146MathSciNetzbMATHGoogle Scholar
  29. Le Peutrec D (2010) Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Ann Fac Sci Toulouse Math (6) 19(3–4):735–809MathSciNetzbMATHCrossRefGoogle Scholar
  30. Lelièvre T, Nier F (2015) Low temperature asymptotics for quasi-stationary distributions in a bounded domain. Anal PDE 8(3):561–628MathSciNetzbMATHCrossRefGoogle Scholar
  31. Lelièvre T, Rousset M, Stoltz G (2010) Free energy computations: a mathematical perspective. Imperial College Press, UKzbMATHCrossRefGoogle Scholar
  32. Maier R, Stein D (1993) Escape problem for irreversible systems. Phys Rev E 48:931–938ADSCrossRefGoogle Scholar
  33. Miron R, Fichthorn K (2003) Accelerated molecular dynamics with the bond-boost method. J Chem Phys 119(12):6210–6216ADSCrossRefGoogle Scholar
  34. Nier F (2018) Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries, vol 252. American Mathematical Society, ProvidencezbMATHGoogle Scholar
  35. Perez D, Cubuk E, Waterland A, Kaxiras E, Voter AF (2015a) Long-time dynamics through parallel trajectory splicing. J Chem Theory Comput 12(1):18–28CrossRefGoogle Scholar
  36. Perez D, Uberuaga B, Voter AF (2015b) The parallel replica dynamics method – coming of age. Comput Mater Sci 100:90–103CrossRefGoogle Scholar
  37. Schütte C, Sarich M (2013) Metastability and Markov state models in molecular dynamics. Courant lecture notes, vol 24. American Mathematical Society, ProvidenceGoogle Scholar
  38. Schütte C, Noé F, Lu J, Sarich M, Vanden-Eijnden E (2011) Markov state models based on milestoning. J Chem Phys 134(20):204105ADSCrossRefGoogle Scholar
  39. Simon B (1984) Semiclassical analysis of low lying eigenvalues, II. Tunneling. Ann Math 120: 89–118MathSciNetzbMATHCrossRefGoogle Scholar
  40. Sørensen M, Voter AF (2000) Temperature-accelerated dynamics for simulation of infrequent events. J Chem Phys 112(21):9599–9606ADSCrossRefGoogle Scholar
  41. Tiwary P, Parrinello M (2013) From metadynamics to dynamics. Phys Rev Lett 111(23):230602ADSCrossRefGoogle Scholar
  42. van Erp T, Moroni D, Bolhuis P (2003) A novel path sampling method for the calculation of rate constants. J Chem Phys 118(17):7762–7774ADSCrossRefGoogle Scholar
  43. Vanden-Eijnden E, Venturoli M, Ciccotti G, Elber R (2008) On the assumptions underlying milestoning. J Chem Phys 129(17):174102ADSCrossRefGoogle Scholar
  44. Voter AF (1997) A method for accelerating the molecular dynamics simulation of infrequent events. J Chem Phys 106(11):4665–4677ADSCrossRefGoogle Scholar
  45. Voter AF (1998) Parallel replica method for dynamics of infrequent events. Phys Rev B 57(22): R13985ADSCrossRefGoogle Scholar
  46. Voter AF (2007) Introduction to the kinetic Monte Carlo method. In: Sickafus KE, Kotomin EA, BP Uberuaga (eds) Radiation effects in solids. Springer/NATO Publishing Unit, Netherlands, pp. 1–23Google Scholar
  47. Wales D (2003) Energy landscapes. Cambridge University Press, CambridgeGoogle Scholar
  48. Wang T, Plechac P, Aristoff D (2018) Stationary averaging for multi-scale continuous time Markov chains using parallel replica dynamics. Multiscale Model. Simul., 16:1–27. https://epubs.siam.org/doi/10.1137/16M1108716 MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.CERMICSÉcole des Ponts ParisTech, INRIAChamps-sur-MarneFrance

Personalised recommendations