Skip to main content
SpringerLink
Log in

Mathematical Foundations of Accelerated Molecular Dynamics Methods

  • Living reference work entry
  • First Online:
Handbook of Materials Modeling

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Similar content being viewed by others

References

  • Allen R, Warren P, ten Wolde P (2005) Sampling rare switching events in biochemical networks. Phys Rev Lett 94(1):018104

    Article  ADS  Google Scholar 

  • Aristoff D, Lelièvre T (2014) Mathematical analysis of temperature accelerated dynamics. SIAM Multiscale Model Simul 12(1):290–317

    Article  MathSciNet  Google Scholar 

  • Aristoff D, Lelièvre T, Simpson G (2014) The parallel replica method for simulating long trajectories of markov chains. AMRX 2:332–352

    MathSciNet  MATH  Google Scholar 

  • 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):4545–4554

    Article  Google Scholar 

  • Binder A, Simpson G, Lelièvre T (2015) A generalized parallel replica dynamics. J Comput Phys 284:595–616

    Article  ADS  MathSciNet  Google Scholar 

  • 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) 6:399–424

    Article  Google Scholar 

  • Bovier A, Gayrard V, Klein M (2005) Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J Eur Math Soc (JEMS) 7:69–99

    Article  Google Scholar 

  • Bowman G, Pande V, Noé F (2014) An introduction to Markov state models and their application to long timescale molecular simulation. Springer, Dordrecht

    Book  Google Scholar 

  • Cameron M (2014) Metastability, spectrum, and eigencurrents of the Lennard-Jones-38 network. J Chem Phys 141(18):184113

    Article  Google Scholar 

  • Cérou F, Guyader A, Lelièvre T, Pommier D (2011) A multiple replica approach to simulate reactive trajectories. J Chem Phys 134:054108

    Article  ADS  Google Scholar 

  • Collet P, Martínez S, San Martín J (2013) Quasi-Stationary distributions. Springer, Berlin/Heidelberg

    Book  Google Scholar 

  • Dellago C, Bolhuis P, Chandler D (1999) On the calculation of reaction rate constants in the transition path ensemble. J Chem Phys 110(14):6617–6625

    Article  ADS  Google Scholar 

  • 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

  • Dickson B (2017) Overfill protection and hyperdynamics in adaptively biased simulations. J Chem Theory Comput 13(12):5925–5932

    Article  Google Scholar 

  • Eckhoff M (2005) Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime. Ann Probab 33(1):244–299

    Article  MathSciNet  Google Scholar 

  • Faradjian A, Elber R (2004) Computing time scales from reaction coordinates by milestoning. J Chem Phys 120(23):10880–10889

    Article  ADS  Google Scholar 

  • Ferrari P, Maric N (2007) Quasi-stationary distributions and Fleming-Viot processes in countable spaces. Electron J Probab 12(24):684–702

    Article  MathSciNet  Google Scholar 

  • Freidlin M, Wentzell A (1984) Random perturbations of dynamical systems. Springer, New York

    Book  Google Scholar 

  • Gelman A, Rubin D (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4):457–472

    Article  Google Scholar 

  • Hänggi P, Talkner P, Borkovec M (1990) Reaction-rate theory: fifty years after Kramers. Rev Mod Phys 62(2):251–342

    Article  ADS  MathSciNet  Google Scholar 

  • 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 105:1–89

    MATH  Google Scholar 

  • Helffer B, Sjöstrand J (1984) Multiple wells in the semi-classical limit I. Commun Partial Diff Equ 9(4):337–408

    Article  Google Scholar 

  • Helffer B, Klein M, Nier F (2004) Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. Mat Contemp 26:41–85

    MathSciNet  MATH  Google Scholar 

  • Hérau F, Hitrik M, Sjöstrand J (2011) Tunnel effect and symmetries for Kramers-Fokker-Planck type operators. J Inst Math Jussieu 10(3):567–634

    Article  MathSciNet  Google Scholar 

  • Kim S, Perez D, Voter A (2013) Local hyperdynamics. J Chem Phys 139(14):144110

    Article  ADS  Google Scholar 

  • Kramers H (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4):284–304

    Article  ADS  MathSciNet  Google Scholar 

  • Kum O, Dickson B, Stuart S, Uberuaga B, Voter A (2004) Parallel replica dynamics with a heterogeneous distribution of barriers: application to n-hexadecane pyrolysis. J Chem Phys 121:9808–9819

    Article  ADS  Google Scholar 

  • Le Bris C, Lelièvre T, Luskin M, Perez D (2012) A mathematical formalization of the parallel replica dynamics. Monte Carlo Methods Appl 18(2):119–146

    MathSciNet  MATH  Google Scholar 

  • Le Peutrec D (2010) Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Ann Fac Sci Toulouse Math (6) 19(3–4):735–809

    Article  MathSciNet  Google Scholar 

  • Lelièvre T, Nier F (2015) Low temperature asymptotics for quasi-stationary distributions in a bounded domain. Anal PDE 8(3):561–628

    Article  MathSciNet  Google Scholar 

  • Lelièvre T, Rousset M, Stoltz G (2010) Free energy computations: a mathematical perspective. Imperial College Press, UK

    Book  Google Scholar 

  • Maier R, Stein D (1993) Escape problem for irreversible systems. Phys Rev E 48:931–938

    Article  ADS  Google Scholar 

  • Miron R, Fichthorn K (2003) Accelerated molecular dynamics with the bond-boost method. J Chem Phys 119(12):6210–6216

    Article  ADS  Google Scholar 

  • Nier F (2018) Boundary conditions and subelliptic estimates for geometric Kramers-Fokker-Planck operators on manifolds with boundaries, vol 252. American Mathematical Society, Providence

    Google Scholar 

  • Perez D, Cubuk E, Waterland A, Kaxiras E, Voter A (2015a) Long-time dynamics through parallel trajectory splicing. J Chem Theory Comput 12(1):18–28

    Article  Google Scholar 

  • Perez D, Uberuaga B, Voter A (2015b) The parallel replica dynamics method – coming of age. Comput Mater Sci 100:90–103

    Article  Google Scholar 

  • Schütte C, Sarich M (2013) Metastability and Markov state models in molecular dynamics. Courant lecture notes, vol 24. American Mathematical Society, Providence

    Google Scholar 

  • Schütte C, Noé F, Lu J, Sarich M, Vanden-Eijnden E (2011) Markov state models based on milestoning. J Chem Phys 134(20):204105

    Article  ADS  Google Scholar 

  • Simon B (1984) Semiclassical analysis of low lying eigenvalues, II. Tunneling. Ann Math 120: 89–118

    Article  MathSciNet  Google Scholar 

  • Sørensen M, Voter A (2000) Temperature-accelerated dynamics for simulation of infrequent events. J Chem Phys 112(21):9599–9606

    Article  ADS  Google Scholar 

  • Tiwary P, Parrinello M (2013) From metadynamics to dynamics. Phys Rev Lett 111(23):230602

    Article  ADS  Google Scholar 

  • van Erp T, Moroni D, Bolhuis P (2003) A novel path sampling method for the calculation of rate constants. J Chem Phys 118(17):7762–7774

    Article  ADS  Google Scholar 

  • Vanden-Eijnden E, Venturoli M, Ciccotti G, Elber R (2008) On the assumptions underlying milestoning. J Chem Phys 129(17):174102

    Article  ADS  Google Scholar 

  • Voter A (1997) A method for accelerating the molecular dynamics simulation of infrequent events. J Chem Phys 106(11):4665–4677

    Article  ADS  Google Scholar 

  • Voter A (1998) Parallel replica method for dynamics of infrequent events. Phys Rev B 57(22):R13 985

    Article  ADS  Google Scholar 

  • Voter A (2005) Introduction to the kinetic Monte Carlo method. Radiation effects in solids. Springer/NATO Publishing Unit, Netherlands

    Google Scholar 

  • Wales D (2003) Energy landscapes. Cambridge University Press, Cambridge

    Google Scholar 

  • Wang T, Plechac P, Aristoff D (2016) Stationary averaging for multi-scale continuous time Markov chains using parallel replica dynamics. https://epubs.siam.org/doi/10.1137/16M1108716

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. CERMICS, École des Ponts, Université Paris-Est, INRIA, Champs-sur-Marne, France

    Tony Lelièvre

Authors
  1. Tony Lelièvre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tony Lelièvre .

Editor information

Editors and Affiliations

  1. Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Wanda Andreoni

  2. Department of Nuclear Science & Engineering , Massachusetts Institute of Technology , Cambridge, MA, USA

    Sidney Yip

Section Editor information

  1. Theoretical Division T-1, Los Alamos National Laboratory, Los Alamos, NM, USA

    Danny Perez

  2. Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA

    Blas Pedro Uberuaga

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Lelièvre, T. (2018). Mathematical Foundations of Accelerated Molecular Dynamics Methods. In: Andreoni, W., Yip, S. (eds) Handbook of Materials Modeling . Springer, Cham. https://doi.org/10.1007/978-3-319-42913-7_27-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-42913-7_27-1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-42913-7

  • Online ISBN: 978-3-319-42913-7

  • eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics

Publish with us

Policies and ethics

Search

Navigation