Transition Path Theory

  • E. Vanden-Eijnden
Part of the Lecture Notes in Physics book series (LNP, volume 703)


The dynamical behavior of many systems arising in physics, chemistry, biology, etc. is dominated by rare but important transition events between long lived states. For over 70 years, transition state theory (TST) has provided the main theoretical framework for the description of these events [17,33,34]. Yet, while TST and evolutions thereof based on the reactive flux formalism [1, 5] (see also [30,31]) give an accurate estimate of the transition rate of a reaction, at least in principle, the theory tells very little in terms of the mechanism of this reaction. Recent advances, such as transition path sampling (TPS) of Bolhuis, Chandler, Dellago, and Geissler [3, 7] or the action method of Elber [15, 16], may seem to go beyond TST in that respect: these techniques allow indeed to sample the ensemble of reactive trajectories, i.e. the trajectories by which the reaction occurs. And yet, the reactive trajectories may again be rather uninformative about the mechanism of the reaction. This may sound paradoxical at first: what more than actual reactive trajectories could one need to understand a reaction? The problem, however, is that the reactive trajectories by themselves give only a very indirect information about the statistical properties of these trajectories. This is similar to why statistical mechanics is not simply a footnote in books about classical mechanics. What is the probability density that a trajectory be at a given location in state-space conditional on it being reactive? What is the probability current of these reactive trajectories? What is their rate of appearance? These are the questions of interest and they are not easy to answer directly from the ensemble of reactive trajectories. The right framework to tackle these questions also goes beyond standard equilibrium statistical mechanics because of the nontrivial bias that the very definition of the reactive trajectories imply – they must be involved in a reaction. The aim of this chapter is to introduce the reader to the probabilistic framework one can use to characterize the mechanism of a reaction and obtain the probability density, current, rate, etc. of the reactive trajectories.


Transition State Theory Probabilistic Framework Transition Path Collective Variable Minimum Energy Path 
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  1. 1.
    C. H. Bennett (1977) In Algorithms for Chemical Computation, eds. A. S. Nowick and J. J. Burton ACS Symposium Series No. 46, 63 Google Scholar
  2. 2.
    R. B. Best, and G. Hummer (2005) Reaction coordinates and rates from transition paths. Proc. Natl. Acad. Sci. USA 102, p. 6732CrossRefADSGoogle Scholar
  3. 3.
    P. G. Bolhuis, D. Chandler, C. Dellago, and P. Geissler (2002) Transition path sampling: Throwing ropes over rough mountain passes, in the dark. Ann. Rev. Phys. Chem. 59, p. 291CrossRefADSGoogle Scholar
  4. 4.
    G. Ciccotti, R. Kapral, and E. Vanden-Eijnden (2005) Blue moon sampling, vectorial reaction coordinates, and unbiased constrained dynamics. Chem. Phys. Chem. 6, p. 1809Google Scholar
  5. 5.
    D. Chandler (1978) Statistical-Mechanics of isomerization dynamics in liquids and transition-state approximation. J. Chem. Phys. 68, p. 2959CrossRefADSGoogle Scholar
  6. 6.
    E. A. Carter, G. Ciccotti, J. T. Hynes, and R. Kapral (1989) Constrained reaction coordinate dynamics for the simulation of rare event. Chem. Phys. Lett. 156, p. 472CrossRefADSGoogle Scholar
  7. 7.
    C. Dellago, P. G. Bolhuis, and P. L. Geissler (2002) Transition Path Sampling. Advances in Chemical Physics 123, p. 1CrossRefGoogle Scholar
  8. 8.
    R. Durrett (1996) Stochastic Calculus. CRC PressGoogle Scholar
  9. 9.
    W. E., W. Ren and E. Vanden-Eijnden (2002) String method for the study of rare event. Phys. Rev. B 66, 052301CrossRefADSGoogle Scholar
  10. 10.
    W. E., W. Ren and E. Vanden-Eijnden (2003) Energy landscape and thermally activated switching of submicron-sized ferromagnetic element. J. App. Phys. 93, p. 2275CrossRefADSGoogle Scholar
  11. 11.
    W. E., W. Ren and E. Vanden-Eijnden (2005) Finite temperature string method for the study of rare events. J. Phys. Chem. B 109, p. 6688CrossRefGoogle Scholar
  12. 12.
    W. E., W. Ren and E. Vanden-Eijnden (2005) Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition tubes. Chem. Phys. Lett. 413, p. 242CrossRefADSGoogle Scholar
  13. 13.
    W. E and E. Vanden-Eijnden (2004) Metastability, conformation dynamics, and transition pathways in complex systems. In: Multiscale Modelling and Simulation, eds. S. Attinger and P. Koumoutsakos (LNCSE 39, Springer Berlin HeidelbergGoogle Scholar
  14. 14.
    W. E and E. Vanden-Eijnden (2006) Towards a Theory of Transition Paths. J. Stat. Phys. 123, p. 503zbMATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    R. Elber, A. Ghosh, and A. Cárdenas (2002) Long time dynamics of complex systems. Account of Chemical Research 35, p. 396CrossRefGoogle Scholar
  16. 16.
    R. Elber, A. Ghosh, A. Cárdenas, and H. Stern (2003) Bridging the gap between reaction pathways, long time dynamics and calculation of rates. Advances in Chemical Physics 126, p. 93CrossRefGoogle Scholar
  17. 17.
    H. Eyring (1935) The activated complex in chemical reactions. J. Chem. Phys. 3, p. 107CrossRefADSGoogle Scholar
  18. 18.
    C. W. Gardiner (1997) Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Springer Berlin HeidelbergzbMATHGoogle Scholar
  19. 19.
    G. Henkelman and H. Jónsson (2000) Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points. J. Chem. Phys. 113, p. 9978CrossRefADSGoogle Scholar
  20. 20.
    G. Hummer (2004) From transition paths to transition states and rate coefficients. J. Chem. Phys. 120, p. 516CrossRefADSGoogle Scholar
  21. 21.
    H. Jónsson, G. Mills, and K. W. Jacobsen (1998) Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions. In: Classical and Quantum Dynamics in Condensed Phase Simulations, ed. by: B. J. Berne, G. Ciccoti, and D. F., Coker, World ScientificGoogle Scholar
  22. 22.
    A. Ma, A. R. Dinner (2005) Automatic Method for Identifying Reaction Coordinates in Complex Systems. J. Phys. Chem. B 109, p. 6769CrossRefGoogle Scholar
  23. 23.
    L. Maragliano, A. Fischer, E. Vanden-Eijnden and G. Ciccotti (2006) String method in collective variables: Minimum free energy paths and isocommittor surfaces. J. Chem. Phys. 125, 024106CrossRefADSGoogle Scholar
  24. 24.
    G. Papanicolaou (1976) Probabilistic problems and methods in singular perturbation. Rocky Mountain Math. J 6, p. 653zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    L. R. Pratt (1986) A statistical-method for identifying transition-states in high dimensional problems. J. Chem. Phys. 9, p. 5045CrossRefADSGoogle Scholar
  26. 26.
    W. Ren (2002) Numerical Methods for the Study of Energy Landscapes and Rare Events. PhD thesis, New York UniversityGoogle Scholar
  27. 27.
    W. Ren (2003) Higher order string method for finding minimum energy paths. Comm. Math. Sci. 1, p. 377Google Scholar
  28. 28.
    W. Ren, E. Vanden-Eijnden, P. Maragakis, and W. E. (2005) Transition pathways in complex systems: Application of the finite-temperature string method to the alanine dipeptide. J. Chem. Phys. 123, 134109Google Scholar
  29. 29.
    M. Sprik and G. Ciccotti (1998) Free energy from constrained molecular dynamics. J. Chem Phys. 109, p. 7737CrossRefADSGoogle Scholar
  30. 30.
    F. Tal and E. Vanden-Eijnden (2006) Transition state theory and dynamical corrections in ergodic systems. Nonlinearity 19, p. 501zbMATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    E. Vanden-Eijnden and F. Tal (2005) Transition state theory: Variational formulation, dynamical corrections, and error estimates. J. Chem. Phys. 123, 184103Google Scholar
  32. 32.
    T. S. van Erp and P. G. Bolhuis (2005) Elaborating transition interface sampling method. J. Comp. Phys. 205, p. 157zbMATHCrossRefADSMathSciNetGoogle Scholar
  33. 33.
    E. Wigner (1938) The transition state method. Trans. Faraday Soc. 34, p. 29CrossRefGoogle Scholar
  34. 34.
    T. Yamamoto (1960) Quantum statistical mechanical theory of the rate of exchange chemical reactions in the gas phase. J. Chem Phys. 33, p. 281CrossRefMathSciNetADSGoogle Scholar

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© Springer 2006

Authors and Affiliations

  • E. Vanden-Eijnden
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York University New York

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