# Towards a Theory of Transition Paths

- 667 Downloads
- 162 Citations

## Abstract

We construct a statistical theory of reactive trajectories between two pre-specified sets *A* and *B*, i.e. the portionsof the path of a Markov process during which the path makes a transition from *A* to *B*. This problem is relevant e.g. in the context of metastability, in which case the two sets *A* and *B* are metastable sets, though the formalism we propose is independent of any such assumptions on *A* and *B.* We show that various probability distributions on the reactive trajectories can be expressed in terms of the equilibrium distribution of the process and the so-called committor functions which give the probability that the process reaches first *B* before reaching *A*, either backward or forward in time. Using these objects, we obtain (i) the distribution of reactive trajectories, which gives the proportion of time reactive trajectories spend in sets outside of *A* and *B;* (ii) the hitting point distribution of the reactive trajectories on a surface, which measures where the reactive trajectories hit the surface when they cross it; (iii) the last hitting point distribution of the reactive trajectories on the surface; (iv) the probability current of reactive trajectories, the integral of which on a surface gives the net average flux of reactive trajectories across this surface; (v) the average frequency of reactive trajectories, which gives the average number of transitions between *A* and *B* per unit of time; and (vi) the traffic distribution of reactive trajectories, which gives some information about the regions the reactive trajectories visit regardless of the time they spend in these regions.

## Key Words

Transition path theory transition state theory transition path sampling matastability reactive trajectories transition pathways## References

- 1.A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein, Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times,
*J. Eur. Math. Soc.***6**:1–26 (2004).MATHMathSciNetGoogle Scholar - 2.A. Bovier, V. Gayrard, and M. Klein, Metastability in reversible diffusion processes. II. Precise estimates for small eigenvalues,
*J. Eur. Math. Soc.***7**:69–99 (2005).MATHMathSciNetCrossRefGoogle Scholar - 3.M. V. Day, Mathematical approaches to the problem of noise-induced exit, pp. 269–287 in:
*Stochastic analysis, control, optimization and application*, Systems Control Found. Appl., Birkhuser Boston, Boston, MA, 1999.Google Scholar - 4.C. Dellago, P. G. Bolhuis, and P. L. Geissler, Transition path sampling,
*Adv. Chem. Phys.*,**123**(2002).A4Google Scholar - 5.R. Durrett.
*Stochastic Calculus.*CRC Press, 1996.Google Scholar - 6.W. E. W. Ren and E. Vanden-Eijnden, Finite temperature string method for the study of rare events,
*J. Phys. Chem. B*,**109**: 6688–6693 (2005).CrossRefGoogle Scholar - 7.W. E. W. Ren and E. Vanden-Eijnden, Transition pathways in complex systems: Reaction coordinates, isocommittor surfaces, and transition tubes,
*Chem. Phys. Lett.***413**: 242–247 (2005).CrossRefADSGoogle Scholar - 8.M. I. Freidlin and A. D. Wentzell,
*Random Perturbations of Dynamical Systems*, 2nd ed. Springer, 1998.Google Scholar - 9.P. Hänggi, P. Talkner, and M. Borkovec, Reaction: rate theory, fifty years after Kramers,
*Rev. Mod. Phys.***62**: 251–342 (1990).CrossRefADSGoogle Scholar - 10.W. Huisinga, S. Meyn, and Ch. Schütte, Phase Transitions and Metastability in Markovian and Molecular Systems,
*Ann. Appl. Probab.***14**: 419–458 (2004).MATHMathSciNetCrossRefGoogle Scholar - 11.G. Hummer, From transition paths to transition states and rate coefficients,
*J. Chem. Phys.***120**: 516–523 (2004).CrossRefADSGoogle Scholar - 12.A. Ma and A. R. Dinner, Automatic Method for Identifying Reaction Coordinates in Complex Systems,
*J. Phys. Chem. B***109**: 6769–6779 (2005).CrossRefGoogle Scholar - 13.R. S. Maier and D. L. Stein, Limiting exit location distributions in the stochastic exit problem,
*SIAM J. Appl. Math.***57**: 752–790 (1997).MATHMathSciNetCrossRefGoogle Scholar - 14.B. J. Matkowsky and Z. Schuss, The Exit Problem for Randomly Perturbed Dynamical Systems,
*SIAM J. App. Math.***33**: 365–382 (1977).MATHMathSciNetCrossRefGoogle Scholar - 15.W. Ren, E. Vanden-Eijnden, P. Maragakis, and W. E, Transition Pathways in Complex Systems: Application of the Finite-Temperature String Method to the Alanine Dipeptide,
*J. Chem. Phys.***123**: 134109 (2005).CrossRefADSGoogle Scholar - 16.Z. Schuss, Singular Perturbation Methods on Stochastic Differential Equations of Mathematical Physics,
*SIAM Review***22**: 119–155 (1980).MATHMathSciNetCrossRefGoogle Scholar - 17.Z. Schuss and B. J. Matkowsky, The Exit Problem: A New Approach to Diffusion Across Potential Barriers,
*SIAM J. App. Math.***36**: 604–623 (1979).MATHMathSciNetCrossRefGoogle Scholar - 18.J. E. Straub, Reaction rates and transition pathways, p. 199 in
*Computational biochemistry and biophysics*, ed. O. M Becker, A. D MacKerell, Jr., B. Roux, and M. Watanabe (Marcel Decker, Inc. 2001).Google Scholar - 19.A.-S. Sznitman,
*Brownian Motion, Obstacles and Random Media*, Springer, 1998.Google Scholar - 20.F. Tal, E. Vanden-Eijnden, Transition state theory and dynamical corrections in ergodic systems.
*Nonlinearity***19**: 501-509 (2006).Google Scholar - 21.E. Vanden-Eijnden, F. Tal, Transition state theory: Variational formulation, dynamical corrections, and error estimates.
*J. Chem. Phys.***123**: 184103 (2005).Google Scholar