Journal of Statistical Physics

, Volume 146, Issue 5, pp 955–974

Γ-Limit for Transition Paths of Maximal Probability

Article

Abstract

Chemical reactions can be modeled via diffusion processes conditioned to make a transition between specified molecular configurations representing the state of the system before and after the chemical reaction. In particular the model of Brownian dynamics—gradient flow subject to additive noise—is frequently used. If the chemical reaction is specified to take place on a given time interval, then the most likely path taken by the system is a minimizer of the Onsager-Machlup functional. The Γ-limit of this functional is determined explicitly in the case where the temperature is small and the transition time scales as the inverse temperature.

Keywords

Onsager-Machlup Gamma-convergence 

References

  1. 1.
    Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Oxford University Press, London (1987) MATHGoogle Scholar
  2. 2.
    Bolhuis, P., Dellago, C., Geissler, P.L., Chandler, D.: Transition path sampling: throwing ropes over rough mountain passes the dark. Annu. Rev. Phys. Chem. 53, 291–318 (2002) ADSCrossRefGoogle Scholar
  3. 3.
    Dürr, D., Bach, A.: The Onsager-Machlup function as Lagrangian for the most probable path of a diffusion process. Commun. Math. Phys. 160, 153–170 (1978) CrossRefGoogle Scholar
  4. 4.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland, Amsterdam (1989) MATHGoogle Scholar
  5. 5.
    Olender, R., Elber, R.: Yet another look at the steepest descent path. J. Mol. Struct., Theochem 63, 398–399 (1997) Google Scholar
  6. 6.
    Pinski, F., Stuart, A.M.: Transition paths in molecules: gradient descent in pathspace. J. Chem. Phys. 132, 184104 (2010) ADSCrossRefGoogle Scholar
  7. 7.
    Braides, A.: Γ-convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications, vol. 22. Oxford University Press, Oxford (2002) MATHCrossRefGoogle Scholar
  8. 8.
    Dal Maso, G.: An Introduction to Γ-convergence. Birkhäuser, Boston (1993) CrossRefGoogle Scholar
  9. 9.
    Voss, J.: Large deviations for one dimensional diffusions with a strong drift. Electron. J. Probab. 13, 1479–1526 (2008) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kohn, R.V., Sternberg, P.: Local minimizers and singular perturbations. Proc. R. Soc. Edinb. 111, 69–84 (1989) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Baldo, S.: Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. Henri Poincaré 7, 67–90 (1990) MathSciNetMATHGoogle Scholar
  12. 12.
    Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, New York (1984) MATHCrossRefGoogle Scholar
  13. 13.
    Ren, W.E.W., Vanden-Eijnden, E.: String method for the study of rare events. Phys. Rev. B 66, 052301 (2002) ADSGoogle Scholar
  14. 14.
    Ren, W.E.W., Vanden-Eijnden, E.: Transition pathways in complex systems: reaction coordinates, isocommitor surfaces and transition tubes. Chem. Phys. Lett. 413, 242–247 (2005) ADSCrossRefGoogle Scholar
  15. 15.
    Reznikoff, E., Vanden Eijnden, E.: Invariant measures of stochastic PDEs. C.R. Acad. Sci., Paris 340, 305–308 (2005) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hairer, M., Stuart, A.M., Voss, A.M.: Analysis of SPDEs arising in path sampling. Part 2: The nonlinear case. Ann. Appl. Probab. 340, 305–308 (2007) MathSciNetGoogle Scholar
  17. 17.
    Chorin, A.J., Hald, O.H.: Stochastic Tools in Mathematics and Science. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 1. Springer, New York (2006) MATHGoogle Scholar
  18. 18.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109–145 (1984) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Physics DepartmentUniversity of CincinnatiCincinnatiUSA
  2. 2.Mathematics InstituteWarwick UniversityCoventryUK

Personalised recommendations