Towards a theory of transition paths

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)503-523
Number of pages21
JournalJournal of Statistical Physics
Volume123
Issue number3
DOIs
StatePublished - 2006

Fingerprint

trajectories
Trajectory
Path
Unit of time
Metastability
Markov processes
Equilibrium Distribution
Hits
metastable state
Markov Process
traffic
proportion
Proportion
Probability Distribution
Traffic
formalism

Keywords

  • Matastability
  • Reactive trajectories
  • Transition path sampling
  • Transition path theory
  • Transition pathways
  • Transition state theory

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Towards a theory of transition paths. / E, Weinan; Vanden Eijnden, Eric.

In: Journal of Statistical Physics, Vol. 123, No. 3, 2006, p. 503-523.

Research output: Contribution to journalArticle

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