# Towards a Theory of Transition Paths

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## 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

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