TY - JOUR

T1 - Towards a theory of transition paths

AU - E, Weinan

AU - Vanden-Eijnden, Eric

N1 - Funding Information:
This work is part of a joint project with Weiqing Ren: we thank him for stimulating discussions. We also thank G. Ben Arous, G. Ciccotti, P. Constantin, A. Fischer, P. Metzner, and C. Schütte for helpful suggestions. The work of E is partially supported by ONR grant N00014-01-1-0674. The work of E. V.-E. is partially supported by NSF grants DMS02-09959 and DMS02-39625, and by ONR grant N00014-04-1-0565.

PY - 2006/5

Y1 - 2006/5

N2 - 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.

AB - 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.

KW - Matastability

KW - Reactive trajectories

KW - Transition path sampling

KW - Transition path theory

KW - Transition pathways

KW - Transition state theory

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U2 - 10.1007/s10955-005-9003-9

DO - 10.1007/s10955-005-9003-9

M3 - Article

AN - SCOPUS:33747589472

SN - 0022-4715

VL - 123

SP - 503

EP - 523

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3

ER -