Studying the motion of Lennard-Jones clusters in an external potential having a very narrow channel passage at the saddle point, we present a one-point boundary scheme to numerically sample transition (reaction) paths. This scheme does not require knowledge of the transition states (saddle points) or that of the final states. A transition path within a given time interval (0,tf) consists of an activation path during (0,tM) and a deactivation path during (tM,tf)(0<tM<tf) joined at an intermediate time tM. The activation path is a solution to a Langevin equation with negative friction, while the deactivation path is that to a regular Langevin equation with positive friction. Each transition path so generated carries a determined statistical weight. Typical transition paths are found for two-particle and three-particle clusters. A two-particle cluster adjusts its orientation to the direction of the narrow channel and then slides through it. A three-particle cluster completes a transition by openning one of its three bonds, becoming linear, and sliding through the channel.

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