  Transition of a system between two states is an important but difficult
problem in natural science. In this article we study the transition problem in
the framework of transition path ensemble. Using the overdamped Langevin
method, we introduce the path integral formulation of the transition
probability and obtain the equation for the minimum action path in the
transition path space. For the effective sampling in the transition path
ensemble, we derive a conditional overdamped Langevin equation. In two exactly
solvable models, the free particle system and the harmonic system, we present
the expression of the conditional probability density and the explicit
solutions for the conditional Langevin equation and the minimum action path.
The analytic results demonstrate the consistence of the conditional Langevin
equation with the desired probability distribution in the transition. It is
confirmed that the conditional Langevin equation is an effective tool to sample
the transition path ensemble, and the minimum action principle actually leads
to the most probable path.

Statistical Mechanics