Finding transition paths and rate coefficients through accelerated Langevin dynamics

We present a technique to resolve the rare event problem for a Langevin equation describing a system with thermally activated transitions. A transition event within a given time interval ${(0,t}_f)$ can be described by a transition path that has an activation part during ${(0,t}_M)$ and a deactivation part during ${(t}_M{,t}_f\left)\right(0<t_M<t_f).\mathrm{}$ The activation path is governed by a Langevin equation with negative friction while the deactivation path by the standard Langevin equation with positive friction. Each transition path carries a given statistical weight from which rate constants and related physical quantities can be obtained as averages over all possible paths. We demonstrate how this technique can be used to calculate activation rates of a particle in a two dimensional potential for a wide range of temperatures where standard molecular dynamics techniques are inefficient.