Langevin equation barrier crossing

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The key quantity in barrier crossing processes in tiiis respect is the barrier curvature Mg which sets the time window for possible influences of the dynamic solvent response. A sharp barrier entails short barrier passage times during which the memory of the solvent environment may be partially maintained. This non-Markov situation may be expressed by a generalized Langevin equation including a time-dependent friction kernel y(t) [ ]... 

In order to complete the above analysis, one needs to solve the full non-Markovian Langevin equation (NMLE) with the frequency-dependent friction for highly viscous liquids to obtain the rate. This requires extensive numerical solution because now the barrier crossing dynamics and the diffusion cannot be treated separately. However, one may still write phenomenologically the rate as [172],... 

Here, we investigate barrier crossing processes in a reaction coordinate x(f) governed by a Langevin equation [Eq. (25)] with white Levy noise ra(f). Now, however, the external potential V(x) is chosen as the (typical) double-well shape... 

Equation (89) is effectively the smallest positive (unstable barrier crossing mode) eigenvalue of the noiseless Langevin equation [Eq. (75) omitting the H (0 term] linearized in terms of the direction cosines about the saddle point. We remark that the influence of the processional term on the longitudinal relaxation is represented by the a term in Eq. (89). Furthermore, the relative magnitudes of... 

The starting point of a non-Markovian theory of barrier crossing is the generalized Langevin equation (cf. Eqs (8.61) and (8.62))... 

---