Stochastic differential equation trajectory model

It is important to emphasize, however, that our model is different from the Langevin equation, which is a stochastic differential equation. Our model has no noise in the limit of small time steps in which the numerical errors approach zero. The noise we introduce is numerical. Once we filter the rapid oscillations, it is impossible for us to recover the tme trajectory using only the low-frequency modes. The noise in the SDE approach is introduced when we approximate a differential equation by a finite difference formula and filter out high-frequency motions. 

This discussion suggests that even the reference trajectories used by symplectic integrators such as Verlet may not be sufficiently accurate in this more rigorous sense. They are quite reasonable, however, if one requires, for example, that trajectories capture the spectral densities associated with the fastest motions in accord to the governing model [13, 15]. Furthermore, other approaches, including nonsymplectic integrators and trajectories based on stochastic differential equations, can also be suitable in this case when carefully formulated. 

Recently Calderon introduced a surrogate process approximation (SPA) to improve the sampling in calculation of the JE. The scheme is applied to the study of the unravelling of deca-alanine at constant temperature in a steered molecular dynamics simulation. The distribution of the work is approximated by developing a model for the dynamics using a relatively small number of real trajectories in conjunction with stochastic differential equations selected to model the process. The... 

The formulation outlined above allows for a simple stochastic implementation of the deterministic differential equation (35). Starting with an ensemble of trajectories on a given adiabatic PES W, at each time step At we (i) compute the transition probability pk k, (h) compare it to a random number ( e [0,1], and (iii) perform a hop if pt t > C- In Ih se of a pure A -level system (i.e., in the absence of nuclear dynamics), the assumption (37) holds in general, and the stochastic modeling of Eq. (35) is exact. Considering a vibronic problem with coordinate-dependent however, it can be shown that the electronic... 

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