Stochastic differential equations generalized coordinates

The transformation rule given in Eq. (2.166) is instead an example to the so-called Ito formula for the transformation of the drift coefficients in Ito stochastic differential equations [16]. ft is shown in Section IX that V q) and V ( ) are equal to the drift coefficients that appear in the Ito formulation of the stochastic differential equations for the generalized and Cartesian coordinates, respectively. 

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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