Onsager-Machlup action

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

The trajectories with the highest probability are those for which the Onsager-Machlup action... 

We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

The square brackets denote a vector, and [ ] a transposed vector. The exact expression for the Onsager-Machlup action is now approximated by... 

Fig. 1. Optimization of the Onsager-Machlup action for the two dimensional harmonic oscillator. The potential energy is U(x,y) = 25i/ ), the mass is 1...

The Onsager-Machlup action mefhodology has a disadvanfage the need to know the total time of fhe frajecfory in advance. Also, low resolution trajectories do not approach a physical limit when the step size increases, in contrast to SDEL as we are going to see below. 

Similarly to the Onsager-Machlup action method the SDEL algorithm is based on the classical action. However, in this case the starting point is the action S parameterized according to the length of the trajectory ... 

Algorithmically, action-based methods are similar to the NEB method since in both cases a path functional is minimized. They differ, however, in the nature of the particular functional. While in the NEB method a path functional is constructed in an ad hoc way such that the path ttaverses the transition state separating reactants fi om products, the functional minimized in action-based methods corresponds, in principle, to the fully dynamical trajectories of classical mechanics. This property, however, is lost if extremely large time steps are used. In this case, the method yields a possible sequence of events that may be encountered by the system as it evolves fi om its initial to its final state, but a dynamical interpretation of such a sequence of states is not strictly permissible any more. Nevertheless, large time step trajectories that minimize the Gauss (Onsager-Machlup) action can provide possible scenarios for transitions that are computationally untreatable otherwise. 

To compute the above expression, short molecular dynamics runs (with a small time step) are calculated and serve as exact trajectories. Using the exact trajectory as an initial guess for path optimization (with a large time step) we optimize a discrete Onsager-Machlup path. The variation of the action with respect to the optimal trajectory is computed and used in the above formula. 

We can most easily see the importance of the A (Xi) by first considering the relationship between forward and reverse mechanical motions. Using the Onsager-Machlup thermodynamic action theory [11], Bier et al. [12] provided explicit relations based on the principle of microscopic reversibility for the relative probability of a down-slide vs an up-slide on an energy surface, which for Fig. 4 can be expressed as... 

The functional above was used already by Gauss [12] to study classical trajectories (which explains our choice of the action symbol). Onsager and Machlup used path integral formulation to study stochastic trajectories [13]. The origin of their trajectories is different from what we discussed so far, which are mechanical trajectories. However, the functional they derive for the most probable trajectories, O [X (t)] is similar to the equation above ... 

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