Exact trajectories

We proposed [7] two possible approaches to estimate these errors. Here we discuss them only briefly. Trajectories that are not too far from the optimal trajectory will have a significant weight. We denote by Xopt t) the optimized trajectory, and by Xexact t) the exact trajectory. The optimal trajectory is not the same as the exact trajectory, since it was computed with a large time step. SjlcP is expanded up to a second order near the optimal trajectory... 

We then require that the exact trajectory will have a weight of at least i, which will make its sampling possible in a search biased by the weight. 

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. 

One property of the exact trajectory for a conservative system is that the total energy is a constant of the motion. [12] Finite difference integrators provide approximate solutions to the equations of motion and for trajectories generated numerically the total energy is not strictly conserved. The exact trajectory will move on a constant energy surface in the 61V dimensional phase space of the system defined by. 

In this approach, the horizontal coordinates x[t] and y[t] both satisfy linear, second order equations. The exact trajectories satisfy Lame s equation of order two, which is extremely difficult to analyse. The corresponding equation for the linear approximation is Mathieu s equation, which is known to have both periodic and aperiodic solutions. 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

With this simplification, Gray, Rice, and Davis obtained reasonably accurate values for the predissociation rate constant as a function of initial vibrational excitation. The rate constant thus obtained is larger than that from exact trajectory calculations by about a factor of two. By contrast, the RRKM theory would give a rate constant that is about three orders of magnitude larger than is observed. 

Comparison of Elementary Rate Constants Calculated from the MRRKM Theory to Those Obtained by Gray and Rice s Exact Trajectory Calculations [55] for System No. 1... 

The focus of this chapter is a recently developed methodology [4 6] that enables us to calculate approximate MD trajectories at extended time scales. The method serves as a bridge between exact trajectories, reaction coordinates, and statistical theories of rate. It has been applied to the investigations of numerous systems [5 7], and we have computed trajectories for durations of nanoseconds [5], microseconds [6], and milliseconds [7]. Of course, it should be noted that the millisecond trajectories are highly approximate, since a very significant fraction of the motions was filtered out (see Section III.C). Nevertheless, they provide a view of the reaction pathway that is useful in interpretations of experimental data [7],... 

It is important to emphasize that the errors so calculated are with respect to the exact trajectory. The computational procedure for the errors is as follows We first compute an exact solution of the Newton s equation of motion,... 

The distribution of errors is a property of the exact trajectories. If we generate an approximate trajectory based on the finite difference formula, we should generate an error distribution that is consistent with what we know about the true solution. 

Figure 1. Distribution of norms of the error vectors computed by the finite difference formula [Eq. (13)] from exact trajectories of valine dipeptide. The dipeptide was initially equilibrated at 300 K. The largest errors are significant and are of the same order of magnitude as the forces.

We have no proof that the above two assumptions are true. All we have are numerical experiments on systems that vary from a dipeptide to a small solvated protein. Our results suggest that the above assumptions are sound for sufficiently large time steps. We note that at the limit of small Af, in which we obtain a nearly exact trajectory, the (much smaller) errors become correlated. For the statistical assumption to be valid the time step needs to be sufficiently large so that correlations will decay rapidly. A few numerical experiments for different step sizes are presented in Fig. 2. 

Are the results of the numerical experiments surprising Let us examine first the second assumption and assume for the moment that the correlation is lost rapidly. Is the normal distribution a surprise It is not. It is a simple demonstration of the Central Limit Theorem (CLT). For sufficiently large systems and after ensemble averaging, the addition of the (nearly uncorrelated) elements of the error vector leads to a normal distribution. Note also that the first and second moments of the errors are bound if the coordinates of the exact trajectory are... 

Figure . The correlation of errors ( (z) (0))/ (0) (0))) estimated from exact trajectories of valine dipeptide and Eq. (13) for three different time steps.

If the errors are zero, we obtain the most probable trajectory within the framework of the stochastic difference equation. This trajectory is not exact and is within cr from the exact trajectory [Eq. (13)]. What are the approximations made In Section III.C we argue that the approximate trajectory is a solution of the slow modes in the system where the high-frequency modes are filtered out. 

At this point we may continue in one of two directions. We may use a single approximate trajectory at the neighborhood of the exact trajectory that is, the trajectory that was obtained by the minimization of the discrete action. Alternatively, we recognize that the exact trajectory deviates from the optimal trajectory by errors distributed normally (keep in mind that the error distribution is a property of the exact trajectory). We may sample errors (and plausible trajectories) from the appropriate distribution of coordinates in the neighborhood of the trajectory with filtered high-frequency modes. The sampling in the neighborhood of the optimized trajectory should be normalized (we approximate one trajectory) ... 

The errors connected with the length formulation are defined as before. An exact trajectory assesses the accuracy of the finite difference formula. The choice of the finite difference formula to use is biased (as before) by the convenience of a constant Jacobian of transformation from the errors to coordinates [see also Eq. (13)] ... 

The notation DX(t) means a path integral (summation over all trajectories, X(t)), that starts at Xq and ends at X,. The action depends on the boundary coordinates as well as on the trajectory. At the limit of a small time step the distribution of the trajectories, X(t), is sharply peaked at the exact trajectory. In fact, in the numerical examples discussed below we only consider exact trajectories. The effects of large steps and the corresponding errors on the conditional probability will be the topics of future work. 

This minimally dynamic approach has been applied to both bimolecular and unimolecular reactions a typical result for the latter case is shown in Fig. 6. In this case we consider the dissociation of CCH on two different potential surfaces due to Wolf and Hase.36 These authors classified the first surface (their case IIC) as yielding RRKM dissociation, whereas their surface IIA yielded non-RRKM dynamics. The exact trajectory results for translational, vibrational, and rotational distributions for these two cases are shown as solid histograms in Fig. 6. The minimally dynamic construction, which requires only short-lived trajectory calculations, are shown as dashed histograms in the same figure and are seen to be in excellent agreement with the exact results. 

This is a remarkable result it is showing that a certain type of continuous column can be modeled exactly using a simple batch experiment. In other words, the exact trajectory mapped out when boiling a liquid of a certain mixture is exactly the same path followed by the liquid down a continuously operated total reflux column. 

To summarize, in this example, with the time interval fixed, the error in Euler s method increases in proportion to the stepsize. On the other hand, the global error grows with the length of the time interval, and at a rapid rate, until it is clear that the numerical trajectory is entirely unrelated to the exact trajectory. Moreover, we observe that the energy errors grow much more slowly than the trajectory errors. 

Consider the level set of potential corresponding to 17 = —0.5. Note that the kinetic energy is everywhere positive, so that a trajectory with total energy E = —0.5 will have to be bounded within the U = —0.5 level set. Would the dynamics (started from an initial condition with total energy E = —0.5) completely fill in this level set over time Because this system is chaotic, we cannot write down a formula for the exact solution, or find exact trajectories of the system, but we can use numerical methods to attempt to shed some light on the question. In Fig. 5.8 we... 

To apply the operational approach with a specific environment, data will be obtained from simulations rather than real operational conditions. This can appear in contradiction with OEF analysis that relies, by definition, on real data. But this way of doing aims only at showing the feasibility of the method and makes possible to focus on a typical environment. With - simulation, the exact trajectory of the mobile is known. In practice, this information can only be obtained with the deployment of a very expensive solution. 

Integrators in molecular dynamics simulations are supposed to be accurate, i.e., they should enforce the exact trajectory being followed as closely as possible. They should provide stability, meaning that the constants of motion, e.g., the total energy in the microcanonical ensemble, are preserved. Nevertheless, the integrators should be efficient, which means that a minimum number of force calculations are needed in order to save computer time. The best numerical methods are based on... 

Fig. 3.4. Time averaged kinetic energy distribution of an ion stored in a 16-pole and in a 32-pole trap. The results have been calculated using numerically exact trajectory calculations. For a better comparison the probability distributions are peak normalized.

---