Time step, 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. 

Figure B3.4.15. A possible Feymnaim path trajectory for a ID variable as a function of time. This trajectory carries an oscillating component with it, where. S is the action of the trajectory. The trajectory is highly fluctuating its values at each time step (v(dt), etc) are not correlated.

Olender and Elber, 1996] Olender, R., and Elber, R. Calculation of classical trajectories with a very large time step Formalism and numerical examples. J. Chem. Phys. 105 (1996) 9299-9315... 

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 ... 

Abstract. A stochastic path integral is used to obtain approximate long time trajectories with an almost arbitrary time step. A detailed description of the formalism is provided and an extension that enables the calculations of transition rates is discussed. 

We have in mind trajectory calculations in which the time step At is large and therefore the computed trajectory is unlikely to be the exact solution. Let Xnum. t) be the numerical solution as opposed to the true solution Xexact t)- A plausible estimate of the errors in X um t) can be obtained by plugging it back into the differential equation. 

It remains to be seen, if the approximation using large time steps is reasonable. We shall show later the effect of the approximation on the power spectrum of the trajectory. More specifically, we shall demonstrate that large time steps filter out high frequency motions. 

Hence, we use the trajectory that was obtained by numerical means to estimate the accuracy of the solution. Of course, the smaller the time step is, the smaller is the variance, and the probability distribution of errors becomes narrower and concentrates around zero. Note also that the Jacobian of transformation from e to must be such that log[J] is independent of X at the limit of e — 0. Similarly to the discussion on the Brownian particle we consider the Ito Calculus [10-12] by a specific choice of the discrete time... 

Since 5 is a function of all the intermediate coordinates, a large scale optimization problem is to be expected. For illustration purposes consider a molecular system of 100 degrees of freedom. To account for 1000 time points we need to optimize 5 as a function of 100,000 independent variables ( ). As a result, the use of a large time step is not only a computational benefit but is also a necessity for the proposed approach. The use of a small time step to obtain a trajectory with accuracy comparable to that of Molecular Dynamics is not practical for systems with more than a few degrees of freedom. Fbr small time steps, ordinary solution of classical trajectories is the method of choice. 

To improve the accuracy of the solution, the size of the time step may be decreased. The smaller is the time step, the smaller are the assumed errors in the trajectory. Hence, in contrast (for example) to the Langevin equation that includes the friction as a phenomenological parameter, we have here a systematic way of approaching a microscopic solution. Nevertheless, some problems remain. For a very large time step, it is not clear how relevant is the optimal trajectory to the reality, since the path variance also becomes large. Further-... 

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... 

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. 

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

One of the advantages of the Verlet integrator is that it is time reversible and symplectic[30, 31, 32]. Reversibility means that in the absence of numerical round off error, if the trajectory is run for many time steps, say nAt, and the velocities are then reversed, the trajectory will retrace its path and after nAt more time steps it will land back where it started. An integrator can be viewed as a mapping from one point in phase apace to another. If this mapping is applied to a measurable point set of states at on(> time, it will... 

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

Successful molecular dynamics simulations should have a fairly stable trajectory. Instability and lack of ec uilibratioii can result from a large time step, treatment of long-range cutoffs, or unrealistic coiiplin g to a temperature bath. ... 

Statistical mechanical averages in a molecular dynamics run are obtained by simply averaging an energetic or structural value over time steps. Thus if the values (xj, i are being computed in a trajectory. the statistical mechanical average is jiist... 

HyperChem run s the molecular dynain ics trajectory, averaging and analyzing a trajectory and creating the Cartesian coordinates and velocities, fhe period for reporting these coordinates and velocities is th e data collection period. At-2. It is a m iiltiplc of the basic time step. At = ii At], and is also referred to as a data step. The value 1I2 is set in the Molecular Dynamics options dialog box. 

Finite difference techniques are used to generate molecular dynamics trajectories with continuous potential models, which we will assume to be pairwise additive. The essential idea is that the integration is broken down into many small stages, each separated in time by a fixed time 6t. The total force on each particle in the configuration at a time t is calculated as the vector sum of its interactions with other particles. From the force we can determine the accelerations of the particles, which are then combined with the positions and velocities at a time t to calculate the positions and velocities at a time t + 6t. The force is assumed to be constant during the time step. The forces on the particles in their new positions are then determined, leading to new positions and velocities at time t - - 2St, and so on. 

Fig. 7.5 Difference between the exact and numerical, trajectories for the approach of two argon atoms with time steps of 1 Off and 50fi.

In the example shown in Figure 3.6, the trajectory passing through point A at the current time is found to originate from the inside of element (e) at the previous time step. 

After identification of the elements that contain feet of particle trajectories the old time step values of F at the feet are found by interpolating (or extrapolating for boundary nodes) its old time step nodal values. In the example shown in Figure 3.6 the old time value of Fat the foot of the trajectory passing through A is found by interpolating its old nodal values within element (e). 

HyperChem includes a number of time periods associated with a trajectory. These include the basic time step in the integration of Newton s equations plus various multiples of this associated with collecting data, the forming of statistical averages, etc. The fundamental time period is Atj s At, the integration time step stt in the Molecular Dynamics dialog box. 

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