Integrating the Equations of Motion

Solution of the equations of motion (either Eq. 8.1 or 8.2) requires three choices  

Whether one uses Newton s or Hamilton s equations of motion, obtaining the atomic positions over time requires numerical integration. Integration of ordinary differential equations (ODE) is a well-traveled territory in numerical analysis. A number of different techniques are routinely used in MD. 

The simplest numerical technique for solving ODE is Euler s method  

The two most widely implemented numerical integration techniques within MD are the Verlet algorithm and the use of instantaneous normal mode coordinates. The Verlet algorithm begins by writing the Taylor expansion for a coordinate at time t+ Af and f- Af  

This algorithm provides very good computational performance. 

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There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

If the magnitudes of the dissipative force, random noise, or the time step are too large, the modified velocity Verlet algorithm will not correctly integrate the equations of motion and thus give incorrect results. The values that are valid depend on the particle sizes being used. A system of reduced units can be defined in which these limits remain constant. 

HyperChem employs the leap frog algorithm to integrate the equations of motion. This algorithm updates the positions of atoms and the velocities for the next time step by this calculation (equation 26). 

Step size is critical in all simulations. This is the increment for integrating the equations of motion. It ultimately determines the accuracy of the numerical integration. For molecules with high frequency motion, such as bond vibrations that involve hydrogens, use a small step size. 

Start with an initial set of positions and momenta and integrate the equation of motion. 

Heat conductivity has been studied by placing the end particles in contact with two thermal reservoirs at different temperatures (see (Casati et al, 2005) for details)and then integrating the equations of motion. Numerical results (Casati et al, 2005) demonstrated that, in the small uj regime, the heat conductivity is system size dependent, while at large uj, when the system becomes almost fully chaotic, the heat conductivity becomes independent of the system size (if the size is large enough). This means that Fourier law is obeyed in the chaotic regime. 

To integrate the equations of motion in a stable and reliable way, it is necessary that the fundamental time step is shorter than the shortest relevant timescale in the problem. The shortest events involving whole atoms are C-H vibrations, and therefore a typical value of the time step is 2fs (10-15s). This means that there are up to one million time steps necessary to reach (real-time) simulation times in the nanosecond range. The ns range is sufficient for conformational transitions of the lipid molecules. It is also sufficient to allow some lateral diffusion of molecules in the box. As an iteration time step is rather expensive, even a supercomputer will need of the order of 106 s (a week) of CPU time to reach the ns domain. 

When integrating the equations of motion, it is important to not impose the shear only at the boundaries because this would break translational invariance. Instead, we need to correct the position in the shear direction at each MD step of size At. This correction is done, for instance, in the following fashion ... 

Since the selection of starting conditions in MD simulations can be somewhat arbitrary, and not necessarily realistic, it is necessary to "equilibrate the system by integrating the equations of motion for some period of time during which the behavior may not... 

Now we are able to substitute B/t) in Eq. 8 from Eq. 9. After replacing the acceleration Rj (t) with the force F/ (t) we finally obtain Eq. 6. There are several others algorithms to integrate the equations of motion (e.g., leapfrog, Verlet). The consequences of different equation of motion integration schemes with regard to AMD are discussed in the excellent review of Remler and Madden (54). 

To study different operating conditions in the pilot plant, a steady-state process simulator was used. Process simulators solve material- and energy-balance, but they do not generally integrate the equations of motion. The commercially-available program, Aspen Plus Tm, was used in this example. Other steady-state process simulators could be used as well. To describe the C02-solvent system, the predictive PSRK model [11,12], which was found suitable to treat this mixture, was applied. To obtain more reliable information, a model with parameters regressed from experimental data is required. 

The trajectory of a particle moving in a gas can be estimated by integrating the equation of motion for a particle over a time period given by increments of the ratio of the radial distance traveled divided by the particle velocity, that is, r/q. Interpreting the equation of motion, of course, requires knowledge of the flow field of the suspending gas one can assume that the particle velocity equals the fluid velocity at some distance r far from the collecting body. 

Since Hamilton s equations imply that pk = 0 if = 0, pk is a constant of motion if qt is such an ignorable coordinate. An ingenious choice of generalized coordinates can produce such constants and simplify the numerical or analytic task of integrating the equations of motion. 

In order to estimate the efficiency of the setup as described above, a Monte Carlo simulation was performed for the whole experiment by numerically integrating the equation of motion... 

Besides the requirement of accurate algorithms to integrate the equation of motions in MD simulations the accuracy of the forces plays a pivotal role. Methodologies to derive intermolecular forces can be divided into two main groups - molecular mechanics (MM) or quantum mechanics (QM). 

On the other hand, by integrating the equation of motion [Eq. (125)], squaring x(f), and making the Gibbs average, under the stationarity assumption, we obtain also [74]... 

An analytical constant of the motion is a precious asset. It can be used to (partially) integrate the equations of motion or to rule out chaos in situations where the dynamics appears to be complicated but is nevertheless regular and integrable (Bliimel (1993a)). 

The use of Verlet and Singer s algorithm makes it necessary to use extra care in integrating the equation of motion. Ciccotti et aL have shown how to do it in the case of Verlet s algorithm. As to the rotational equation of motion, we followed a similar procedure using the quantities... 

The drag coeflBcients for small particles have been tabulated (12) as a function of particle radius. By integrating the equation of motion, it can be shown that a particle moving in the flame as described would adjust to within 10% of the gas velocity in 20 mm if it had a radius of 1 X 10 m or of 50 mm if the radius were 10 X 10 m. These distances are consistent both with the particles attaining the maximum rather than the mean velocity in the burner tubes and with the production of 10-mm long vapor trails during evaporation. 

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