Simulation Convergence

Another check of convergence in MD simulations, as alluded to above, is to ensure that the sampling length is longer than the autocorrelation decay time for a particular property by several multiples of that time. In practice, this analysis is performed with less regularity than is the simple monitoring of individual property values. 

It must be borne in mind, however, that the typical simulation lengths that can be achieved with modem hardware and software are very, very rarely in excess of 1 ps. It is thus quite possible that the simulation, although it appears to be converged with respect to the analyses noted above, is trapped in a metastable state having a lifetime in excess of 1 ps, and as a 

A more detailed decomposition of macromolecular dynamics that can be used not only for assessing convergence but also for other purposes is principal components analysis (PCA), sometimes also called essential dynamics (Wlodek et al. 1997). In PCA the positional covariance matrix C is calculated for a given trajectory after removal of rotational and translational motion, i.e., after best overlaying all structures. Given M snapshots of an N atom macromolecule, C is a 3N X 3A matrix with elements 

Returning to the issue of convergence, as noted above the structure of each snapshot in a simulation can be described in the space of the PCA eigenvectors, there being a coefficient for each vector that is a coordinate value just as an x coordinate in three-dimensional Cartesian space is the coefficient of the i Cartesian basis vector (1,0,0). If a simulation has converged, die distribution of coefficient values sampled for each PCA eigenvector should be normal, 

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For the random walk in energy space, two histograms are recorded, //+ E) and H- E), which for sufficiently long simulations converge to steady-state distributions which satisfy H E) + H E) = H E) = W E)g E). For each energy level the fraction of random walkers which have label plus is then given by fiE) = H+(E)/H(E). The above-discussed boundary conditions dictate /(E ) = 0 and f E+) = 1. 

The bottom product rates were specified in molar units. For column 1 the side stream flow rate also was specified in molar units. Specification of the molar flow rates makes the simulations converge more easily. Nonstandard specifications can be harder for a nonequilibrium model to converge than for a corresponding equilibrium model because nonstandard specifications are more likely to lead to large variations in vapor and liquid flows from iteration to iteration. Since mass transfer coefficient and pressure drop calculations may... 

Figure 5 1 Activated mesocarbon microbead RAIC model, (a) Structural representation of the converged structure. The spheres represent carbon atoms that are shown at a scale much less than their van der Waals radii for reasons of clarity, (b) C-C radial distribution functions (RDF). The experimental RDF (solid line), the simulated, converged RDF (long-dashed line), and the initial simulated RDF (dotted line) are shown. The numbers indicate the different peaks of the RDF. (Adapted from Ref. [33].)...

To initiate the global simulation, the CHEOPS server as well as the wrappers of all participating simulators are started. Next, the simulation expert specifies the input file for CHEOPS and runs the simulation. CHEOPS solves the simulation in a sequential-modular mode Each simulator is run in turn, where the simulation results are passed from one simulator to the next. In the case of feedback loops, this process is iterated until a steady state is reached (i.e., the global simulation converges). 

Because the receptor conformations are changed concurrently with the ligand conformations, a 4D simulation convergence time is comparable with that of a single receptor docking, and is significantly shorter than for the traditional ensemble docking ... 

It is well known that the error estimate from any MC calculations (or in general from a simulation) converges as N 112, where N is the number of... 

Transfer Required Data after Simulation Converged... 

If we go back to Setup in the Cl block, change the reflux ratio to 3, and click the N button, the simulation converges with the results shown in Figure 3.36. The distillate impurity has decreased to about 2 mol% /C4, and the bottoms now has about 1.5 mol% C3. So we are getting pretty close to the desired purities. We could continue to manually change the reflux ratio and the distillate flow rate to attempt to achieve the desired product purities by trial and error. However, there is a much easier way, as discussed in the next section. 

Everything is ready to run again. Clicking the blue N button executes the program. The simulation converges in three iterations. 

As suggested future work, more scenarios can be investigated for better comparison. Ammonia production rate can be varied to get a better idea about the relationship between increasing/decreasing production rate and total Carbon Footprint. Amine solution can be varied in type and composition in order to look for optimum CO2 recovery scenario. Variation in process conditions is also suggested for better simulation convergence and results and also for optimum energy utilization. 

Next a second Vary ( 2 ) is set up similarly with reflux ratio selected. The upper and lower bounds are set at 0.1 and 1.0, respectively. Everything is ready to run again. Clicking the blue A button executes the program. The simulation converges in 16 iterations. 

In terms of fundamental studies, modeling of polymer chains is stiU in its infancy. Although there are plenty of instances where experiments and simulation converge, there is no general purpose, all-encompassing simulation approach that works universally. The primary diffi-... 

The value of may initially be a poor estimate of Feq. nevertheless, it should point in the right direction and successive, periodic refinements could then be implemented (e.g., by updating the initial point to the current conditions) so that the simulation converges to Feq and the corresponding equilibrium density. 

Those simulations beginning from the intermediate structures (rms 5-10 A) also collapsed and became more nativelike in terms of a number of criteria, but they did not refold. The radius of gyration and solvent accessible surface area of the protein in these simulations converged with those values obtained from simulation of the crystal structure at the same temperature. There was preferential burial of hydrophobic groups, some of the native tertiary contacts were formed, but the main-chain contacts characterizing secondary structure... 

Figure 8.17 shows the column specification page with the unrealistic column specifications removed and the specifications reassigned to simulate a control strategy where the tray 20 temperature and the reflux volumetric flow are controlled. The simulation converges quickly with these two specifications active, indicating that the two control objectives are relatively decoupled. 

Pumparound duties/ Side stripping steam rates These values are not routinely measured during true column operation and may vary significantly. It is generally inadvisable to rely on these values to make a simulation converge. 

First of all, it was verified that the dynamic simulations converged to the steady-state data obtained in the bench-scale unit in order to check the consistency of the model. The energy balance was not considered in the simulations because this unit operates in isothermal mode. Figure 8.26 shows the transient concentration profiles of sulfur and nitrogen in the liquid and solid phases at stabilized catalyst state. 

Figure 16.3 analyzes the dependence of the calculated a s for simulations converged for infinite chain length as a function of the ratio. 

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