Computer simulation convergence

To illustrate how stratification works in the context of free energy calculations, let us consider the transformation of state 0 into state 1 described by the parameter A. We further assume that these two states are separated by a high-energy barrier that corresponds to a value of A between Ao and Ai. Transitions between 0 and 1 are then rare and the free energy estimated from unstratified computer simulations would converge very slowly to its limiting value, irrespective of the initial conditions. If, however, the full range of A is partitioned into a number of smaller intervals, and... 

Iterative deconvolution is the original deconvolution method and remains quite reliable. The method relies on the synthesis of the library by the divide, couple, and recombine method to prepare a series of mixtures each with one residue of a selected diversity position being unique to each mixture. An active mixture(s) is selected and a resynthesis is performed whereby a second diversity position is defined. This is repeated until the resynthesis produces individual compounds. The highly active individual compounds this yields are the actives observed in the original active pool(s) ofthe library. The iterative method has been modeled by computer simulations. The results reported indicate that, even when accounting for experimental variability, an iterative deconvolution will converge to a molecule(s) that is the most active or very close to the most active (within 1 kcal) even for very large pools ( 65 000 compounds/pool) [18,19],... 

Construction. Once a converged computer simulation is available, construction of an x-y diagram is far easier than constructing one from scratch. Many computer simulations provide the option of plotting an x-y diagram, If this option is unavailable, the following sequence of steps can be followed ... 

Heat duty (or internal How) specification. A composition or product rate specification may be substituted by a heat duty or internal flow (e.g., reflux) specification. This is done either to improve convergence in a computer simulation (especially if compositions are in the part per million levels), or in a revamp when the column or its exchangers are at a capacity limit. The mass, component, and energy balance equations translate this specification into a composition or product rate specification. Sections 4.2.3 and 4,3.1 have some further discussion. 

Computer simulations of the fully converged, high-temperature limit spectra are shown at the bottom of each data set in Fig. 14.2. The hyperfine coupling constants used for each simulation are nearly identical and will be discussed in the next section. 

Fig. 3.20. Computer simulation results for the ratio C j x versus the fraction of unbroken springs in a triangular lattice network with different bond-bending forces (/3 = 0, 0.01, 0.3 and 1), having uniform distribution of bond-breaking thresholds. The ratio Cn/ji seems to converge to an universal value (c 1.25) as the complete fracture point is approached (Sahimi and Arbabi 1992).

It is easier to formulate and converge a computer simulation for a sequence of simple columns than a complex column. 

Taylor series at second order is a questionable approximation. This approximation is justified if the third- and higher-order direct correlation functions of the liquid phase are negligibly small, but this does not appear to be the case. In particular, Curtin [130] and Cerjan et al. [132] have studied the effect of including third-order terms in the perturbation series, and have found that the agreement with computer simulations is significantly worse than for the second-order theory. This clearly shows that third-order (and perhaps higher-order) terms are important, and that the convergence properties of the perturbation series are poor. 

Of course, the above considerations may not be relevant to the problem at hand, since in solving the OZ equation, the important functions are y(l, 2), its Fourier transform and B(l,2). ° It is to be expected that y(l,2) will vary less quickly between different orientations and will be continuous even for hard core potentials. Thus, its expansion in spherical harmonics should be better behaved than that of gf(l,2). Computer simulation cannot be used to obtain y(l,2) but Lado has presented some evidence based upon his solution of the RHNC approximation for a hard diatomic fluid using a spherically averaged bridge function that the convergence is good. Nevertheless, the results he presents are, in our view, for a rather short diatomic bond length and may not be conclusive. 

M. Tanaka, M. Terauchi, K. Tsudaand K. Saitoh, Convergent Beam Electron Diffraction IV , JEOE Etd, Tokyo and earlier volumes. An outstanding collection of CBED patterns and application examples, including detailed description of the techniques used for analysis. J. Spence and J. M. Zuo, Electron Microdiffraction , Plenum Publishing, 1992. An advanced text on quantitative analysis of CBED with useful examples, tables and computer simulation programs for electron diffraction analysis. 

The simulation is converged if all units, tear streams and overall material balance have converged. The convergence of overall heat balance is optional. The computation of convergence can be expressed mathematically as the minimisation of the residuals between X, the estimated values at the beginning of an iteration, and X, the calculated values after a pass through the computational sequence ... 

Two important difficulties arise in the computation of an average property as a sum. First, the number of molecules that can be handled in a computer is of the order of a few hundred. Secondly, flic number of configurations needed to reach the convergency in the sum can be too great. The first problem can be solved by different computational strategies, such as the imposition of periodic boundary conditions. The solution of the second problem differs among die main used techniques in computer simulations. 

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