Histogram extrapolation

Fig. 22a. Plot of SCO li(q- 0) vs e/kBT for the model of Fig. 19b, N = 128, L = 80 and several choices of m = < M ) as indicated in the figure. For m = 0 the extrapolated curve for L- oo is shown as a full curve, while the linear extrapolations are shown as dash-dotted straight lines. For m = 0.3 and m = 0.S the linear extrapolations ate also shown, the actual temperatures of the coexistence curve being shown by stars in all three cases. Note that in this immediate vicinity of Tc all the curvature seen in the data (which are generated by histogram extrapolation) is due to finite size effects, b Plot of (1 — ()>,) 2/Scoii(q = 0) vs e/kBT for the model of Fig. 3, N = 32, < >v = 0.6, and various choices of the volume fraction <)>a/(1 — v) as indicated. Curves are a guide to the eye only. Since data over a very wide regime of temperatures are shown, curvature is due to an effective renormalization of the effective chi-parameter with temperature. Both the location of Tc and of the spinodal temperatures are shown with arrows. From Sariban and Binder [265]...

Fig. 24a. Phase diagram of the asymmetric polymer mixture (A. = 2.0, NA = NB = N = 32, 4, = 0.5) in the plane of variables reduced temperature and relative concentration 4a/(4a + 4b) of component A. The dashed lines are the histogram extrapolations for three simulated system sizes, the full line denotes the binodal, and the circle denotes the critical point. From Deutsch and Binder [93]. b Phase diagram of asymmetric polymer mixtures for NA = NB = N = 32, 4 = 0.5 in the (T, Ap) plane. Three choices of the asymmetry parameter A are shown as indicated. The first order transitions are shown as a full line, the critical points as circles. Temperature is normalized such that in the Flory-Huggins-approximation the critical temperature would occur for the same abscissa value. From Deutsch [266]...

This single histogram extrapolation is only practical in use for such a range of parameters where the exponentials in eq. (7.12) do not emphasize values of E, m far out on the wings of the histogram H, where the statistical accuracy is bad. But, for polymer mixtures near Tc, the number of chains n in the simulated boxes is often rather small, and then the distributions Pr Aii (E, wt) are all rather broad and this reweighting works nicely over a reasonably broad range of T and (Fig. 7.6). 

The final washability curve and histogram (Figure 5) are of arsenic in a sample from the Herrin (No. 6) Coal Member. The washability curve shows an intense concentration of arsenic in the heavier fractions. The curve approaches the abscissa and if extrapolated would intersect the ordinate near the origin. Arsenic is one of the least organically related elements in all four of the coals studied. 

Normative data using 40-min trials terminated after sleep onset have been published with extrapolations to estimate results if 20- and 30-min trials had been used. Figure 2 is a frequency histogram and Table 1 is a summary of parameters for normative MWT data gathered in the international, multisite cooperative... 

The purpose of statistical evaluation of sample data is to extrapolate from a collection of individual events (e.g., 30 min of process time) to the entire population of events (e.g., 8-h shift). Because microbial monitoring data usually measure the impact of human activity, which is not reproducible exactly from one event to the next, results usually do not fit standard statistical models for normal distributions. In spite of this limitation, it is necessary to summarize the data for comparison to limits. The best statistical methods of evaluation are determined by the nature of the data. Wilson suggests that microbial monitoring data histograms generally resemble Poisson or negative... 

I Values obtained by using a histogram width O.OIRm and an extrapolation using a harmonic fit... 

Figure 1. Histogram of frequencies of various yearly exposure times for hazard struck by moving vehicle . Extrapolated to the total Dutch working population.

It is often desirable to combine histograms from several simulation runs to get extrapolations over a wider parameter range and to increase the accuracy. If s simulations are performed at parameters Ti, Afii), / = 1,..., j, the generalization of eqs (7.10) and (7.12) for the density of states is is now the number of states recorded in simulation / for histogram Ht, (E, m) ... 

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