Finite histograms

However, it does have some disadvantages. The / and g histograms obtained from a finite-length simulation contain errors themselves. The histogramming and graphical analysis require additional steps in the calculation, and can introduce new sources of errors [54]. Better alternatives, such as Bennett s method and the overlap sampling method, will be discussed further in this chapter. 

For the moment estimates, we have seen that the composition PDF, /, (delta functions (i.e., the empirical PDF in (6.210)). However, it should be intuitively apparent that this representation is unsatisfactory for understanding the behavior of fyiir) as a function of fj. In practice, the delta-function representation is replaced by a histogram using finite-sized bins in composition space (see Fig. 6.5). The histogram h, (k) for the /ctli cell in composition space is defined by... 

Figure 4.5 An example of a histogram for a finite number of measurements.

Table HI compiles MC results obtained over the years for the critical temperature and critical density of the RPM. Table in includes also results from the cluster calculations of Pitzer and Schreiber [141]. In a critical assessment of earlier work [40, 141, 179-181, 246], Fisher deduced in 1994 that T = 0.052-0.056 and p = 0.023-0.035 represent the best values [15]. Since then, however, the situation has substantially changed. Caillol et al. [53,247] performed simulations of ions on the surface of a four-dimensional hypersphere and applied finite-size corrections. Valleau [248] used his thermodynamic-scaling MC for systems with varying particle numbers to extract the infinite-size critical parameters. Orkoulas and Panagiotopoulos [52] performed grand canonical simulations in conjunction with a histogram technique. All studies indicate an insufficient treatment of finite-size effects in earlier work. While their results do not agree perfectly, they are sufficiently close to estimate T = 0.048-0.05 and p = 0.07-0.08, as already quoted in Eq. (6). Critical points of some real Coulombic systems match quite well to these figures [5]. The coexistence curve derived by Orkoulas and Panagiotopoulos [52] is displayed in Fig. 9.

For polymodal or wide distributions the histogram method 03-9u (or the exponential sampling method) is more representative. In this method, the particle size distribution is presented by a finite number of discrete sizes, each of them an adjustable fraction of the total concentration. Then, the... 

Figure 2.11 Histogram of the finite-time Lyapunov exponents calculated from 8000 trajectories in the chaotic sine-flow of Eq. (2.66) and Fig. 2.9, for t = 50,100,150 and 200. The distribution becomes increasingly concentrated around A°°.

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]...

We are of the opinion that the tendency of a system to display unilateral or bilateral phase changes is indicative of more serious finite-size effects. Application of the GDI method (or the Gibbs ensemble for that matter) under such circumstances can provide only a qualitative picture of the phase behavior. Histogram-reweighting methods have been developed and refined to the point where they now provide a very precise description of the critical region in the thermodynamic limit. These methods should be applied if a quantitative characterization of the critical region is desired [52-57]. 

The quantity/ (x) dr is, by definition, the probability of occurrence of the random variable within the interval of width dr around point x. In practical terms, this means that if we randomly extract an x value, the likelihood that it falls within the infinitesimal interval from x to x+dr is f(x)dx. To obtain probabilities corresponding to finite intervals — the only ones that have physical meaning — we must integrate the probability density function between the appropriate limits. The integral is the area below the f(x) curve between these limits, which implies that Fig. 2.3 is also a histogram. Since the random variable is now continuous, the... 

In the next step, a facies type has to be assigned to each finite-difference model cell within the simulation domain. For this task, the experimental histogram of the clusters (clusterl - clusterM) and the cluster variogram models are used to generate conditioned equiprobable three-dimensional realizations of the facies fields described by categorical variables. For the facies-based approach used here, the three-dimensional conditional sequential indicator simulation method (SIS) for categorical variables... 

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