Histogram equations

Although these equations are expressions for the density of states as a function of both energy and reaction coordinate, the free energies are identical to those obtained from the standard one-dimensional multiple histogram equation. 

Once the mean and standard deviation have been determined, the frequency distribution determined from the PDF can be compared to the original histogram, if one was constructed, by using a scaling factor in the PDF equation. For example, the expected frequency for the Normal distribution is given by ... 

We ean now plot the Normal frequeney distribution superimposed over the histogram bars for eomparison. The eurve is generated using equation 15, where the variables of interest, x, are values in steps of 10 on the x-axis from, say, 380 to 540. The Normal frequeney equation is given below, and Figure 6 shows the histogram and the Normal... 

Consider the union of only two components (equation 2.7-28), where x, and are each distributed as Figure 2.7-2. If there were only one bar in each histogram, the values Xj and x would be added and the probabilities multiplied as the probability of joint occurrence (equation 2.7-29). 

Copper-zinc alloy, analysis, 177-179 Corrosion, of Monel metals, study by x-ray emission spectrography, 231 of stainless steels, study by x-ray emission spectrography, 230, 231 Counter, windowless, 55, 222 Counterfeit bank notes, histograms from, 225-227 Counters, see Detectors Counting, electronic, 46 Counting error, 65, 66 equation, 66, 278... 

Fig. 9-8 Histogram of dissolved solids of samples from the Orinoco and Amazon River basins and corresponding denudation rates for morpho-tectonic regions in the humid tropics of South America (Stal-lard, 1985). The approximate denudation scale is calculated as the product of dissolved solids concentrations, mean armual runoff (1 m/yr), and a correction factor to account for large ratios of suspended load in rivers that drain mountain belts and for the greater than average annual precipitation in the lowlands close to the equator. The correction factor was treated as a linear function of dissolved solids and ranged from 2 for the most dilute rivers (dissolved solids less than lOmg/L) to 4 for the most concentrated rivers (dissolved solids more than 1000 mg/L). Bedrock density is assumed to be 2.65 g/cm. (Reproduced with permission from R. F. Stallard (1988). Weathering and erosion in the humid tropics. In A. Lerman and M. Meybeck, Physical and Chemical Weathering in Geochemical Cycles," pp. 225-246, Kluwer Academic Publishers, Dordrecht, The Netherlands.)...

This equation provides a prescription for calculating AA. The probability distribution function, P(A), for the range of A comprised between A0 and A is obtained from computer simulations, usually as a histogram. The ratio P /Po is then estimated. This generic idea has been implemented in various, creative ways, yielding a class of techniques called probability distribution or histogram methods. These methods are discussed in Chap. 3. 

A brief sketch of the derivation of the WHAM equations follows we note that a detailed explanation is available in the book by Frenkel and Smit [14], Consider the canonical reweighing (3.5). Our goal will be to combine the histograms pi(U) from several runs at different temperatures T to predict the distribution of potential energies at a new temperature T. Individually, each run enables us to reweight its histogram to obtain the distribution at T... 

Figure 19.6 Basis for histogram method of area determination, by equations 19.3-11 to -14, based on points marked x (curve unnecessary-drawn for illustration only)...

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