Histogram Methods

The diserete data ean be analyzed from the residenee time distribution by using either the histogram method or the trapezoidal rule... 

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. 

Troyer, M. Wessel, S. Alet, F., Flat histogram methods for quantum systems algorithms to overcome tunneling problems and calculate the free energy, Phys. Rev. Lett. 2003,12, 120201... 

These bounds are the nonequilibrium equivalents of the Gibbs-Bogoliubov bounds discussed in Chap. 2. Having the free energy now bounded from above and below already demonstrates the power of using both forward and backward transformations. Moreover, as was shown by Crooks [18, 19], the distribution of work values from forward and backward paths satisfies a relation that is central to histogram methods in free energy calculations... 

In the sections below, we describe several studies in which flat-histogram methods were used to examine phase equilibria in model systems. The discussion assumes the reader is familiar with this general family of techniques and the theory behind them, so it may be useful to consult the material in Chap. 3 for background reference. Although the examples provided here entail specific studies, their general form and the principles behind them serve as useful templates for using flat-histogram methods in novel phase equilibria calculations. 

Other variations on these basic free energy methods have been published, although for various reasons they have not yet been widely adopted. These methods include MD/MC methods,38 the acceptance ratio method,39, 40 the weighted histogram method,41 the particle insertion method,42 43 and the energy distribution method.39 The reader is referred to the original publications for additional discussion of these approaches. 

In the histogram method, Figure 19.6, the area under the response curve is the sum of the rectangular areas whose heights are the nc individual measured responses... 

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)...

Since the response data are discrete, we may use either the histogram method or the trapezoid rule. We use the trapezoid rule for illustration, but both methods give the same results. 

Table 19.4 Spreadsheet calculations for Example 19-3, using the histogram method...

Table 19.4 gives the calculations of E(t), t, and of based on the histogram method. Column (4) lists the values of c j calculated from (A). Column (5) gives At, required for the calculations in subsequent columns, from equations 19.3-12 to -14. Column (6) gives values for the tracer material-balance (see below). Column (7) gives values of E t) from equation 19.3-4 with C(t) = E(t). Columns (8) and (9) give values required for the calculation of Tin equation 19.3-7, and of of in 19.3-8, respectively. 

The value of N obtained if the histogram method is used is 2.6. By comparison, nonlinear regression with E-Z Solve (file exl9-3.msp) predicts that N = 3.0. 

Different estimates of Pe, are obtained if values of i (21.8 s) and of(180s2) fiom the histogram method are used... 

A statistical analysis of light-scattering data can compensate for polydispersity. In cumulant analysis, lng(z) is expanded in a power series and coefficients of the different terms are evaluated against the experimentally obtained t, in search of the closest-fitting average selected by the smallness of the standard deviation. In a histogram method, the experimental t is... 

IV.3. Monte-Carlo data analysis with the weighted histogram method... 

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