Histogram Analysis

Beyond histogram analysis, one can also use integrated forms of the Crooks relation, (5.35), to estimate free energies. Formally, we can rewrite (5.35) as an average 

Note that on the right-hand side, the work W is that of the reversed path [i.e., has the opposite sign see (5.33)]. Equation (5.50) has to be solved numerically (e.g., by using a Newton-Raphson solver) for the free-energy difference A A. 

A word of caution should be made here the assumption in both Bennett s original calculation [21] and in the maximum-likelihood formalism [22] is that all work values are statistically independent. This may not be the case, for instance, if consecutive initial configurations are close in time along an equilibrium trajectory. If results 

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Example 23 A histogram analysis. (See Table 1.23 and data file HISTO.dat.)... 

Kumar, S. Bouzida, D. Swendsen, R. H. Kollman, P. A. Rosenberg, J. M., The weighted histogram analysis method for free energy calculations on biomolecules. I. The method, J. Comput. Chem. 1992,13, 1011-1021... 

Analyze the work distributions. Use the histogram analysis, optimal and cumulant estimators of Sect. 5.7 to determine the free energy difference. 

In some cases, one may not be able to collect enough trajectories for a histogram analysis. Instead, one can use moments [15]. In particular, the derivative of the free energy is approximately given by the average force (weighted by the Boltzmann factor of the accumulated work)... 

Umbrella Sampling and Weighted Histogram Analysis Umbrella Sampling (US)... 

Multidimensional free-energy calculations using the weighted histogram analysis method, J. Comput. Chem. 16 1339 (1995). 

Chodera, J.D., Swope, W.C., Pitera, J.W., Seok, C., Dill, K.A. Use of the weighted histogram analysis method for the analysis of simulated and parallel tempering simulations. J. Chem. Theory Comput. 2007, 3, 26-41. 

Graham MV, Purdy JA, Emami B, et al. Clinical dose-volume histogram analysis for pneumonitis after 3D treatment for non-small cell lung cancer (NSCLC). Int J Radiat Oncol Biol Phys 1999 45(2) 323-329. 

Figure25. Histogram analysis of conductances obtained for total 9095 s observations by the combination of frans-azo-ammonium and oligoether carboxylate ion pair 21.

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