Flat-histograms

In other words, we simply introduce a factor in the microstate probabilities which is inversely proportional to the conventional macroscopic distribution. As a result, this factor cancels the integrated macroscopic probabilities and leaves the distribution constant - exactly the flat-histogram scenario of interest. 

In contrast to the weights formalism, the partition function approach directly employs the ideal flat-histogram expression in (3.36). Its goal is not to determine q but Q(N. V, T) directly, or more precisely in this case, the N dependence of Q. Due to numerical reasons, we usually work instead with the associated thermodynamic potential which is the logarithm of the partition function of interest in this case it is In Q = — 7/1 =. A, where we have used script i as an abbreviation. Thus our sampling scheme becomes... 

Table 3.1. Common flat-histogram ensembles and their reweighing procedures...

Not surprisingly, the essential component of flat-histogram algorithms is the determination of the weights, r/, or the thermodynamic potential, e.g., / or /. There exist a number of techniques for accomplishing this task. The remainder of this section is dedicated to reviewing a small but instructive subset of these methods, the multicanonical, Wang-Landau, and transition-matrix approaches. We subsequently discuss their common and sometimes subtle implementation issues, which become of practical importance in any simulation. 

Wang, F. Landau, D. P., Determining the density of states for classical statistical models a random walk algorithm to produce a flat histogram, Phys. Rev. E 2001, <54, 05 6101... 

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... 

A number of textbooks and review articles are available which provide background and more-general simulation techniques for fluids, beyond the calculations of the present chapter. In particular, the book by Frenkel and Smit [1] has comprehensive coverage of molecular simulation methods for fluids, with some emphasis on algorithms for phase-equilibrium calculations. General review articles on simulation methods and their applications - e.g., [2-6] - are also available. Sections 10.2 and 10.3 of the present chapter were adapted from [6]. The present chapter also reviews examples of the recently developed flat-histogram approaches described in Chap. 3 when applied to phase equilibria. 

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. 

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