Flat-Histogram Methods

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. 

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. 

One particular case of Eq. (A 12) has attracted considerable attention. If one sets M = E and considers the infinite temperature limit, the probabilities of the macrostates ) and Ej can be replaced by the associated values of the density-of-states function G(Ej) and G(Ej). The resulting equation has been christened the broad-histogram relation [128] it forms the core of extensive studies of transition probability methods referred to variously as flat histogram [129] and transition matrix [130]. Applications of these formulations seem to have been restricted to the situation where the energy is the macrovariable, and the energy spectmm is discrete. 

Because it is a flat histogram method, it tries to sample the whole CV space. This can push the simulated system toward states with nonphysically high free energy and might drift the simulation toward thermodynamically nonrelevant configurations. 

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