The Multicanonical Method

The procedure implied by (3.46) is simple we start off with any weights of our choice, although often the most convenient choice is simply r/i (N) = 0. We perform 

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The density of states increases rapidly with energy but the Boltzmann factor decrease exponentially, meaning that Pcanon(T, E) is bell-shaped, with values that can vary by mar orders of magnitude as the energy changes. In the multicanonical method the simulatic... 

One of the drawbacks of the multicanonical method is that, during the simulations tc derive the weight factor, the energy distribution in H(E) can oscillate rather than steadilj approaching a limiting distribution. Another drawback is that it can fail to properlj... 

Fig. 3.4. Evolution of the weights i],(N) and the histograms fi(N) in a grand-canonical implementation of the multicanonical method for the Lennard-lones fluid al V 125. The temperature is T = 1.2 and the initial chemical potential is /./ = —3.7. The weights are updated after each 10-million-step interval, and the numbers indicate the iteration number. The second peak in the weights at large particle numbers indicates that the initial chemical potential is close to its value at coexistence...

In this section we describe several methods not pertaining to the techniques described earlier. We discuss the multicanonical method of Berg as applied to macromolecules, " " - and the adiabatic switching procedure of Rein-hardt, which is related to the thermodynamic integration approach but is based on different grounds. For completeness, we mention four additional techniques, three of which were developed originally for spin models. 

In summary, the multicanonical method provides unique features that are important for investigating first-order phase transitions and phenomena that occur over a large range of energies, such as protein folding. Upon further optimization, the method is expected to become an important tool for conformational search and for studying phase transitions in macromolecular systems. 

Another class of methods that has been used to remove sampling difficulties is based on what is often called the multicanonical ensemble These methods have also been called entropy sampling methods,for reasons that are made clear below. It is easiest to understand the multicanonical methods by considering the full classical canonical coordinate-momentum distribution... 

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. 

Fig. 3.7. Schematic comparison of the multicanonical and iterative transition-matrix methods. The light gray box indicates a single iteration of several thousand or more individual Monte Carlo steps. The acceptance criterion includes the configuration-space density of the original ensemble, p, and is presented for symmetric moves such as single-particle displacements...

Sayano et al. [97] generated the potential of mean force for interactions between amino acids and base pairs. Okamoto and coworkers [98, 99] have also developed replica-exchange multicanonical methods for proteins. Numerous additional biological examples can be found among the references within [94]. 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

The ESPS method draws on and synthesizes a number of ideas in the extensive free-energy literature, including the importance of representations and space transformations between them [63, 68, 69], the utility of expanded ensembles in turning virtual transitions into real ones [23], and the general power of multicanonical methods to seek out macrostates with any desired property [27],... 

We now introduce the idea of the multicanonical technique into the isobaric-isothermal ensemble method and refer to this generalized-ensemble algonthnmsthemultibaric-multithermalalgorithm(MlJBAT i) [62,63,123-125]. 

In order to estimate the free energy many canonical simulations at different temperatures are necessary furthermore, it is often difEcult to define a suitable reference state with a known entropy Sq. Two alternatives can be followed to overcome these difficulties (i) expanded ensemble methods and (ii) multicanonical methods. 

The transitions from the supersaturated vapor to the drop and from the drop to the slab configurations represent barriers in the configuration space which are not removed by the multicanonical reweighting scheme. Hence, these shape transitions limit the applicability of reweighting methods in the study of phase equilibria. 

Since the density of states and thus the multicanonical weights are not known initially, a scalable algorithm to estimate these quantities is needed. The Wang-Landau algorithm [13, 14] is a simple but efficient iterative method to obtain good approximations of the density of states g E) and the multicanonical weights TTmulticanonical(F ) [Pg.599]

Extended ensemble methods, such as the multicanonical ensemble, Wang-Landau sampling or parallel tempering can also be generalized to quantum systems [35,36], as we will show in the next two sections. 

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