Multicanonical method

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...

Bhattacharya, K.K. Sethna, J.P., Multicanonical methods, molecular dynamics, and Monte Carlo methods comparison for Lennard-Jones glasses, Phys. Rev. E 1998, 57, 2553-2562... 

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]. 

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],... 

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. 

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... 

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