Multicanonical histogram

Multicanonical histogram hmua (f) and logarithm of the density of states g E) tor sequence 20.1. From [47]. 

Multicanonical histograms Wmuca (f/ 0) of energy f and angular overlap parameter Q and free-energy landscapes F Q) at different temperatures for the three seguences (see Table 8.1) (a) 20.1, (b) 20.4, and (c) 20.3. The reference folds reside at (3 = landf = fmin. Pseudophases are symbolized by D (denatured states), N (native folds),... 

Since the multicanonical ensemble contains all thermodynamic information, including the conformational transitions, it is quite useful to measure within the simulation the multicanonical histogram (cp. also Section 9.2)... 

Representatives and schematic characteristics of entangled and spherically capped conformations dominating the lowest-energy branches In the multicanonical histogram shown in Fig. 11.1(b). Red spheres correspond to hydrophobic (/I), light ones to polar (fi) residues. From [255]. 

Recently, Orkoulas and Panagiotopoulos [161] have shown that it is possible to use histogram reweighting and multicanonical simulations, starting with individual simulations near the critical point, to map out the liquid-vapour coexistence curve in a very efficient way. 

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. 3.4. 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...

Among the methods discussed in this book, FEP is the most commonly used to carry out alchemical transformations described in Sect. 2.8 of Chap. 2. Probability distribution and TI methods, in conjunction with MD, are favored if there is an order parameter in the system, defined as a dynamical variable. Among these methods, ABF, derived in Chap. 4, appears to be nearly optimal. Its accuracy, however, has not been tested critically for systems that relax slowly along the degrees of freedom perpendicular to the order parameter. Adaptive histogram approaches, primarily used in Monte Carlo simulations - e.g., multicanonical, WL and, in particular, the transition matrix method - yield superior results in applications to phase transitions,... 

A more robust (though computationally more intensive) alternative to GDI is provided by a synthesis of extended (multicanonical) sampling and histogram reweighting techniques. The method is bootstrapped by an ES measurement of the full canonical distribution of a suitable order parameter, at some point on the coexistence curve [identified by the equal areas criterion specified in Eq. (10)]. 

Fig. 4.2. Time series of potential energy of the C-peptide system from the multi-canonical MD production run (a) and the average potential energy as a function of temperature (b). The latter was obtained from the trajectory of the multicanonical MD production run by the single-histogram reweighting techniques...

One choice of generalized weights is the multicanonical ensemble [11,12] where the weight of a configuration c is defined as Wmuiticanonicai(c) oc l/g (F (c)). The multicanonical ensemble then leads to a flat histogram in energy space... 

Fig. 3. Scaling of round-trip times for a random walk in energy space sampling a flat histogram open squares) and the optimized histogram solid circles) for the two-dimensional fully frustrated Ising model. While for the multicanonical simulation a power-law slowdown of the round-trip times 0 N L ) is observed, the round-trip times for the optimized ensemble scale like 0([A ln A ] ) thereby approaching the ideal 0(A )-scaling of an unbiased Markovian random walk up to a logarithmic correction...

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