Multicanonical results

In a multicanonical simulation, the phase space is sampled in such a way that the energy distribution gets as flat as possible. Thermodynamically, this means that the sampling of the 

2 As in the previous chapters, we denote by monomer a single chemical unit within a single chain. However, in the aggregation literahire, the notion monomer is often also used for entire chains. 

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) 

The multicanonical partition function is also trivially constant with respect to temperature, 

It is obvious that integrating Hm sca E, F) over F recovers the uniform multicanonical energy distribution  

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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,... 

Figure 5. Results from a multicanonical simulation of the 3D Lennard-Jones fluid at a point on the coexistence curve. The figure shows both the multicanonical sampling distribution PAP) (symbol o) and the corresponding estimate of the equilibrium distribution Ro(p) with p = N/V the number density (symbol ). The inset shows the value of the equilibrium distribution in the...

The first example is the results of the calculation of the residual entropy of the ordinary ice [126,127], This calculation shows how accurate the density of states can be obtained by multicanonical simulations from the reweighting formula of (4.24). 

At j3 = 0, the number of states is 6N for the six-state model and 22N for the two-state model. Once these normalizations at j3 = 0 are given, the proportionality constant C can be determined from the results of the multicanonical simulations [24]. Hence, one can obtain an accurate estimate of the number of the lowest-energy state, f2(Eo), where Eq is the energy of the lowest-energy state. 

Using periodic boundary conditions (BCs), our simulations are based on a lattice construction set up earlier by Berg [135]. We have performed multicanonical MC simulations for the two models with the lattice sizes that correspond to the number of water molecules N = 128, 360, 576, 896, and 1,600. Combining the two fit results in the thermodynamic limit (N —> oo) leads to our final estimate... 

As a production run, we carried out a 15 ns multicanonical MD simulation and the results of this production run were analyzed in detail. 

Although the derivation of the recursion is somewhat tricky, the initial idea, the final result and its implementation are rather simple. The first step is to assume that we already know the optimized estimator E) for the microcanonical temperature, obtained after the ( — 1 )th recursion. Running the nth multicanonical recursion with it, we can replace Eq. (4.105) by... 

The following results were obtained in multicanonical computer simulations that yielded precise estimates for... 

The following results for a chain with N=20 monomers were obtained in multicanonical simulations [304, 307,308],... 

The simulation results for this model that will be discussed in the following section were obtained by multithreaded, multicanonical, and parallel tempering Monte Carlo simulations imder the assumption that the peptide only interacts with the surface layer (/= 1). A simulation box of dimension [50 A] with periodic boundary conditions parallel to the substrate was used. In perpendicular direction, peptide mobility is restricted by the Si substrate residing by definition atz = 0. The influence of the wall parallel to the substrate is simply steric, i.e., the atoms experience hard-wall repulsion atz = Zmax = 50 A [340]. 

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