Multicanonical simulations

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. 

The average value of any property A at a temperature T C2m be determined from the multicanonical simulation using the following formula ... 

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...

In general, the multicanonical weight factor Wmu(E), or the density of states n(E), is not a priori known, and one needs its estimator for a numerical simulation. This estimator is usually obtained from iterations of short trial multicanonical simulations. The details of this process are described, for instance, in [24,33]. However, the iterative process can be nontrivial and very tedius for complex systems. 

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. 

A multicanonical simulation generates configurations according to the partition function... 

Using multicanonical simulation techniques we can sample all configurations in the pertinent interval of density (order parameter) and determine their free energy. It is important to note that the simulation samples all states at a fixed order parameter with the Boltzmann weight of the canonical ensemble. What are the typical configurations that a finite system of volume V adopts inside the miscibility gap in the canonical ensemble [48-55] ... 

Note that the conjugated field, Ap, does not have a direct interpretation in a multicanonical simulation. 

B. A. Berg (2003) Multicanonical simulations step by step. Comp. Phys. Comm. 153, pp. 397-406 ibid. (2004) Markov Chain Monte Carlo Simulations and their Statistical Analysis. World Scientific P. 380... 

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...

The multicanonical simulation method [94-96] is the most consequent realization of the ideas developed in the context of microcanonical statistical analysis. This method... 

In multicanonical simulations, the weight functions are updated after each iteration, i.e., the weight and thus the current estimate of the density of states are kept constant at a given recursion level. For this reason, the precise estimation of the multicanonical weights in combination with the recursion scheme (4,105)-(4.108) can be a complex and not very efficient procedure. In the method introduced by Wang and Landau [99], the density of states estimate is changed by a so-called modification factor c after each sweep, g(E) —> c " g E), where > 1 is kept constant in the nth recursion, but it is reduced from iteration to iteration. A frequently used ad hoc modification factor is given by = (c ) / ,... 

For the qualitative discussion of the folding behavior it is useful to consider the histogram of energy E and angular overlap parameter Q in the multicanonical ensemble as it is directly obtained from multicanonical simulations,... 

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... 

In Fig. 11.1(a), HraucaiE, F) is shown for the two-peptide system 2xFl as a density plot in the E-Y plane, which is the direct output obtained in a multicanonical simulation. Qualitatively, we observe two separate main branches (which are channels in the corresponding free-energy landscape), between which a noticeable transition occurs. In the vicinity of the energy iisep -3.15, both channels overlap, i.e., the associated macrostates... 

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

We will now take a closer look at the adsorption transition in the phase diagram (Fig. 13.12) and we do this by a microcanonical analysis [307, 308]. As we have discussed in detail in Section 2,7, the microcanonical approach allows for a unique identification of transition points and a precise description of the energetic and entropic properties of structural transitions in finite systems. The transition bands in canonical pseudophase diagrams are replaced by transition lines. Figure 13.15 shows the microcanonical entropy per monomer s e)=N lng e) as a function of the energy per monomer e=EfN for a polymer with N=, 20 monomers and a surface attraction strength = 5, as obtained from multicanonical simulations of the model described in Section 13.6. 

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