Multicanonical

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. 

Berg B A and Neuhaus T 1992 Multicanonical ensemble—a new approach to simulate Ist-order phase transitions Phys. Rev.L 68 9-12... 

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... 

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

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... 

Okamoto Y and U H E Hansmann 1995. Thermodynamics of Helix-coil Transitions Studied 1 Multicanonical Algorithms. Journal of Physical Chemistry 99 11276-11287. 

Senderowltz H, F Guarnieri and W C Still 1995. A Smart Monte Carlo Technique for Free Enerj Simulations of Multicanonical Molecules. Direct Calculations of the Conformational Populatioi of Organic Molecules. Journal of the American Chemical Society 117 8211-8219. 

H and B J Berne 1999. Multicanonical Jump Walking A Method for Efficiently Sampling Rough Energy Landscapes. Journal of Chemical Physics 110 10299-10306. 

Higo, J., Galzitskaya, O. V., Ono, S., and Nakamura, H. (2001). Energy landscape of a /3-hairpin peptide in explicit water studied by multicanonical molecular dynamics. Chem. Phys. Lett. 337, 169-175. 

Berg, B. A. Neuhaus, T., Multicanonical ensemble a new approach to simulate first-order phase transitions, Phys. Rev. Lett. 1992, 68, 9-12... 

Smith, G. R. Bruce, A. D., Multicanonical Monte Carlo study of solid-solid phase coexistence in a model colloid, Phys. Rev. E1996, 53, 6530... 

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...

The WL simulation terminates when the last stage takes g below some cutoff value at that point, the output is the calculated 5 . Contrary to the multicanonical... 

However, since and -5 asymptote to the same function, one might approximate (U) = S dJ) in (3.57) so that the acceptance probability is a constant.3 The procedure allows trial swaps to be accepted with 100% probability. This general parallel processing scheme, in which the macrostate range is divided into windows and configuration swaps are permitted, is not limited to density-of-states simulations or the WL algorithm in particular. Alternate partition functions can be calculated in this way, such as from previous discussions, and the parallel implementation is also feasible for the multicanonical approach [34] and transition-matrix calculations [35],... 

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...

Flattening the Energy Distribution Multicanonical Sampling and Related Methods... 

It is ironic that in the N —> oo limit the Tsallis sampling algorithm has the main feature for which the multicanonical algorithm was designed - it performs a random walk in energy - while the multicanonical algorithm loses this feature. 

---