Multicanonical algorithm

In this article, we describe the generalized-ensemble algorithms mentioned earlier. Namely, we first review the two familiar methods MUCA and REM. We then describe multidimensional extensions of these methods. Examples of the results by some of these algorithms are then presented. 

In the canonical ensemble at temperature T, each state x = ( /, p) with the Hamiltonian H(q,p) is weighted by the Boltzmann factor 

Because the coordinates q and momenta p are decoupled in (4.1), we can suppress the kinetic energy part and can write the Boltzmann factor as 

The canonical probability distribution of potential energy Pnvt( o T) is then given by the product of the density of states n(E) and the Boltzmann weight factor Wb(E T)  

Because n(E) is a rapidly increasing function and the Boltzmann factor de- 

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

Berg, B. Neuhaus, T., Multicanonical algorithms for first order phase transitions, Phys. Lett. B 1991, 267, 249-253... 

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. 

Bandyopadhyay, P. Ten-No, S. Iwata, S., Ab initio Monte Carlo simulation using multicanonical algorithm temperature dependence of the average structure of water dimer, Mol. Phys. 1999, 96, 349-358... 

The multicanonical algorithms are powerful, but the probability weight factors are not a priori known and have to be determined by iterations of short trial simulations. This process can be nontrivial and very tedius for complex systems with many degreees of freedom. 

Detailed formulations of the two well-known generalized-ensemble algorithms, namely, multicanonical algorithm (MUCA) and replica-exchange method (REM), were given. We then introduced further extensions of the above two methods. We have shown the effectiveness of these algorithms by applying them to various biomolecular systems. 

The entropic sampling version of the multicanonical algorithm, as applied to a discrete lattice model of a chain with attracting interactions, is of interest and is described here. Let the number of different energies of the system be L, and let i andbe chain conformations, with the degeneracy of energy E denoted by (compare with Eqs. [3], [13], and [14]). The multicanonical probability of conformation i is... 

The ability of the multicanonical algorithm to cover the whole energy... 

B. A. Berg and T. Neuhaus, Phys. Lett., B267, 249 (1991). Multicanonical Algorithms for First Order Phase Transition. 

Y. Okamoto and U. H. E. Hansmann,/. Phys. Chem., 99,11276 (1995). Thermodynamics of Helix-Coil Transitions Studied by Multicanonical Algorithms. 

Figure 3.3. The lowest-energy conformation of Met-enkephalin obtained in vacuum (the top left). Superposition of the eight lowest-energy conformations obtained from the combination of the RISM theory and the simulated annealing (the top right) and from the multicanonical algorithm combined with the RISM theory (the bottom).

Helix-Coil Transitions Studied by Multicanonical Algorithms. 

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