Multicanonical probability

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... 

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],... 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

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,... 

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. 

A multicanonical MC simulation is performed, for instance, with the usual Metropolis criterion [118] The transition probability of state x with potential energy E to state x with potential energy E is given by... 

---