Wang-Landau 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 ) / , 

Monte Carlo and chain growth methods for molecular simulations 

---

Yamaguchi, C. Kawashima, N., Combination of improved multibondic method and the Wang-Landau method, Phys. Rev. E. 2002, 65, 056710... 

Calvo, F., Sampling along reaction coordinates with the Wang-Landau method, Mol. Phys. 2002, 100, 3421-3427... 

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]. 

Shell, M.S., Debenedetti, P.G., Panagiotopoulos, A.Z. Generalization of the Wang-Landau method for off-lattice simulations. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 2002, 66, 056703. 

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. 

Since the density of states and thus the multicanonical weights are not known initially, a scalable algorithm to estimate these quantities is needed. The Wang-Landau algorithm [13, 14] is a simple but efficient iterative method to obtain good approximations of the density of states g E) and the multicanonical weights TTmulticanonical(F ) [Pg.599]

Extended ensemble methods, such as the multicanonical ensemble, Wang-Landau sampling or parallel tempering can also be generalized to quantum systems [35,36], as we will show in the next two sections. 

Wang-Landau recursion method, 57 Weighted histogram analysis method (WHAM), 274... 

F. Wang and D. Landau (2001) Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86, p. 2050 G. Bussi, A. Laio, and M. Parrinello (2006) Equihbrium free energies horn non-equilibrium metadynamics. Phys. Rev. Lett. 96, p. 090601 R. Martonak, A. Laio, and M. Parrinello (2003) Predicting crystal structures The parrineUo-rahman method revisited. Phys. Rev. Lett. 90, p. 75503... 

---