Wang—Landau sampling

The modification factor plays a central role in a WL simulation and has several effects. First, its presence violates microscopic detailed balance because it continuously alters the state probabilities, and hence acceptance criterion. Only for g = 0 do we obtain a true Markov sampling of our system. Furthermore, we obviously cannot resolve entropy differences which are smaller than g, yet we need the modification factor to be large enough to build up the entropy estimate in a reasonable amount of simulation time. Wang and Landau s resolution of these problems was to impose a schedule on g, in which it starts at a modest value on the order of one and decreases in stages until a value very near to zero (typically in the range 10 5-10 8). In this manner, detailed balance is satisfied asymptotically toward the end of the simulation. 

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 

The same approach applies to alternate partition functions the WL output is used directly in the macroscopic probability scheme. For example, in the previous grand canonical scenario, the WL simulation would yield which would be substituted directly into the macrostate probabilities as Q exp(J ) for subsequent results generation. 

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], 

The principle benefits of the WL method are its wide applicability, ease of implementation, and rapid initial sampling of the macrostate space of interest. It 

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Wiist, T., Landau, D.P. The HP model of protein folding a challenging testing ground for Wang-Landau sampling. Comput. Phys. Commun. 2008, 179, 124-7. 

B. J. Schulz, K. Binder, M. Muller, and D. P. Landau (2003) Avoiding boundary effects in Wang-Landau sampling. Phys. Rev. E 67, 067102... 

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 Sampling for Quantum Systems High Temperature Expansion... 

It should now be obvious that Wang-Landau sampling and similar algorithms can be applied to quantum systems, by using the generalized density of states for quantum systems introduced in Sect. 5.1. 

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