Multicanonical recursion

Although the derivation of the recursion is somewhat tricky, the initial idea, the final result and its implementation are rather simple. The first step is to assume that we already know the optimized estimator E) for the microcanonical temperature, obtained after the ( — 1 )th recursion. Running the nth multicanonical recursion with it, we can replace Eq. (4.105) by... 

Looking more carefully at the recursive scheme in the form given by the set of equations (4.105) 4.108) is instructive, as it shows how the multicanonical weights are naturally coimected to microcanonical thermodynamic quantities such as temperature, entropy, and Iree energy as functions of energy. However, by making use of Eq. (4.102), we can reduce this scheme by establishing a relationship between W acaiE) and fi [95]. We simply consider the ratio... 

Thus, the weights wo,i determine the optimized estimator completely. Suppose we use this optimized j3 to determine the multicanonical weights according to the multicanoni-cal recursion relations (4.105)-(4.108) and then perform a multicanonical run which yields Amnna( ) or, equivalently, y E). If we now want to combine and the naive estima-... 

Since the optimized, error-weighted recursion is much more powerful than the standard recursion in that it provides a smoother and faster convergence in the recursive process of estimating the multicanonical weights, it should generally be favored, even more so as the additional implementation effort is minimal. 

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 ) / ,... 

In the following, we will describe the recursion method for the multicanonical weights, from which we obtain an estimate for the density of states. Since there is no information about an appropriate choice for the multicanonical weights in the beginning, we set them in the zeroth iteration for all chains 2[Pg.130]

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