Multicanonical weight

In the multicanonical ensemble [11,12], on the other hand, each state is weighted by a non-Boltzmann weight factor Wmn E) (which we refer to as the multicanonical weight factor), so that a uniform potential energy distribution Pmu E) is obtained ... 

The definition in (4.16) implies that the multicanonical weight factor is inversely proportional to the density of states, and we can write it as follows ... 

Here, S(E) is the entropy in the microcanonical ensemble. Since the density of states of the system is usually unknown, the multicanonical weight factor has to be determined numerically by iterations of short preliminary runs [11,12]. 

If the exact multicanonical weight factor Wmu(E) is known, one can calculate the ensemble averages of any physical quantity A at any temperature T =l/kB(3) as follows ... 

In general, the multicanonical weight factor Wmu(E), or the density of states n(E), is not a priori known, and one needs its estimator for a numerical simulation. This estimator is usually obtained from iterations of short trial multicanonical simulations. The details of this process are described, for instance, in [24,33]. However, the iterative process can be nontrivial and very tedius for complex systems. 

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]

A new simulation is now initiated using this multicanonical weight factor (rather than the Boltzmann factor), from which new values of S( ) and thus H(E) can be determined. This cycle is continued until the distribution in H( ) is reasonably flat in the energy range being considered. 

Once the final multicanonical weight factor has been derived it provides the distribution for the production simulation in which high-energy configurations will be sampled adequately and high-energy barriers can be crossed with ease. Moreover, from this single simulation it is possible to derive the canonical distribution f canon(7 ,E) at any temperature (hence the name multicanonical ) ... 

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... 

Finally, after the best possible estimate for the multicanonical weight function is obtained, a long multicanonical production ran is performed, including all measurements of quantities of interest. From the multicanonical trajectory, the estimate of the canonical expectation value of a quantity O is then obtained at any (canonical) temperature Thy. 

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 ) / ,... 

Before we go into the technical aspects of this method, let us first discuss it more formally, The energy-dependent multicanonical weights (4,153) are trivially introduced into the partition sum as suitable decomposition of unity in the following way ... 

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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