Multicanonical bias parameters

When creating and using our bias function there are many options available to us in regards to the implementation. The most obvious choices are the initial value of the refinement factor and our choice for the flatness criterion. The choice of flatness criterion, especially, can have a large effect on the simulation. Many studies have looked at the error between the calculated density of states from a Wang-Landau simulation compared with the real density of states for that system. However, some articles show that the method converges to the real result [122, 32, 76] while other work shows a saturation in the error [119, 112]. This contradiction in literature on the convergence of this method is somewhat clarified by Belardinelli and Pereyra

Figure 4.7: The simulation box is divided into cells of with rc, particles in the blue shaded cell will only interact with particles in that cell and the green shaded neighbouring cells.

[7] who showed mathematically that if the refinement parameter reduces faster that 1/t, wheretis the Monte Carlo time, then the error will saturate; this could explain some of the discrepancies between the work on this subject.

The difficulty with continuous systems is the lack of exact densities of states with which to compare the simulation results. Most work on error saturation is done on discrete Ising-like models; these are a long way from continuous simulations and are much more simplified. However, Li et al. [63] suggested the application of the Wang-Landau method to numerical integration, which is the simplest contin- uous system available. Belardinelli et al. [8] looked at the saturation of error in continuous numerical integration systems and confirmed that the 1/t adaptation is more accurate for all systems studied. They also showed that the bin width used for storing the density of states introduces a saturation of error in the 1/talgorithm as well as the original algorithm.

For the hard sphere systems we have a discrete order parameter due to the discrete nature of the hard sphere system. This means that we can avoid this saturation of error if we stick to the natural resolution of our system, that is we have a bin width of one. We also must be aware of the possibility of a saturation

Figure 4.8: The Verlet list contains all particles within a distance rv of particle i, particles withinrc of particle iwill interact with it.

of error in the method, and we can reduce this error by increasing the strictness of our flatness criterion. We can also test the error in our calculations against previous work from different methods and if the error between them is too great then this is a possible cause and could be alleviated by using the 1/t adaptation. We only need to ensure that our biasing function is good enough to allow us to sample the full order parameter space during a simulation with fixed biasing function; the more accurate it is though, the shorter our production runs can be.

Schulz et al. [92] discussed issues in regards to the treatment of the boundaries of the order parameter or energy range with Wang-Landau style multicanonical methods. When a boundary is reached a choice is made about how trial moves, that would take the simulation outside of the allowed range, are treated.

In the original Wang-Landau method they rejected the trial move without updating the current histogram or density of states function. It was shown that this leads to a systematic underestimation of the density of states at the bound- aries. Instead, when proposing a trial move that would take the simulation outside the allowed range, we should reject the move and then update the histogram and biasing function for the current order parameter value; this was shown to produce no systematic errors.

where the order parameter range is split up in to ‘windows’ with overlapping sec- tions, Figure 4.9. Separate simulations are run in each window and this means that the range for each simulation is smaller and more manageable. These windows are then joined together to form the complete function or density of states. If there are lots of boundaries then more care is needed when dealing with possible errors in the boundaries. Even though Schulz et al. [92] showed that there was no systematic error in the boundaries once they updated the histograms when a rejection occurred, this was only for a simple Ising model system and it may be more complicated for complex, continuous systems.

Cunha-Netto et al. [26] suggested that having fixed windows was the issue and led to a build up in error around these fixed boundaries. They suggested a method with adaptive windows where the boundaries change whenever the refine- ment parameter is reduced. They showed that this was an effective method for large continuous systems such as polymers [27].

Figure 4.9: The allowed order parameter range is split into overlapping windows, labeled in different colours in the diagram. A separate simulation is run within each window.

However, since the hard sphere system is so simple, these extensions of the multicanonical method may not be necessary.

When using the multicanonical biasing in the hard sphere system, the initial order parameter is at the gateway state that we want to bias towards. This is

because both crystals are initialised to allowed configurations and hence the order parameter is zero. Therefore we let the system equilibrate to higher order parameter values for 1,000 MC cycles before building up the weighting. This ensures that the gateway states are not over sampled at the beginning leading to long convergence times as this is corrected.

Another possible option to speed up the early parts of the convergence is whenever a new bin is visited, that has never been visited in this simulation before, we increase it by the smallest non-zero value in the weighting [60]. This should speed up the coverage of the entire space, but obviously this does not affect the simulation once all the range has been explored. However, this is more useful when there is not a fixed range of order parameter space or it is very hard to traverse. Neither of these situations occur in our simulations, and it was deemed an unnecessary computational effort as the simulations run fast enough without it.

The final parameter that needs to be decided upon is the initial value to use for the refinement parameter. The larger the starting value the faster the simulation will be pushed around the space but the longer it will take to get to a small enough value to say that the simulation is converged. However, if we start with a very small refinement value then the time taken to build up enough bias to push the simulation around the entire space will be very long. Therefore, a balance needs to be struck between these two factors.