Summary: In Wang Landau method, you have to calculate the probability $\exp (- \Delta S)$ but $\Delta S$ is generally large when the size of the system is not small. How can I make $\Delta S$ small?
I'm writing a monte carlo simulation with Wang and Landau algorithm. When the size of the system is small, the density of states is correctly gotten. However, when the size is large (*1), much more iterations are needed to flatten the energy histogram and thus the entropy of each energy bin becomes large. For example, when the entropy refinement parameter $f = 1.0$ (note this is the initial value), the energy histogram $H$ is given as below.
H = 618479 H = 619151 H = 620313 H = 620378 H = 620409 H = ...
And entropy $S$ is also given as below. (Since $f = 1.0$ now, $H$ and $S$ have the same values.)
S = 618479 S = 619151 S = 620313 S = 620378 S = 620409 S = ...
These histograms seem flat enough, but the difference of elements is of order $O(1000)$. We have to decide whether or not to accept the suggested new states with probability $\exp (- \Delta S)$, but this is zero when $\Delta S$ is large. So the only states whose $S$ is small are accepted and the algorithm doesn't proceed at all.
How can I solve this problem? I think it is not practical to wait until the histogram becomes completely flat. I wonder if there is a more feasible solution.
(*1): I assume $L \times L$ grid graph. I can compute $L \leq 9$ but not $L \geq 10$. You may think $10 \times 10$ grid graph is "not large", though.
Actually I'm trying to estimate the number of rook paths of a grid graph, using Wang Landau method rather than strict enumeration. The system is $(L + 1)^2$ nodes placed on a grid (Fig.1). Nearest neighbors are randomly linked. We propose a new state by choosing one (virtual) edge, and flip it (Fig.2) (i.e. if there is a edge there, cut it, and if there isn't, create a new edge). In the ground state, the edges constitute a path which links first node (upper left) and last node (lower right). This is just my challenge, and not an existing model.
The histograms above were gotten in a few minutes on my personal computer. I don't know this is long or short, but at least in $L \leq 9$ case (in which histograms are gotten in dozens of seconds or so), the estimated number of solutions are very similar to the true one. So I think my model isn't bad...