How to normalize the density of states in JDoS Wang-Landau?

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The discussion centers on the normalization of the Joint Density of States (JDoS) in the Wang-Landau algorithm, particularly in the context of a bi-dimensional Ising model. The user seeks guidance on updating the logarithm of the density of states, ln[Ω(E,m)], noting that for ground states, the density must equal Q (with Q being 2 for the Ising model). They mention that for ground states, there is only one state for maximum and minimum magnetization, which could complicate the normalization process. A key point raised is that normalization may not be necessary for updating the density of states, as Monte Carlo steps primarily consider relative densities, allowing for non-normalized generation of the density of states during the algorithm's execution. Normalization can be performed after the algorithm concludes. The user also seeks clarification on what JDoS entails.
UFSJ
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Hi guys.

I want some help understanding how I can make the normalization of the JDoS density of states (Ω[E,m]) in the Wang-Landau algorithm. When I am working with DoS (Ω[E]) I use the knowledge that the value of the density of states in the ground states must be
equal to Q (Q = 2 for the Ising model), that is, I make the update ln[Ω(E)] = ln[Ω(E)] - ln[Ω(Eground state )] + ln[2]. However, I don't know how to update the ln[Ω(E,m)] in the JDoS algorithm. I am working with a bi-dimensional space of energy (E) and magnetization (m) in an Ising spin-lattice.
 
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I assume that you talk about an Ising like model. If you sample in E and m, then the number of states with E_groundstate is one for m=+N and m=-N (where N is the number of spins total and m assumed to be in units of spins pointing up). The number of states with E_groundstate and any other magnetization is zero, which may actually become a problem for you if you do not handle this situation.

I think normalization is irrelevant for updating your estimate for the density of states (DOS). Your Monte-Carlo steps (assuming you do MC) only take the relative DOS into account, and the normalization drops out of the calculation. So you can just generate your DOS estimation non-normalized and normalize after the algorithm is terminated.

What is JDoS?
 

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