What's not touched on in this article and arguably explains the ubiquity of the Ising model: it's the maximum entropy model given a mean and pairwise correlations
I would not describe the Ising model as simple, not even in the 2d case and especially not the general spin glass (where W_ij can be anything).
[1] https://en.wikipedia.org/wiki/Principle_of_maximum_entropy
Statistical mechanical entropy is information entropy (and related to thermodynamical entropy). From the link: “The principle was first expounded by E. T. Jaynes in two papers in 1957[1][2] where he emphasized a natural correspondence between statistical mechanics and information theory. In particular, Jaynes offered a new and very general rationale why the Gibbsian method of statistical mechanics works. He argued that the entropy of statistical mechanics and the information entropy of information theory are basically the same thing. Consequently, statistical mechanics should be seen just as a particular application of a general tool of logical inference and information theory.”
Simplicity is a relative thing, I guess. Let’s say the interestingness/complexity ratio is high.
The Shannon entropy of the distribution of the positions and velocities is precisely the Boltzmann entropy of the same, but there is no statistical mechanical analog to the Shannon entropy of a (non-Boltzmann) distribution, to my knowledge.
While Boltzmann entropy was first, and an important breakthrough, it has some limitations. Statistical mechanics is actually based on Gibbs/von Neumann entropy, which is equivalent to Shannon entropy. There are many subtleties indeed but I think the relationship between statistical mechanics and information theory may be closer than you think.
“To simulate the system, waiting times w_kj are generated for all candidate states k, and the shortest waiting time i = argmin_k w_kj is chosen. We call this shortest waiting time the holding time. A transition is then performed to state i after a delay of length w_ij.”
Say there are three states A, B, C and the transition rate to C from A or B is low. Wouldn’t your “simulation” jump endlessly between A and B without ever (or never again) visiting C?
I may be missing something.
The transition rates for each individual state transition are drawn from an exponential distribution with a rate dependent on the energy difference between the states.
So, no, C will be visited with some probability.
There are many problems with that paper, but validity of the sampling algorithm is not one of them. It is an unbiased sampler and will eventually converge to the sampled distribution (as proved in the paper). How long that convergence will take is another store. It remains unclear whether this approach converges faster than your garden-variety Markov chain and if so when and why.