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Setup

Let the Hamiltonian of the Ising model be

$$ H_{J,h}(\sigma) = \Sigma_{i, j} J\sigma_i \sigma_j + \Sigma_{i} h \sigma_i.$$

Then the Gibbs partition function for the pair $(J,h)$ is given by

$$ Z_{J,h} = \Sigma_\sigma \exp(-H_{J,h}(\sigma)),$$

which describes the equillibrium distribution of the system.


I hope to understand Ising model from a non-equillibrium point of view. More precisely, given that the system is at a specific state $\sigma_0$, I'd like to know the probability for it to evolve into another specific state $\sigma_1$ at the next step, with respect to the given pair $(J,h)$. Namely, what is the evolving policy conditioned by $(J,h)$?

Questions

More formally put,

  1. What is, if any, an irreducible policy whose equillibrium state is $Z_{J,h}$?
  2. Is such policy unique?
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    $\begingroup$ There are many. The most common ones are the Glauber dynamics (which does not preserve magnetization) and its many variants (see Metropolis algorithm) and the Kawasaki dynamics (which does). Other include cluster algorithms, such as Swendsen-Wang. $\endgroup$ – Yvan Velenik Nov 28 '20 at 9:32
  • $\begingroup$ Why are there so many? Doesn't that mean there are free parameters unfixed? $\endgroup$ – Student Nov 28 '20 at 14:38
  • $\begingroup$ There is no canonical way of choosing a dynamics associated to the Ising model. Essentially, any Markov chain which has the corresponding Gibbs measure as stationary distribution will do. There are, of course, infinitely many such chains. $\endgroup$ – Yvan Velenik Nov 28 '20 at 14:49
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    $\begingroup$ That's why people usually consider various dynamics that present desirable features (conservative or not, reversible or not, etc.), chosen among the infinitely many Markov chains having this Gibbs measure as stationary distribution... This is sufficient to derive useful insights into coarsening dynamics, metastability, and so on... $\endgroup$ – Yvan Velenik Nov 28 '20 at 16:46
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    $\begingroup$ I don't think that there is a natural way to enforce uniqueness of the dynamics. However, you can analyze this type of questions for any fixed choice of the dynamics. I'd expect the result to depend on rough features of the dynamics (reversible, conservative, etc.), but otherwise the qualitative properties should be the same (but I may be completely wrong about that). Some (not very recent) papers dealing with this type of questions: this one and this one. $\endgroup$ – Yvan Velenik Nov 29 '20 at 9:10

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