Suppose we have a 2D Ising model in physical world, at finite temperature, the system have to obey the Boltzmann distribution: $$P(\{s_i\})=\frac{1}{Z}\exp^{-\beta E(\{s_i\})}$$ where $Z$ is the partition function of the system and $E(\{s_i\})$ is the energy of the particular configuration $\{s_i\}$. My question is that what's could be a dynamics of the system such that it can give rise to the particular distribution?
The question is naive in the sense that maybe there is not classical Ising model in the real world at all, the model is just an effective description of some system in which classical spins are not the fundamental degree of freedom.
But let's suppose we have it, what could be a physical dynamic for it? Just like that molecular dynamics can lead to the Boltzmann distribution for hard disk system; in which the physical law is just Newton mechanics.
