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The Ising model is a statistical mechanical model of ferromagnetism that defines the energy of a collection of magnetic dipoles arranged in a lattice, hence, through the Boltzmann distribution, also the (relative) probability of encountering the system in one configuration or the other.

To sample this, the Metropolis algorithm is often used, in the following form: a random site is selected, at which the spin is flipped with a certain probability depending on the temperature and the change in energy, in such a way that in the (very) long run, all configurations are generated with a probability distribution that is as prescribed.

The model doesn't describe any dynamics at all, but seeing this algorithm in action (visually in 2D), it is unavoidable to imagine this to be the time evolution of the system.

I know there is no a priori reason why it should be so, but does this algorithmic evolution anyhow resemble the actual dynamical evolution of spin states of particles in a real ferromagnetic material?

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Yes. Kinetic Monte Carlo (KMC) methods are similar to the Metropolis algorithm. The difference is that KMC assigns some time interval to each spin flip. Length of this time interval depends on energy barrier heights that needs to be overcomed for spin flipping. Of course Metropolis completely unaware of energy barriers. But if it flipps spins one by one, then yes, it really reproduce the same orders of spin flipping as KMC would

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