# Monte Carlo steps in Ising model Metropolis algorithm

In K.Binder's book Monte Carlo Simmulation in Statistical Physics 4th ed., one Monte Carlo step is defined as "one sweep through the lattice". However, in many other books and papers, the Monte Carlo step is used to refer sweep on one single spin. Are these two method -- selecting spin on the lattice randomly one by one or traversing over the lattice (e.g. using a for loop) -- equivalent?

Actually, these two methods will give similar results in my simulation, but is there any (mathematical) proof for that?

• Yes, these are equivalent and the proof is an easy application of the theory of finite-state Markov chains. Mar 13, 2018 at 17:30
• Equivalent in expressiveness is one thing. Equivalent in speed of convergence is another. Isn't it? I have to confess that I've never looked into the theory. I can certainly make and argument that one is the low spin-flip probability limit of the other. Mar 13, 2018 at 17:58
• @dmckee : Yes, relaxation times will be different, although I doubt that there are large differences in term of efficiency (this has probably been studied, but I don't know the literature). In fact, both are pretty inefficient algorithms to simulate the Ising model. (Of course, they are at least essentially trivial to implement.) Mar 13, 2018 at 18:54