The Markov Chain of Spin Configurations

It is useful to think about the generation of the Markov chain of spin configurations as a physical thermalization process. We should be able to start from any choice of an initial spin configuration, e.g. all spins aligned, and eventually arrive at the desired sequence. However, the initial configuration may not be at all representative of configurations in the converged sequence. For example, if the temperature is high, it would be highly unlikely to find all the spins aligned. So it is necessary to carry out a sequence of many sweeps to make sure the sequence has had a chance to converge. In practice, we may use any observable at our disposal to monitor convergence, but prudence dictates that the observable that is slowest to converge is the safest indicator.

In a Monte Carlo simulation the expectation of an observable, computed through Eq (2), is only a statistical estimate. We might think to estimate the error in the mean by computing the standard deviation in the usual way and dividing by $\sqrt{N}$. However, that method works only if the measurements are statistically independent. Since we generate new spin configurations by updating the previous configuration, there is no guarantee that some memory of the previous configuration won't survive the updating process. That is, there may be some correlation from configuration to configuration in the Markov chain, spoiling statistical independence. The remedy is to make measurements at more widely spaced intervals to assure statistical independence. How wide this must be depends on the correlations, which can be tested statistically for various observables.