Monte Carlo simulations of the Ising model

Text sample:
Chapter 2.2.4. Initialization bias:
In the beginning of our simulation the system is out of equilibrium. As long as the system is out of equilibrium our data will have a systematic error since the equilibrium distribution was not yet reached. After a time _eq the system is in equilibrium. Unfortunately in general we do not know this time. One possibility is to estimate the time is by plotting observables like magnetization, internal energy...as functions of time in one diagram. After a sufficient time we expect that the observables do not change. Now they have reached approximately equilibrium. But it may be just a local minimum. Therefore we'll perform at least two simulations with different initial conditions, e.g. T=0 and T=_. We'll plot the respective observables of both simulations in one diagram. The cross-over point indicates the equilibration time. This has to be done for every observable, i.e. two observables in one plot since different equilibration times may exist for different observables. There is still the danger that the system reached only metastability. Therefore it is advisable to perform simulations in the in these regions with different initial values and test the consistency. Most Monte Carlo simulations simulations display metastability near a first-order transition. If we know the equilibration time we'll discard date performed before equilibration was achieved. The statistical errors associated with an out-of-equilibrium system are of order _/n as opposed to statistical errors after the initial transient being of order (_/n)1/2.
2.2.5. Autocorrelation in equilibrium:
Since Monte Carlo methods sample new states from old states there is a high correlation between subsequent states. To get rid of this correlation we have to wait a time _auto before taking the next data point so the points are uncorrelated. This means that we have only n/2_auto independent data points if we conduct a run of length n.
To quantify the correlation