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Next: Exercise 13.4: The correlation Up: Monte Carlo Simulation Previous: Exercise 13.3: Equilibration of

Measuring observables

Note that after reaching equilibrium we wish to compute the mean values of several physical quantities of interest. In general this is quite time consuming and hence we do not want to calculate their values more that necessary. After a single spin flip of one spin, the values of the observables in the two configurations will not differ much, and will be almost the same. Ideally, we wish to compute the observables for configurations that are statistically independent. This means that we have to run the simulation severals steps in between measurements. This number of steps is a typical quantity that depends on the physics of the model, the parameters used, the temperature, and in particular, the ``update dynamics'' used in the algorithm, as we shall see below. This ``correlation time'' is not known a priori, and we have to estimate it with a preliminar test run.

One way to determine the time intervals over which configurations are correlated is to compute the time-dependent autocorrelation functions defined by:

\begin{displaymath}
C_M(t)=\langle M(t)M(0) \rangle - \langle M \rangle ^2
\end{displaymath}

and

\begin{displaymath}
C_E(t)=\langle E(t)E(0) \rangle - \langle E \rangle ^2.
\end{displaymath}

Note that at $t=0$, $C_M$ is proportional to the susceptibility, and $C_E$ is proportional to the heat capacity. For sufficient large $t$, $M(t)$ and $M(0)$ will become uncorrelated, and

\begin{displaymath}
\langle M(t)M(0) \rangle \rightarrow \langle M(t) \rangle \langle M(0)
\rangle = \langle M \rangle ^2
\end{displaymath}

, and the same will occur with $C_E$. Hence $C_M$ and $C_E$ should vanish for $t \rightarrow \infty$. In general, we expect these quantities to decay exponentially with time. The time is takes $C(t)$ to decay to $1/e$ of its value at $t=0$ is an estimate of the autocorrelation time $\tau$. Since configurations separated by times less that $\tau$ are statistically correlated, we will compute the desired physical quantities for times intervals of the order of $\tau$ rather than after each MC step.



Subsections
next up previous
Next: Exercise 13.4: The correlation Up: Monte Carlo Simulation Previous: Exercise 13.3: Equilibration of
Adrian E. Feiguin 2009-11-04