# How to calculate the autocorrelation function of magnetic susceptibility for the Ising model?

In the paper Wolff U. 1989. Physics Letters B. 228(3):379–82, the autocorrelation time of susceptibility, $$\tau_\chi$$ was calculated, but the way to do so was not clearly explained in the paper.

To my understanding, the (normalized) autocorrelation function is defined as

$$C_A(t)=\frac{\langle (A(0)-\langle A\rangle)(A(t)-\langle A\rangle)\rangle}{\langle A^2\rangle-\langle A\rangle^2}$$ where $$A$$ is a quantity like energy $$E$$ and magnetization $$M$$. These two observables can be measured in each step of a Monte Carlo simulation. However, quantities like susceptibility and heat capacity are not instantaneously measurable in a simulation. Therefore it looks impossible to obtain a function like $$\chi(t)$$ and calculate the autocorrelation using the above equation.

How to calculate the autocorrelation function of magnetic susceptibility for the Ising model?

• maybe you have to use the fluctuation-dissipation theorem? You know the temperature of the system you are simulating, and you can calculate the bandwidth of the power spectral density of the magnetization after you do simulation for a long time. – wcc Jun 12 at 0:18
• Thank you @AmIAStudent. That's a good way to calculate a dynamic quantity. However, Because I calculate under equilibrium (so did Wolff), I don't think the magnetic susceptibility $\chi$ is dynamic at all. I would say that's a mistake. What's your opinion? – Joshua_whi Jun 13 at 19:06
• well, because of causality $\chi(t)$ is zero for $t<0$, so in that sense it is not a static quantity...Also any frequency dependence of $\chi(\omega)$ (Fourier transform of $\chi(t)$) implies $\chi(t)$ will vary as a function of time. – wcc Jun 13 at 21:07
• I think reviewing the Wikipedia articles on linear response function and fluctuation-dissipation theorem may help remove the doubt you have. – wcc Jun 13 at 21:13