I ran into the following statement here and here but I believe it's more general.

Let's suppose we're running a simulation of a system and we are interested in the distribution of a quantity (say $M$). One would then run the simulation for a long time and construct a histogram of $M$. The counts in the histogram corresponding to a value of $M$ are $N(M)$. One can denote by $\langle N(M) \rangle$ the ensemble average of $N(M)$, over all simulations of a given length. The statement is that the error in the histogram goes like:

$\delta^2N(M) = (1+2 \tau) \langle N(M) \rangle$

Here $\tau$ is the correlation time of the system, or, of the quantity of interest. My first question is, is $\delta^2N(M)$ the variance or the standard deviation from the ensemble average? Second, how would you prove the above relation?

As an example, one could think of a Monte Carlo simulation of an Ising system, trying to obtain a distribution of the magnetization.

  • 1
    $\begingroup$ Take a look at Appendix B of this paper. $\endgroup$ – user8153 Apr 2 '19 at 0:05
  • $\begingroup$ @user8153, almost there. The precise step I need is cited from a paper I have no access to, but it's a good starting point. Thank you. $\endgroup$ – Botond Apr 19 '19 at 17:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.