Disclaimer: This question is cross posted in here because the answer might be field dependent.

Let $h: [0, L]\times \mathbb{Z}^1 \to \mathbb{Z}^1$ be a function, called the height function, and lets denote the mean value of $h$ at time $t$ be $\bar h(t)$.

In the paper Anomaly in numerical integrations of the Kardar-Parisi-Zhang equation by Chi-Hang Lam and F. G. Shin, Physical Review E, VOLUME 57, NUMBER 6, June 1998, at page 56, it is given that

$$w := \left < \frac{ 1 \sum_{x=1}^L (h(x,t) - \bar h(t))}{ L} \right >^{1/2},$$

where $L$ is the lattice size used in the numerical integration [...], The brackets 􏰒 􏰲 denote ensemble averaging, which is equivalent to averaging over time when steady state is being considered.

However, what do they mean by "ensemble average" and "averaging over time" ?

  • 1
    $\begingroup$ what can it possibly mean other than taking the integral of the function between $[0,T]$? :p $\endgroup$ – gented Jun 21 at 9:17
  • $\begingroup$ @gented Yes, but as a function of $T$ ? i.e for every T, we need to take the "ensemble average over time" between [0,T] ? $\endgroup$ – onurcanbektas Jun 21 at 9:21
  • $\begingroup$ Well, all integrals are a function of the domain, after all. $\int_0^x dt f(t)$ is a function of $x$. $\endgroup$ – gented Jun 21 at 9:25

This sounds like the ergodic hypothesis in statistical mechanics. It is the hypothesis that in a system in equilibrium (steady state), averages of a given observable over many-system at one fixed time would be equal to the average of the same quantity on one system but over a long period of time.


Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.