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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" ?

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    $\begingroup$ 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
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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.

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