# What does averaging over time means in literature?

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" ?

• what can it possibly mean other than taking the integral of the function between $[0,T]$? :p – gented Jun 21 at 9:17
• @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] ? – onurcanbektas Jun 21 at 9:21
• Well, all integrals are a function of the domain, after all. $\int_0^x dt f(t)$ is a function of $x$. – gented Jun 21 at 9:25