# What does $\delta$ represents in FLUCTUATION-DISSIPATION THEOREM?

i am trying to follow the following tutorial. I keep seeing $$\delta$$ over functions such as $$\delta F(x)=F(x)-\langle F(x)\rangle_t$$ (Eq 14.4) in this and in other tutorials and questions here. What does the $$\delta$$ represent here?

Also, is it correct to say that the autocorrelation of such a function $$\delta F(x,t)$$ Is given by $$\langle \delta F(x,t),\delta F(x,t+\tau)\rangle$$? If so, how do I develop this? Is that the covariance somehow?

Googled it and could not find anything.

The $$\delta$$ is just a common notation for indicating that the quantity after the $$\delta$$ is a deviation from an average; it doesn't really mean or do anything other than indicate that.
If the original quantity under consideration (say $$F(x)$$) is a random variable, then $$\delta F(x)$$ is also a random variable, which is usually the case—unfortunately the document you cite appears to use this notation to describe deterministic quantities in the form of differences between different averages.
In any case, $$\langle \delta F(x,t),\delta F(x,t+\tau)\rangle$$ is the auto-covariance of $$F(x)$$ whereas $$\langle F(x,t),F(x,t+\tau)\rangle$$ is the auto-correlation of $$F(x)$$.
• So, as autocorrelation, shouldn't it be $\langle F(r,t+\tau)F(r,t)\rangle$? why is it $\langle \delta F(r,t+\tau)\delta F(r,t)\rangle$? – havakok Jan 1 '19 at 9:50
• The auto-covariance of $F(x)$ is the auto-correlation of $\delta F(x)$. – GiorgioP Jan 1 '19 at 10:24