# 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 caseâ€”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