Ginzburg Criterion - why average over the Correlation Length? To derive the Ginzburg criterion for the upper-critical dimension the fluctuations are averaged over a volume set by the correlation length. Why is this done? i.e. why do we average over a volume in the first place and why when we do not the entire volume?
 A: Here is a simple way to think about it: Consider the Feynman diagrammatic expansion of the statistical field theory. The propagator for the order parameter field is $1/(p^2+m^2)$, where $m=1/\xi$ is the inverse correlation length. We can now study the scaling behavior of diagrams as follows
$$
 \int \frac{1}{p^2+m^2} \sim \int_m^\Lambda \frac{1}{p^2}
$$
where $\Lambda$ is a UV cutoff. In coordinate space $\xi=1/m$ becomes an IR (long distance) cutoff. This corresponds to averaging fluctuations over a volume of size $V\sim \xi^d$. 
A: Mean field theory ignores fluctuations of the field and is valid if these fluctuations are smaller then mean field contribution.  You thus need to sum over all fluctuations or in real space you need to consider fluctuations between any two parts of the system.  Since the correlation length sets the length scale in which you know fluctuations have decayed to 0.  You in fact only need to integrate over a volume set by the correlation length.  This is what is expressed by the ~ in Thomas's response.
