I have one question about calculation rules for means.
Say I have an equation of the form \begin{align*} \langle H(a,b) \rangle = \langle H_1(a,b) \rangle + \langle H_2(a,b)\rangle \end{align*} where $\langle ... \rangle$ is a mean (physical, i.e. integral over time / space).
Is then \begin{align*} \langle \frac{\partial H(a,b)}{\partial a} \rangle = \langle \frac{\partial H_1(a,b)}{\partial a} \rangle + \langle \frac{\partial H_2(a,b)}{\partial a}\rangle \end{align*} also true? (a is a variable, but not the one considered in the mean integral, i.e not time / space).
Sure, if
$$ \langle f(a)\rangle = \int{\rm d^3}{\bf x}~{\rm d}t ~f(a, {\bf x}, t) $$
then
$$ \frac{{\rm d}}{{\rm d}a}\langle f(a)\rangle = \frac{{\rm d}}{{\rm d}a} \int{\rm d^3}{\bf x}~{\rm d}t ~f(a, {\bf x}, t) = \int{\rm d^3}{\bf x}~{\rm d}t ~\frac{\partial}{\partial a}f(a, {\bf x}, t) $$