\sect{Baruch's  C25.}

Consider a system with random impurities. An experiment measures
one realization of the impurity distribution and many experiments
yield an average denoted by $\langle ...\rangle$. Consider the
free energy as being a sum over $N$ independent subsystems, i.e. parts
of the original system, with
average value $F =(1/N) \sum_{i=1}^N F_i$ ; the subsystems are
identical in average, i.e. $\langle F_i\rangle =\langle F\rangle$.
\begin{itemize}
\item[(a)] The subsystems are independent, i.e. $\langle F_iF_j\rangle
=\langle F_i\rangle \langle F_j\rangle$ for $i\neq j$, although
they may interact through their surface. Explain this.
\item[(b)]  Show that $\langle (F- \langle F\rangle)^2\rangle \sim 1/N $
so that even if the variance $\langle (F_i- \langle F\rangle)^2\rangle $
may not be small any measurement of F is typically near its average.
\item[(c)] Would the conclusion (b) apply to the average of the
partition function Z, i.e. replacing $F_i$ by $Z_i$ ?\\
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%