two particles]{}]{}. For a detailed discussion of the Deam-Edwards distribution, see Ref. [@Deam1976; @Broderix2002]. By using this distribution of the quenched disorder, we can perform the disorder average of the Helmholtz free energy via the replica technique, thus obtaining $$\begin{aligned}
\label{EQ:HFReplica}
    \lda \HelmFreeEner \rda
    &\equiv& \sum_{\RealDiso} \DEDist(\RealDiso)
        \HelmFreeEner_{\RealDiso} (\Volu) \nonumber\\
    &=& -\BoltCons T \sum_{\RealDiso} \DEDist(\RealDiso) \ln Z_{\RealDiso} (\Volu)
        \nonumber\\
    &=& -\BoltCons T  \lim _{n \to 0} \sum_{\RealDiso} \DEDist(\RealDiso)
        \frac{Z_{\RealDiso}(\Volu)^{n}-1}{n}.\end{aligned}$$ We now insert the Deam-Edwards distribution to get $$\begin{aligned}
\label{EQ:HFReplicaDE}
    \lda \HelmFreeEner \rda
    &=& -\BoltCons T  \lim _{n \to 0} \sum_{\RealDiso}
        \frac{
            \Big(\frac{\LinkDens \Volu_0}{2\PartNumb \LinkPote_0}\Big)^{\LinkNumb}
            Z_{\RealDiso}(\Volu_0)}{\LinkNumb ! Z_1} \nonumber\\
    &&\quad\times   \frac{Z_{\RealDiso}(\Volu)^{n}-1}{n}.\end{aligned}$$ This disorder-averaged free energy differs from the form traditionally obtained via the replica technique, in that there is an extra replica $Z_{\RealDiso}(\Volu_0)$, which originates in the Deam-Edwards distribution. We shall call this extra replica the $0^{\textrm{th}}$ replica, and note that it represents the *preparation state* of the system. [^3] The summation over the realizations of the quenched disorder $\RealDiso$ can be performed, following the calculation in Appendix \[APP:DisoAver\]; thus we arrive at the form $$\begin{aligned}
\label{EQ:HFEDisoAver}
    \lda \HelmFreeEner \rda
    &=& -\BoltCons T  \lim _{n \to 0} \frac{1}{n}\Big( \frac{Z_{1+n}}{Z_1}-1 \Big),\end{aligned}$$ which can also be expressed as $$\begin{aligned}
\label{EQ:HFEDisoAverDeri}
    \lda \HelmFreeEner \rda 