after linking*, which has the same volume $\Volu_0$ as the liquid state, as the elastic reference state. This issue of the state right after linking and the elastic reference state will be discussed in detail in Section \[SEC:PhenFE\].

In light of this construction of the pressure ensemble, we have the capability of learning about the bulk modulus of the system, and to characterize volume changes caused by linking, a process that has the effect of eliminating translational degrees of freedoms.

To establish an appropriate statistical mechanics for the fixed-pressure ensemble, we shall make the following Legendre transformation of the Helmholtz free energy, which leads to the Gibbs free energy $\GibbFreeEner (\Pres,T)$:

\[EQ:LegeTran\] $$\begin{aligned}
    \Pres &=& -\frac{\partial \HelmFreeEner(\Volu,T)}
        {\partial \Volu}\big\vert_{T} \label{EQ:LegeTran1}  \, , \\
    \GibbFreeEner (\Pres,T) &=& \HelmFreeEner(\Volu,T) + \Pres \Volu \, . \label{EQ:LegeTran2}\end{aligned}$$

In Eq. (\[EQ:LegeTran2\]) the volume $\Volu$ takes the value (in terms of $\Pres$) that satisfies Eq. (\[EQ:LegeTran1\]), i.e., the volume that minimizes the Gibbs free energy at a given pressure $p$.

In the following sections, we shall first calculate the disorder-average of the Helmholtz free energy, and then make this Legendre transformation to obtain the disorder-averaged Gibbs free energy. This will allow us to explore the elasticity 