as a projection of the density onto a basis function $\phi(r)$, i.e., $\rho_{\phi}=<\rho|\phi>$. It weights the density fluctuations according to their spatial position. The application of the external potential will induce an average change $\delta \rho_{\phi}$ in $\rho_{\phi}$: $$\delta \rho_{\phi}=\langle\rho_{\phi}\rangle_{U}-
\langle\rho_{\phi}\rangle_{U=0}
\label{eq:deltarho}$$ where the thermal average $\langle\rho_{\phi}\rangle_{U}$ is given by $$\langle \rho_{\phi}\rangle_{U}={1\over Z}\hbox{\rm Tr}
\left[ e^{-\beta (H+U)}\rho_{\phi}\right]\quad 
\label{eq:AV}$$ The partition function $Z=\hbox{\rm Tr}e^{-\beta (H+U)}$ and $\beta$ is the inverse temperature. For small values of $\Delta P$, this change can be calculated using perturbation theory [@Toda91]. Up to third order in $\Delta P$, we find $$\begin{aligned}
\label{eq:expansion}
\nonumber
\delta \rho_{\phi}&=& 
        -\frac{\beta\Delta P}{\rho_o}\langle \rho_{\phi}^2\rangle_c +
        {\beta^2\Delta P^2\over 2\rho_o^2}\langle 
        \rho_{\phi}^3\rangle_c\\ 
        && -{\beta^3\Delta P^3\over 6\rho_o^3}
        \langle \rho_{\phi}^4\rangle_c,\end{aligned}$$ where the cumulant averages are $$\begin{aligned}
\langle \rho_{\phi}^2\rangle_c&=&\langle \rho_{\phi}^2\rangle -
        \langle \rho_{\phi}\rangle^2 
\label{eq:CUMU2}\\
\langle \rho_{\phi}^3\rangle_c&=&\langle \rho_{\phi}^3\rangle -
        3\langle \rho_{\phi}\rangle\langle \rho_{\phi}^2\rangle +
        2\langle \rho_{\phi}\rangle^3 \label{eq:CUMU3}\\ \nonumber
\langle \rho_{\phi}^4\rangle_c&=&\langle \rho_{\phi}^4\rangle -
        4\langle \rho_{\phi}\rangle\langle \rho_{\phi}^3\rangle -
        3\langle \rho_{\phi}^2\rangle^2 + \\
        && 12\langle \rho_{\phi}\rangle^2\langle \rho_{\phi}^2\rangle -
        6\langle \rho_{\phi}\rangle^4
\label{eq:CUMU4}\end{aligned}$$ with the thermal average $\langle \rho_{\phi}^n\rangle=\langle \rho_{\phi}^n\rangle_{U=0}$. The third order cumulant, eq.(\[eq:CUMU3\]), is zero in the liquid phase because for every configuration there exists an equivalent configuration with the opposite sign of $\left(\rho_{\phi}-\langle \rho_{\phi}\rangle\right)$ and so we will not consider this term 