\label{eq:cosine}$$ Since the system is isotropic, we compute the compressibilities for each direction and then average over the direction $\mu$.

In most of our calculations we work in the canonical ensemble where we fix the volume $V$, the number $N$ of particles and the density $\rho_o$. However, it is straightforward to generalize our results to the grand canonical ensemble where the number of particles is not fixed. We simply replace the thermal average defined in equation (\[eq:AV\]) by $$\langle \rho_{\phi}\rangle_{U}={1\over {\cal Z}}\sum_{N}
e^{\mu N}\hbox{\rm Tr}
\left[ e^{-\beta (H_{N}+U_{N})}\rho_{\phi}\right]\quad
\label{eq:grandcanonicalAV}$$ where $\mu$ is the chemical potential, $H_N$ is the Hamiltonian with $N$ particles, $U_N$ is given by eq. (\[eq:U\]) for a system with $N$ particles, and ${\cal Z}$ is the grand canonical partition function given by $${\cal Z}=\sum_{N}e^{\mu N}\hbox{\rm Tr}
\left[e^{-\beta (H_{N}+U_{N})}\right]$$ The generalized compressibilities can be defined using equations (\[eq:CUMU2\]) through (\[eq:susceptibilities\]) with the thermal averages $\langle \rho_{\phi}^n\rangle=\langle \rho_{\phi}^n\rangle_{U=0}$ defined in the grand canonical ensemble.

Results for Linear Generalized Compressibility
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We now turn to our results for the binary glass forming liquid.

### $\chi_{l}$ from Absolute Value of Positions versus Temperature

We will first discuss the linear generalized compressibility calculated from the absolute values of the particle positions using 