heat does not exhibit any size dependence. However, we cannot rule out the possibility that a thermodynamic phase transition occurs at temperatures below where we fall out of equilibrium. Indeed theories which postulate a thermodynamic transition put the transition temperature well below the mode coupling $T_C$.

[Fig6.eps]{}

Specific Heat Versus Density
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In Figure \[fig:sphtdens\] we show the specific heat as a function of density. As the density increases, the specific heat rises to a peak at $\rho_{o}^{\rm peak}=0.8$. This corresponds to $\Gamma=1.44$ which is in good agreement with the $\Gamma$ value of 1.46 that we found for the specific heat peak when we varied the temperature. Going to higher densities corresponds to going to lower temperatures. At densities higher than 0.8, the system falls out of equilibrium.

[Fig7.eps]{}

Generalized Compressibilities
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As we mentioned in the introduction, the generalized compressibilities are thermodynamic probes that are a function of the microscopic structure of the system. They are solely a function of the positions of the particles and do not depend on their histories. So one could take snapshots of the configurations of the particles at different instances, scramble the order of the snapshots, and still be able to calculate the 