Frequency-dependent specific heat of viscous silica

We apply the Mori-Zwanzig projection operator formalism to obtain an expression for the frequency dependent specific heat $c\left(z\right)\mathrm{}$ of a liquid. By using an exact transformation formula due to Lebowitz et al., we derive a relation between $c\left(z\right)\mathrm{}$ and $K\left(t\right),$ the autocorrelation function of temperature fluctuations in the microcanonical ensemble. This connection thus allows to determine $c\left(z\right)\mathrm{}$ from computer simulations in equilibrium, i.e., without an external perturbation. By considering the generalization of $K\left(t\right)\mathrm{}$ to finite wavevectors, we derive an expression to determine the thermal conductivity $\lambda$ from such simulations. We present the results of extensive computer simulations in which we use the derived relations to determine $c\left(z\right)\mathrm{}$ over eight decades in frequency, as well as $\lambda .$ The system investigated is a simple but realistic model for amorphous silica. We find that at high frequencies the real part of $c\left(z\right)\mathrm{}$ has the value of an ideal gas. $c^{\prime }\left(\omega \right)$ increases quickly at those frequencies which correspond to the vibrational excitations of the system. At low temperatures $c^{\prime }\left(\omega \right)$ shows a second step. The frequency at which this step is observed is comparable to the one at which the α-relaxation peak is observed in the intermediate scattering function. Also the temperature dependence of the location of this second step is the same as the one of the $\alpha$ peak, thus showing that these quantities are intimately connected to each other. From $c^{\prime }\left(\omega \right)$ we estimate the temperature dependence of the vibrational and configurational part of the specific heat. We find that the static value of $c\left(z\right)\mathrm{}$ as well as $\lambda$ are in good agreement with experimental data.