The temperature dependence of the thermal conductivity is linked to the nature of the energy transport at a frequency omega, which is quantified by thermal diffusivity d(omega). Here we study d(omega) for a poorly annealed glass and a highly stable glass prepared using the swap Monte Carlo algorithm. To calculate d(omega), we excite wave packets and find that the energy moves diffusively for high frequencies up to a maximum frequency, beyond which the energy stays localized. At intermediate frequencies, we find a linear increase of the square of the width of the wave packet with time, which allows for a robust calculation of d(omega), but the wave packet is no longer well described by a Gaussian as for high frequencies. In this intermediate regime, there is a transition from a nearly frequency independent thermal diffusivity at high frequencies to d(omega) ~ omega^(-4) at low frequencies. For low frequencies the sound waves are responsible for energy transport and the energy moves ballistically. The low frequency behavior can be predicted using sound attenuation coefficients.
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