We present a rigorous and efficient approach to the calculation of classical lattice-dynamical quantities from simulations that do not require an explicit solution of the time evolution. We focus on the temperature-dependent vibrational spectrum. We start from the moment expansion of the relevant time-correlation function for a many-body system, and show that it can be conveniently rewritten by using a basis in which the low-order moments are diagonal. This allows us to compute the main spectral features (e.g., position and width of the phonon peaks) from thermal averages available from any statistical simulation. We successfully apply our method to a model system that presents a structural transition and strongly temperature-dependent phonons. Our theory clarifies the status of previous heuristic schemes to estimate phonon frequencies.
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