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The phase field crystal (PFC) approach extends the notion of phase fieldmodels by describing the topology of the microscopic structure of a crystallinematerial. One of the consequences is that local variation of the interatomicdistance creates an elastic excitation. The dynamics of these excitations posesa challenge: pure diffusive dynamics cannot describe relaxation of elasticstresses that happen through phonon emission. To this end, several differentmodels with fast dynamics have been proposed. In this article we use theamplitude expansion of the PFC model to compare the recently proposedhydrodynamic PFC amplitude model with two simpler models with fast dynamics. Wecompare these different models analytically and numerically. The resultssuggest that in order to have proper relaxation of elastic excitations, thefull hydrodynamical description of the PFC amplitudes is required.

Long wavelength properties of phase field crystal models with second order dynamics