  The phase field crystal (PFC) approach extends the notion of phase field
models by describing the topology of the microscopic structure of a crystalline
material. One of the consequences is that local variation of the interatomic
distance creates an elastic excitation. The dynamics of these excitations poses
a challenge: pure diffusive dynamics cannot describe relaxation of elastic
stresses that happen through phonon emission. To this end, several different
models with fast dynamics have been proposed. In this article we use the
amplitude expansion of the PFC model to compare the recently proposed
hydrodynamic PFC amplitude model with two simpler models with fast dynamics. We
compare these different models analytically and numerically. The results
suggest that in order to have proper relaxation of elastic excitations, the
full hydrodynamical description of the PFC amplitudes is required.
