Two possible types of superfluidity in crystals

Using a modified procedure to derive hydrodynamics, it is shown from symmetry considerations that a superfluid velocity ${\overrightarrow{\mathrm{v}}}^s$ which is invariant under a Galilean transformation will be driven by the temperature: ${\stackrel{\stackrel{•}{\to }}{v}}^s+\left(\frac{k_B}{m}\right)\overrightarrow{\nabla }T=0\mathrm{}$. A crystal with such a velocity is able to sustain a persistent entropy flux rather than a mass current and has a propagating mode connected to temperature fluctuations. On the other hand, only a crystal with a ${\overrightarrow{\mathrm{v}}}^s$ which behaves like a true velocity under a Galilean transformation, will be able to sustain a persistent mass current and have vacancy propagation as its Goldstone mode. This spectrum differs, however, from that given by previous authors.