One of the key features of a pure superfluid is that it is irrotational. Bulk rotation is mimicked by the formation of vortices, slim “tornadoes” representing regions where the superfluid degeneracy is broken. In practice, this means that one would often, e.g. when modeling global neutron star oscillations, consider a macroscopic model where one “averages” over a large number of vortices. The resulting model would closely resemble the standard fluid model. Of course, it is important to remember that the vortices are present on the microscopic scale, and that they may affect the values of various parameters in the problem. There are also unique effects that are due to the vortices, e.g. the mutual friction that is thought to be the key agent that counteracts relative rotation between the neutrons and protons in a superfluid neutron star core [79].

For the present discussion, let us focus on the simplest model problem of superfluid . We then have two fluids, the superfluid Helium atoms with particle number density and the entropy with particle number density . From the derivation in Section 6 we know that the equations of motion can be written

and To make contact with other discussions of the superfluid problem [25, 26, 27, 29], we will use the notation and . Then the equations that govern the motion of the entropy obviously become Now, since the superfluid constituent is irrotational we will have Hence, the second equation of motion is automatically satisfied once we impose that the fluid is irrotational. The particle conservation law is, of course, unaffected. This example shows how easy it is to specify the equations that we derived earlier to the case when one (or several) components are irrotational. It is worth emphasizing that it is the momentum that is quantized, not the velocity.It is instructive to contrast this description with the potential formulation due to Khalatnikov and colleagues [62, 69]. We can obtain this alternative formulation in the following way [26]. First of all, we know that the irrotationality condition implies that the particle momentum can be written as a gradient of a scalar potential (say). That is, we have

Here is the mass of the Helium atom and is what is traditionally (and somewhat confusedly) referred to as the “superfluid velocity”. We see that it is really a rescaled momentum. Next assume that the momentum of the remaining fluid (in this case, the entropy) is written Here is Lie transported along the flow provided that (assuming that the equation of motion (343) is satisfied). This leads to There is now no loss of generality in introducing further scalar potentials and such that , where the potentials are constant along the flow-lines as long as Given this, we have Finally, comparing to Khalatnikov’s formulation [62, 69] we define and let and . Then we arrive at the final equation of motion Equations (345) and (350), together with the standard particle conservation laws, are the key equations in the potential formulation. As we have seen, the content of this description is identical to that of the canonical variational picture that we have focussed on in this review. We have also seen how the various quantities can be related. Of course, one has to exercise some care in using this description. After all, referring to the rescaled momentum as the “superfluid velocity” is clearly misleading when the entrainment effect is in action.http://www.livingreviews.org/lrr-2007-1 |
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