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16.1 Superfluids

Neutron star physics provides ample motivation for the need to develop a relativistic description of superfluid systems. As the typical core temperatures (below 8 10 K) are far below the Fermi temperature of the various constituents (of the order of 1012 K for baryons) neutron stars are extremely cold on the nuclear temperature scale. This means that, just like ordinary matter at near absolute zero temperature, the matter in the star will most likely freeze to a solid or become superfluid. While the outer parts of the star, the so-called crust, form an elastic lattice, the inner parts of the star are expected to be superfluid. In practice, this means that we must be able to model mixtures of superfluid neutrons and superconducting protons. It is also likely that we need to understand superfluid hyperons and color superconducting quarks. There are many hard physics questions that need to be considered if we are to make progress in this area. In particular, we need to make contact with microphysics calculations that determine the various parameters of such multi-fluid systems. However, we will ignore this aspect and focus on the various fluid models that have been used to describe relativistic superfluids.

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 4 He. We then have two fluids, the superfluid Helium atoms with particle number density nn and the entropy with particle number density ns. From the derivation in Section 6 we know that the equations of motion can be written

∇ μn μx = 0 (341 )
n μ∇ μx = 0. (342 ) x [μ ν]
To make contact with other discussions of the superfluid problem [2526Jump To The Next Citation Point2729], we will use the notation sμ = n μ s and Θ ν = μs ν. Then the equations that govern the motion of the entropy obviously become
μ ρ ∇μs = 0 and s ∇ [ρΘν] = 0. (343 )
Now, since the superfluid constituent is irrotational we will have
n ∇[σμν] = 0. (344 )
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 [62Jump To The Next Citation Point69Jump To The Next Citation Point]. 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 α (say). That is, we have

n V = − μρ-= − ∇ α. (345 ) ρ m ρ
Here m is the mass of the Helium atom and V ρ 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
s μρ = Θ ρ = κρ + ∇ ρφ. (346 )
Here κρ is Lie transported along the flow provided that sρκ ρ = 0 (assuming that the equation of motion (343View Equation) is satisfied). This leads to
ρ ρ s ∇ ρφ = s Θ ρ. (347 )
There is now no loss of generality in introducing further scalar potentials β and γ such that κ ρ = β ∇ ργ, where the potentials are constant along the flow-lines as long as
sρ∇ β = sρ∇ γ = 0. (348 ) ρ ρ
Given this, we have
Θρ = ∇ ρφ + β ∇ ργ. (349 )
Finally, comparing to Khalatnikov’s formulation [6269] we define Θρ = − κw ρ and let φ → κζ and β → κβ. Then we arrive at the final equation of motion
Θ − --ρ= wρ = − ∇ ρζ − β∇ ργ. (350 ) κ
Equations (345View Equation) and (350View Equation), 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.
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