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Entropy entrainment and dissipation in superfluid Helium

Building on a general variational framework for multi-fluid dynamics, we discuss finite temperature effects in superfluids. The main aim is to provide insight into the modelling of more complex finite temperature superfluid systems, like the mixed neutron superfluid/proton superconductor that is expected in the outer core of a neutron star. Our final results can also (to a certain extent) be used to describe colour-flavour locked quark superconductors that may be present at the extreme densities in the deep neutron star core. As a demonstration of the validity of the model, which is based on treating the excitations in the system as a massless ``entropy'' fluid, we show that it is formally equivalent to the traditional two-fluid approach for superfluid Helium. In particular, we highlight the fact that the entropy entrainment encodes the ``normal fluid density'' of the traditional approach. We also show how the superfluid constraint of irrotationality reduces the number of dissipation coefficients in the system. This analysis provides insight into the more general problem when vortices are present in the superfluid, and we discuss how the so-called mutual friction force can be accounted for in our framework. The end product is a hydrodynamic formalism for finite temperature effects in a single superfluid condensate. This framework can readily be extended to more complex situations.


Introduction
Low temperature physics continues to be a vibrant area of research, providing a number of interesting and exciting challenges. Many of these are associated with the properties of superfluids/superconductors, either created in the laboratory or in the cores of mature neutron stars. Basically, matter appears to have two options when the temperature decreases towards absolute zero. According to classical physics one would expect the atoms in a liquid to slow down and come to rest, forming a crystalline structure. It is, however, possible that quantum effects become relevant before the liquid solidifies leading to the formation of a superfluid condensate (a quantum liquid). This will only happen if the interaction between the atoms is attractive and relatively weak. The archetypal superfluid system is Helium. It is well established that He 4 exhibits superfluidity below T = 2.17 K. Above this temperature liquid Helium is accurately described by the Navier-Stokes equations. Below the critical temperature the modelling of superfluid He 4 requires a "twofluid" description (Khalatnikov, 1965;Wilks, 1967;Putterman, 1974). Two fluid strong interaction develops an attractive component (which is necessary for the BCS mechanism to operate) and these neutrons will be superfluid and may flow through the lattice. When the density reaches nuclear saturation, the crust lattice gives way to the fluid core of the star. In the outer parts of this core, the neutron fluid will co-exist with protons, electrons and perhaps muons. At low temperatures, both neutrons and protons are expected to form condensates. Hence one is forced to consider the dynamics of a neutron superfluid coexisting with a proton superconductor and a relativistic gas of electrons/muons. Modelling an entire star, with all these different phases, is a serious challenge. Yet, the nature of the outer core is relatively well understood. The physics of the deep core is much less certain (Haensel et al, 2007). One alternative is that the composition continues to change as the presence of more massive baryons (hyperons) becomes energetically favourable. Another possibility is that the ground state of matter at high density corresponds to a plasma of deconfined quarks and gluons. The different phases of matter provide a number of different channels for Cooper pairing, leading to many potential "superfluid" components. In order to develop a moderately realistic model for a neutron star core we need to improve our understanding of tricky issues concerning hyperon superfluidity and quark colour superconductors . Neutron star observations may provide the only way to constrain our models for this extreme sector of physics.
The rapid spin-up and subsequent relaxation associated with radio pulsar glitches (Lyne et al, 2000) provides strong, albeit indirect, evidence for neutron star superfluidity. The standard model for these events is based on, in the first instance, the pinning of superfluid vortices (e.g. to the crust lattice) which allows a rotational lag to build up between the superfluid and the part of the star that spins down electromagnetically, and secondly the sudden unpinning which transfers angular momentum from one component to the other leading to the observed spin-change. Key to the modelling of these events is the vortex pinning and the mutual friction (see  for a recent discussion) between the two components in the star.
The modelling of superfluid neutron star oscillations has also received considerable attention. It is known that different classes of pulsation modes can be, more or less clearly, associated with different aspects of the neutron star model. As an example, a superfluid star has a set of oscillation modes that arise because of the existence of the second sound (Epstein, 1988;Mendell, 1991a;Lee, 1995;Comer et al, 1999;Andersson & Comer, 2001). The hope is that one will be able to use future observations, e.g. via gravitational waves, to learn more about the interior composition of the star (Andersson & Kokkotas, 1998). Particularly interesting in this respect is the possibility that various oscillation modes may be unstable. The most promising such instability is (according to current thinking) associated with the inertial r-modes, see Andersson & Kokkotas (2001); Andersson (2003) for reviews of the relevant literature. The r-mode instability is expected to be active provided that the gravitation radiation reaction, which drives the instability, is more efficient than the different damping mechanisms that suppress the growth of the mode. On the one hand, this is interesting because it makes the instability window sensitive to the detailed composition of the star. On the other hand, it makes realistic modelling of the instability exceedingly difficult. Having said that, progress has been made on understanding the nature of the r-modes in a superfluid neutron star (Lindblom & Mendell, 2000;Yoshida & Lee, 2003;Prix et al, 2004), in particular the role of the vortex mediated mutual friction damping. We also know that the bulk viscosity associated with hyperons and deconfined quarks can affect the results significantly (Nayyar & Owen, 2006;. In all cases the effects of superfluidity will be considerable. So far, studies of the dynamics of superfluid neutron stars have almost exclusively considered the zero temperature problem. This is an obvious starting point since i) it simplifies the analysis and ii) mature neutron stars tend to be "cold", with core temperature below 10 8 K. However, this logic has an obvious flaw. The critical temperature at which the different phases of matter become superfluid is density dependent (Andersson et al, 2005). For instance, singlet state pairing of neutrons is expected to be present from just beyond the neutron drip to some point in the fluid core. For any given stellar temperature there must therefore exist transition regions where thermal effects play a dominant role. A detailed model ought to account for these regions. This involves understanding the dynamical role of the thermal excitations. The aim of the present paper is to take some steps towards such an understanding. We will demonstrate the close connection between the variational multi-fluid framework (Prix, 2004;) that we have previously used to model the outer neutron star core Glampedakis et al, 2007Glampedakis et al, , 2008Glampedakis & Andersson, 2009), and the classic two-fluid model for He 4 at finite temperatures (Khalatnikov, 1965;Wilks, 1967;Putterman, 1974). This is an important contribution which clearly establishes the viability of the variational multi-fluid approach, and lays the foundation for future applications to problems of astrophysical relevance.
A similar comparison between the corresponding non-dissipative relativistic formulations has already been carried out by Carter & Khalatnikov (1992). They demonstrate how the convective variational multi-fluid formalism developed by Carter -see Carter (1989); Andersson & Comer (2007) for detailed discussions -on which our multi-fluid formalism is based, can be translated into the model developed by Khalatnikov & Lebedev (1982). Our analysis provides additional insight into how thermal excitations should be accounted for, as well as an idea of the dissipation coefficients that are needed to complete a finite temperature model. Even though our aim is not to reformulate the modelling of superfluid Helium, we believe that our discussion should be of some interest also in that context. The most relevant contributions may be the variational derivation of the hydrodynamic equations (and the associated use of truly conserved flux quantities) and the analysis of the superfluid irrotationality constraint. It should also be noted that our formalism is spiritually close to the extended thermodynamics approach (this point will be discussed in detail elsewhere (Andersson & Comer, 2009)). This is an interesting reflection of the universality of conducting multi-fluid models.
Finally, it is worth noting that even though the single particle species model we consider here is not relevant for the conditions in the outer neutron star core it may nevertheless be of use for astrophysics modelling. It could be relevant for a low (but finite) temperature quark core in the colour-flavour-locked phase (where a single condensate co-exists with a phonon gas (Manuel & Llanes-Estrada, 2007)). Because of the potential relevance for gravitational-wave astronomy, studies of the oscillations and instabilities of such a model would be very interesting. This is a highly relevant problem since observations of these phenomena may shed light on the fundamental ground state of matter at extreme densities.

Flux-conservative two-fluid model
We take as our starting point the flux-conservative multi-fluid framework developed by . We consider the simplest system corresponding to a single particle, heat conducting fluid that can undergo a transition to a superfluid state. In the canonical framework, such systems have two degrees of freedom -the atoms are distinguished from the massless "entropy". In the following, the former will be identified by a constituent index n, while the latter is represented by s. This description is different (in spirit) from the standard two-fluid model for Helium, and it is relevant to investigate how the two models are related. In particular, we want to understand better the various dissipative terms that arise when the system is out of equilibrium. That is, we want to be able to compare our dissipative formalism to the results in the standard literature (Khalatnikov, 1965;Putterman, 1974). Our hope is that this will improve our understanding of the role of the thermal excitations. This would be an important step toward more realistic modelling of the various condensates that are expected to be present in a neutron star core.
Our flux-conservative model  combines the usual conservation laws for mass, energy and momentum with the results from a variational analysis (Prix, 2004). The latter is based on using the particle fluxes n x i as the main variables and deducing the associated chemical potentials µ x and the conjugate momenta p i x †. The variational analysis defines the canonical momentum associated with each flux, in the usual way. However, because of the so-called entrainment effect each momentum does not have to be parallel to the associated flux. In the case of a two-component system, with a single species of particle flowing with n n i = nv n i and a massless entropy with flux n s i = sv s i , where n is the particle number density and s represents the entropy per unit volume, the momentum densities are and To complete the model we need to provide an energy functional E = E(n, s, w 2 ns ), which then determines the chemical potential and the entrainment coefficient; These relations highlight one of the main questions considered in this work. We want to understand the role and physical nature of the entrainment between particles and thermal excitations represented by the entropy fluid. This is very different from the entrainment that has so far been considered in neutron star models. In the most commonly studied context, the entrainment between neutrons and protons arises because of the strong nuclear interaction. Each neutron (say) is endowed with a virtual cloud of protons, leading to an effective mass different from the bare neutron mass (Sauls, 1989;Comer & Joynt, 2003). In the dynamical description, this effect is represented by the entrainment. This mechanism is familiar from lowtemperature systems, e.g. He 3 where entrainment couples the two spin populations, and is well explained by Landau Fermi liquid theory (Andreev & Bashkin, 1975;Borumand et al, 1996;Chamel, 2008;Gusakov et al, 2009). In this context it may seem somewhat unorthodox to consider "entrainment" between particles and entropy. However, such a mechanism arises naturally in the variational model, and it is clearly relevant to ask whether it plays an important role. In fact, if we consider the entrainment as altering the effective mass of a constituent, then it would be very natural for this mechanism to affect also the entropy. The entropy entrainment would simply represent the inertia associated with the heat flow. In our view, this interpretation is conceptually quite elegant and we want to understand to what extent it is useful in practice.
As discussed by , the associated momentum equations can be written † where we have used the fact that the temperature follows from µ s = T . In these expressions, D x ij represent the viscous stresses while the "forces" f x i allow for momentum transfer between the two components. In the following we will assume that the system is isolated, which means that f n i + f s i = 0. We want to deduce the general form for the dissipative terms in the equations. To do this we follow the procedure discussed by , i.e. we combine the standard conservation laws with the Onsager symmetry principle. In the present context, when there is no particle creation, mass conservation leads to At the same time entropy can increase, so we have From general principles one can show that the energy loss or gain due to external influences follows from (cf. Eq. (33) of ) (2.8) † Throughout this paper we use a coordinate basis to represent tensorial relations. This means that we distinguish between co-and contra-variant objects, v i and v i , respectively. Indices, which range from 1 to 3, can be raised and lowered with the (flat space) metric g ij , i.e., v i = g ij v j . Derivatives are expressed in terms of the covariant derivative ∇ i which is consistent with the metric in the sense that ∇ i g kl = 0. This formulation of what is, essentially, a fluid dynamics problem may seem somewhat unfamiliar to some readers, but it has great advantage when we want to discuss the geometric nature of the different dissipation coefficients. We will then also use the volume form ǫ ijk which is completely antisymmetric, and has only one independent component (equal to √ g in the present context).

Article submitted to Royal Society
In the case of an isolated system ε ext = 0 so the above relation can be recast as (2.10) The above results are taken, more or less directly, from . At this point we recognize a conceptual mistake in our previous analysis. When identifying the thermodynamical forces and the associated fluxes that are needed to complete the dissipative model from (2.9), we omitted a number of terms related to ∇ j v i s . As a result, the models discussed by  are not as general as they could have been. In fact, if we were to compare our original formulation to the standard dissipative model for superfluid Helium (Khalatnikov, 1965;Putterman, 1974) several bulk viscosity terms would be missing.
Let us rework, and correct, the analysis of  in the particular case of two fluids. From (2.9) we identify the three thermodynamic forces w i ns , ∇ j v i s and ∇ j w i ns . The associated fluxes are −f n i , −D j i and −D nj i . Following the strategy set out by , the fluxes will be formed from linear combinations of the forces in such a way that (the notation here may seem somewhat elaborate, but it is chosen in order to make the inclusion of additional fluids in the framework straightforward) and In these expressions we have made use of the Onsager symmetry principle. Limiting the model to the inclusion of quadratic terms in the forces in (2.9), we find that L nn ij = 2R nn g ij , (2.14) and L ijkl = ζg ij g kl + η g ik g jl + g il g jk − 2 3 g ij g kl + 1 2 σǫ ijm ǫ m kl . (2.19) We can reduce the number of unspecified dissipation coefficients by noting that the conservation of total angular momentum requires D ij to be symmetric, cf. eq (22) of . This means that we must have (2.20) We are then left with a system that has 9 dissipation coefficients; R nn , S nn , ζ n , η n , ζ nn , η nn , σ nn , ζ and η.
To conclude the general analysis, let us write down the final expressions for the dissipative fluxes. To do this we use the decomposition where we have introduced the expansion and the "vorticity" associated with the entropy flow. We will use analogous expressions for gradients of the relative velocity. The definition of the various quantities should be obvious. We finally arrive at

The superfluid constraint
Let us now assume that we are considering a superfluid system. For low temperatures and velocities the fluid described by (2.4) should then be irrotational. To impose this constraint we need to appreciate that it is the momentum that is quantised in a rotating superfluid, not the particle velocity (Prix, 2004). This means that we require ǫ klm ∇ l p n m = 0 . (3.1) To see how this affects the equations of motion, we rewrite (2.4) as That is, using (3.1) we have If we take the curl of this equation we see that the dissipative fluxes must satisfy In other words, we should have for some scalar Φ. This constraint ensures that the superfluid remains irrotational, i.e., there is no generation of turbulence or vorticity.
In order to satisfy this constraint, it is useful to express (2.9) in terms of the variables j i ns = nw i ns and v i s rather than the variables used in the previous section. This means that we have where we have defined and It follows that (3.5) becomes Repeating the analysis from the previous section in terms of the new variables, we see that the thermodynamic fluxes will now be formed from and −D ij =L n ijk j k ns +L n ijkl ∇ k j l ns + L ijkl ∇ k v l s . (3.12) Recall that the conservation of total angular momentum requires D ij to be symmetric.
Let us now consider the constraint (3.9). We need That is we must haveL n ijk = 0 , (3.14) and L nn ij = ∇ kLnn ikj .
(3.15) This leaves us with (3.16) In other words, we must have L nn jikl =ζ nn g ji g kl , (3.17) and L n ijkl =ζ n g ij g kl , (3.18) which means that Φ =ζ nn ∇ l j l ns +ζ n Θ s . Finally, it is straightforward (given the results in the previous section) to show that (3.20) That is, only four dissipation coefficients remain once we impose the superfluid constraint.
We want to compare this result to the standard two-fluid model for Helium, e.g. the results discussed in chapter 9 of Khalatnikov (1965). In order to do this, we need to translate our variables into those that are usually considered. In addition to providing a useful "sanity check" on our analysis, this will give us a direct translation between the various coefficients. This should be useful for future modelling of superfluid neutron stars. After all, the Helium dissipation coefficients have been studied in detail both experimentally and theoretically (mainly through kinetic theory models).

Translation to the orthodox framework
The relationship between our framework and the traditional non-dissipative twofluid model for Helium has already been discussed by Prix (2004). To extend the discussion to the dissipative problem is, as we will now demonstrate, straightforward.

(a) Non-dissipative case
It is natural to begin by identifying the drift velocity of the quasiparticle excitations in the two models. After all, this is the variable that leads to the "two-fluid" dynamics. Moreover, since it distinguishes the flow that is affected by friction it has a natural physical interpretation. In the standard two-fluid model this velocity, v i N , is associated with the "normal fluid" component. In our framework, the excitations are directly associated with the entropy of the system, which flows with v i s . These two quantities should be the same, and hence we identify The second fluid component, the "superfluid", is usually associated with a "velocity" v i S . This quantity is directly linked to the gradient of the phase of the superfluid condensate wave function. This means that it is, in fact, a rescaled momentum. As discussed by Prix (2004) where m is the atomic mass. These identifications lead to Article submitted to Royal Society where ε = 2α/ρ, with ρ the total mass density. We see that the total mass current is If we introduce the superfluid and normal fluid densities, we have the usual result; Obviously, it is the case that ρ = ρ S + ρ N . This completes the translation between the two formalisms. Comparing the two descriptions, it is clear that the variational approach has identified the natural physical variables; the average drift velocity of the excitations and the total momentum flux. Since the system can be "weighed" the total density ρ also has a clear interpretation. Moreover, the variational derivation identifies the truly conserved fluxes, cf. (2.6). In contrast, the standard model uses quantities that only have a statistical meaning (Landau & Lifshitz, 1959). The density ρ N is inferred from the mean drift momentum of the excitations. That is, there is no "group" of excitations that can be identified with this density. Since the superfluid density ρ S is inferred from ρ S = ρ − ρ N , it is a statistical concept as well. Furthermore, the two velocities, v i N and v i S , are not individually associated with a conservation law. From a practical point of view, this is not a problem. The various quantities can be calculated from microscopic theory and the results are known to compare well to experiments. At the end of the day, the two descriptions are (as far as applications are concerned) identical and the preference of one over the other is very much a matter of taste (or convention). Having said that, we believe that it is easier to adapt the variational model to more complex systems, e.g. the mixed superfluids that will be present in a neutron star core [where key general relativistic effects can be naturally incorporated in our framework (Andersson & Comer, 2007)].
The above results show that the entropy entrainment coefficient follows from the "normal fluid" density according to This shows that the entrainment coefficient diverges as the temperature increases towards the superfluid transition and ρ N → ρ. At first sight, this may seem an unpleasant feature of the model. However, it is simply a manifestation of the fact that the two fluids must lock together as one passes through the phase transition. The model remains non-singular as long as v n i approaches v s i sufficiently fast as the critical temperature is approached.
Having related the main variables, let us consider the form of the equations of motion. We start with the inviscid problem. It is common to work with the total momentum. Thus we combine (2.4) and (2.5) to get Here we have π n i + π s i = ρv n i ≡ j i (4.9) which defines the total momentum density. From the continuity equation (2.6) we see that (4.10) The pressure Ψ follows from (4.11) We also need the relation where we have defined (4.14) Putting all the pieces together we have The second equation of motion follows directly from (3.3); where we have defined Prix (2004) (4.17) The above relations show that our inviscid equations of motion are identical to the standard ones, cf. Khalatnikov (1965) and Putterman (1974). The identified relations between the different variables also provide a direct way to translate the quantities in the two descriptions. In particular, we have demonstrated how the "normal fluid density" corresponds to the entropy entrainment in our model. This answers one of our initial questions: We now understand the role of the entropy entrainment that arises in a natural way within the variational framework.

(b) The dissipative case
Let us now move on to the dissipative problem. From (3.20) we immediately see that in the dissipative case we need to augment (4.15) by the divergence of This result should be compared to the dissipative equations in, for example, Khalatnikov (1965). In that description, the dissipation in the total momentum flux follows from the divergence of That is, (4.20) First of all we see that the two shear viscosity coefficients are the same. Secondly, we identify ζ = ζ 2 ,ζ n = mζ 1 . (4.21) Moving on to the second momentum equation we need the gradient of, cf. (3.19), From Khalatnikov (1965) we see that we should compare this to h where Once we identifyζ nn = m 2 ζ 3 ,ζ n = mζ 4 (4.25) we see that the two formulations agree perfectly. Moreover, it is obvious that ζ 1 = ζ 4 as required by the Onsager symmetry. In order to complete the comparison of the two models, we need to comment on the (perhaps surprising) absence of dissipative heat flux terms in our model. At first sight, this would seem to be at odds with the traditional description (Khalatnikov, 1965) which contain Fourier's law for the heat conductivity, q i = κ∇ i T . For consistency, our model requires κ = 0, i.e. the thermal conductivity must vanish. Is this an unattractive feature of our model? In fact, it is not. First of all, it should be noted that the heat flux is intimately related to the entropy flow. In a two-component model one does not have the freedom to introduce an "independent" heat flux in addition to the massless entropy flux n i s , without at the same time introducing a new dynamical degree of freedom. Essentially, the model given by Khalatnikov (1965) is a three component model (it certainly identifies three fluxes). That this makes sense physically is clear from the fact that the thermal conductivity in Helium arises from the interaction between phonons and rotons (Khalatnikov, 1965), which can drift at different rates. Our two-fluid model would be a valid representation of the cold regime where the condensate coexists with a single excitation component (the thermal phonons). It is well-known that, the thermal conductivity κ vanishes in this case. The model is therefore relevant below 0.8 K or so, in the regime where the phonon dispersion relation is very close to linear.
It is also relevant to comment on the well-known problems associated with Fourier's law, i.e. the fact that it leads to a non-causal behaviour of thermal signals. This issue was one of the main motivations for the development of extended irreversible thermodynamics (Jou et al, 1983;Müller & Ruggeri, 1993). A truly sound model for superfluid Helium ought to reflect these developments. Even though such a model is yet to be formulated, it is clear that our approach will allow us to make progress in this direction (by introducing an additional component representing the rotons). This follows naturally from the discussion of Andersson & Comer (2009) where we demonstrate that the relaxation time associated with the entropy flux in heat conductivity problems is intimately related to the entrainment.
We have now achieved the main objective of this work. We have demonstrated that our dissipative two-fluid formulation, with one of the fluids being associated with the massless entropy flow, reproduces the orthodox model for superfluid Helium. This comparison is valuable since it enables us to draw experience for available results for the various dissipation coefficients, e.g. in terms of their effect on sound waves. It also demonstrates that it is straightforward to relate the variational formulation to standard microphysical calculations. Perhaps, the most practical insight is that our analysis has explicitly shown that a full variational treatment of Helium requires three, not two, fluid degrees of freedom. It remains to be seen how this will impact the variational formalism that has been much used to model superfluid neutron star dynamics.

Vortices and mutual friction
The analysis in the previous two sections provides useful insights into the dynamics of a single component superfluid at finite temperatures. From a conceptual point of view, it is obviously important to understand how the superfluid irrotationally constraint simplifies the dynamics of the two-fluid system, i.e. that the number of dissipation coefficients is reduced from nine to four. However, the final model may be of rather limited practical use.
In reality, the superfluid constraint is too severe. A superfluid can rotate by forming an array of vortices. To describe such a system, we must revert to the dissipative fluxes (2.25)-(2.27). However, this more general description still fails to account for all dissipative channels in the problem. In particular, it does not easily accomodate the vortex mediated mutual friction force. In the simplest description Hall & Vinen, 1956) we expect a force f mf i = B ′ ρ n n v ǫ ijk κ j w k ns + Bρ n n v ǫ ijk ǫ klmκj κ l w ns m , (5.1) to act on the particles (with a balancing force affecting the excitations). Here n v is the vortex area density and κ i represents the orientiation of the vortices (the hat represents a unit vector) (Sidery et al, 2008). This force follows after averaging over a locally straight vortex array. In  we discussed how this force could be accounted for in our dissipative model. This analysis was not entirely successful. The main reason for this is that the variational description assumes that the system is isotropic. This is obviously no longer the case when one introduces an array of vortices with a preferred direction. This problem can be resolved in different ways. One can either add an additional "fluid" degree of freedom, representing the averaged vorticity, to the variational discussion (see Geurst (1989); Yamada et al (2007) for interesting discussions and Carter & Langlois (1995) for a relativistic account). Formally, this may be the most natural approach. In particular, since the mutual friction then arises as a linear friction associated with the drift of vortices relative to the excitations.
A more direct alternative would be to augment the analysis of the dissipative fluxes with the preferred directionκ j . This leads to quite a large number of possible extra dissipative terms. To see this, let us briefly return to (2.14)-(2.19). These relations followed the assumption that the dissipative fluxes must be linear in the thermodynamical forces. As a result, a two index coefficient like L nn ij can only be constructed out of the metric g ij . If we have an additional vector in the problem, then a number of additional two-index objects can be written down. We can then have L nn ij = 2R nn g ij + R 1κiκj + R 2 ǫ ijkκ k . (5. 2) The force resulting from this expression can be written In order to compare this to (5.1) we rewrite the latter as and we see that we should identify 2R nn = −R 1 = Bρn v κ , and R 2 = B ′ ρ n n v κ .
( 5.5) This provides a simple and natural generalisation of the dissipative framework discussed in this paper. Of course, it was designed only to account for the standard form of the vortex mutual friction. It does not in any way provide a completely general description of a system with vortices. Such a model would allow a (possibly quite large) number of additional dissipative terms, and would be much more complicated. This would nevertheless be an interesting problem to consider. After all, we have not yet accounted for the vortex tension etcetera (Bekarevich & Khalatnikov, 1961;Mendell, 1991a;Donnelly, 1991).

Discussion
In this paper we have developed a dissipative two-fluid model, based on distinguishing the particle flux from a massless entropy flow. Correcting a conceptual mistake in a previous analysis  we have formulated a general model for an isotropic system, which requires the determination of nine dissipation coefficients. We then demonstrated how imposing the constraint of irrotationality, which is expected for a (pure) superfluid, reduces the complexity of the problem. The final model is in one-to-one correspondence with the classic two-fluid model for He 4 and we have provided a translation between the different variables. This comparison highlights the link between the entropy entrainment in our model and the "normal fluid density" in the standard description (see also Prix (2004)). Finally, we discussed how the presence of vortices in the superfluid affects the model. In particular, we indicated how one may account for the vortex mediated mutual friction force. Our final model should be directly applicable to low temperature, single "particle species" systems, ranging from laboratory systems to astrophysical objects.
There is considerable scope for future developments in this problem area. First of all, it would be relevant to allow for causal dissipative heat flux terms, building on the discussion of Andersson & Comer (2009). Secondly, we want to use the experience gained here to develop finite temperature models for the different superfluid components expected to be present in a neutron star core. The final model discussed in this paper, essentially representing superfluid Helium at low temperatures, may be immediately relevant (in a certain temperature regime) for a compact star with a colour superconducting quark core . Further work is required to formulate a model for the coexisting neutron superfluid and proton superconductor expected in the outer core of a neutron star, as well as the neutron superfluid that penetrates such star's elastic inner crust. Other exotic phases, like hyperon superfluids, may be even more complex. However, by demonstrating the intimate link between the entropy entrainment and the thermal excitations, the present analysis has provided a key ingredient for such models.
NA acknowledges support from STFC via grant number PP/E001025/1. GLC acknowledges partial support from NSF via grant number PHYS-0855558.