We analyze the effect of a gravitational field on the sound modes of
superfluids. We derive an instability condition that generalizes the well known
Jeans instability of the sound mode in normal fluids. We discuss potential
experimental implications.Comment: 9 pages, Late

Hason, Itamar

Oz, Yaron

English

arXiv

Itamar Hason

Yaron Oz

Springer - Publisher Connector

Jeans instability in superfluids

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Eur. Phys. J. C (2014) 74:3183DOI 10.1140/epjc/s10052-014-3183-xRegular Article - Theoretical PhysicsJeans instability in superfluidsItamar Hasona , Yaron OzbRaymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, 69978 Tel Aviv, IsraelReceived: 25 August 2014 / Accepted: 14 November 2014 / Published online: 26 November 2014© The Author(s) 2014. This article is published with open access at Springerlink.comAbstract We analyze the effect of a gravitational field onthe sound modes of superfluids. We derive an instability con-dition that generalizes the well-known Jeans instability of thesound mode in normal fluids. We discuss potential experi-mental implications.1 IntroductionThe Jeans instability of the sound mode in normal fluidsthat is caused by a gravitational field is a well-known phe-nomenon [1]. It exhibits itself in astrophysical scenarios ofaggregation of masses and galaxy formation. The Jeans dis-persion relation of a normal fluid sound mode is obtained bylinearizing the fluid equations in the presence of a gravita-tional field. It modifies the sound mode dispersion relationω2 = u21k2 toω2 = u21k2 − 4πGρ. (1)ω is the frequency, k is the momentum vector, u1 is the speedof sound in a normal fluid u21 = ∂p∂ρ evaluated at fixed entropy,p is the pressure, ρ the local mass density and G is the grav-itational coupling. Instability occurs when the RHS of (1) isnegative,k2 < 4πGρu21. (2)It determines a Jeans length scale λJ , all scales larger thanthat being unstable to a gravitational collapse.In this letter we will analyze the effect of a gravitationalfield on the sound modes in superfluids. Superfluids are quan-tum fluids, i.e. fluids at temperatures close to zero wherequantum effects are of primary importance [2]. They exhibitremarkable properties, such as the ability to flow without vis-cosity in narrow capillaries. Superfluidity/superconductivitya e-mail: itamarhason@gmail.comb e-mail: yaronoz@post.tau.ac.ilis also expected to be realized in neutron stars (see e.g. [3]),as well as in high density phases of QCD [4].The hydrodynamic description of superfluids consists oftwo separate motions, a normal flow and a super flow withdensities ρn and ρs, respectively, ρn+ρs = ρ [5,6]. The superflow moves without viscosity, the normal flow is viscous, andthe two flows do not exchange momentum between them. Wedenote by vn and vs the velocities of the normal and superflows, respectively. The superfluid velocity vs correspondsto the gradient of the condensate phase that breaks sponta-neously the global/local symmetry, which results in super-fluidity/superconductivity, respectively. Thus, the superfluidmotion is irrotational∇ × vs = 0. (3)There are two sound modes in superfluids: the first soundu1 which is a density wave as in normal fluids, and the sec-ond sound u2, which is a temperature wave and is unique tosuperfluids. Its dispersion relation readsω2 = u22k2, u22 =ρss2ρn∂s∂T, (4)where T is the temperature, s is the entropy density per parti-cle, and the derivative is taken at fixed pressure. As expected,the second sound vanishes in the limit ρs → 0. In generalthe waves can be a superposition of the two.As we will show, in the absence of density fluctuations thepure second sound (4) remains stable, while density fluctu-ations imply a stability criterion for both sound modes andtheir superpositions. It readsk2 < 4πGρu21 + 1(1−J )u22, (5)where J =∂s∂p∂ρ∂T∂s∂T∂ρ∂p. This condition generalizes the ordinaryJeans instability (2) and reduces to it in the limit ρs → 0.The RHS of (5) depends on the details of the system’s ther-modynamic properties. An interesting limit to take is that of1233183 Page 2 of 5 Eur. Phys. J. C (2014) 74:3183very low temperatures T → 0, where the super flow com-ponent is dominant. In this limit one has u21 = 3u22 [6] andJ → 0 [7]. Thus, at very low temperature we have a largerJeans length than that of a normal fluid by a factor of 2√3.The letter is organized as follows. We will introduce agravitational potential to the superfluid hydrodynamics equa-tions and study fluctuations at the linear order. We will thenanalyze the instability conditions and derive (5), which is ourmain result. Finally, we will discuss potential experimentalimplications.2 Gravitational instability in superfluid hydrodynamicsWe will consider the evolution equations of a superfluid in thepresence of a gravitational field. We denote the gravitationalpotential by φ. It satisfies Gauss’s law,∇2φ = 4πGρ. (6)2.1 Self-gravitating superfluid hydrodynamicsWe will consider an ideal superfluid. The superfluid densitycurrent readsj = ρnvn + ρsvs, (7)and it satisfies a continuity equation, which is not affected bythe gravitational field∂ρ∂t+ ∇ · j = 0. (8)Similarly, the entropy conservation is not affected by thegravitational field and reads∂∂t(ρs) + (ρs) ∇ · vn = 0. (9)Note that entropy is carried only by the normal flow and notby the super flow.The energy and momentum conservation equations, how-ever, are modified in the presence of the gravitational fieldand take the form∂ ji∂t+ ∂ik∂xk+ ρ ∂φ∂xi= 0, (10)∂ E∂t+ ∇ · Q + j · ∇φ = 0. (11)ik is the momentum flux tensor and Q is the energy flux ascalculated without a gravitational field in [6].In order to derive the modified equations in the pres-ence of the gravitational field, one uses thermodynamics andthe Galilean principle [6]. One denotes by K0 the referenceframe, where the super flow velocity is zero. The velocity ofthe normal flow in this frame is vn − vs. In this frame one hasE0 = −p + Tρs + μρ + φρ + (vn − vs) · j0 (12)andd(μ + φ) = −sdT + 1ρdp − ρnρ(vn − vs)d(vn − vs). (13)E0 and j0 are the energy and density current in the systemK0, respectively. There are two effects of the gravitationalfield. First, to replace the chemical potential μ by μ + φ.Second, to introduce a new term in the fluid flow: in additionto the force term −∇ p, we have −ρ ∇φ.Finally, the super flow being a potential flow satisfies∂ vs∂t+ ∇(12v2s + μ + φ)= 0. (14)It is straightforward to see that the hydrodynamics equationstogether with the thermodynamic relations constitute a com-plete set that determines all the charge densities and veloci-ties.2.2 Self-gravitating superfluid soundLinearizing the above ideal superfluid hydrodynamics equa-tions in a presence of the gravitational field we get∂ρ∂t+ ∇ · j = 0∂∂t(ρs) + (ρs) ∇ · vn = 0∂ j∂t+ ∇ p + ρ ∇φ = 0∂ vs∂t+ ∇μ + ∇φ = 0,(15)from which we derive∂2ρ∂t2= ∇2 p + ρ∇2φ,∂2s∂t2= ρsρns2∇2T + ρsρns∇2φ.(16)Perturbing1 around (16), using Gauss’s law for the gravita-tional potential, and expressing all quantities in terms of thepressure p and the temperature T we get∂ρ∂p∂2δp∂t2+ ∂ρ∂T∂2δT∂t2= ∇2δp + 4πGρ(∂ρ∂pδp + ∂ρ∂TδT)∂s∂p∂2δp∂t2+ ∂s∂T∂2δT∂t2= ρsρns2∇2δT + 4πG ρsρns(∂ρ∂pδp + ∂ρ∂TδT). (17)1 As in the ordinary Jeans instability analysis, we perturb around astable state and therefore ignore the zeroth order contribution of theLaplacian of the gravitational potential ∇2φ0.123Eur. Phys. J. C (2014) 74:3183 Page 3 of 5 3183This system of equations for a wave of the form exp(i k ·x − iωt) reads(∂ρ∂p(ω2 + 4πGρ) − k2)δp + ∂ρ∂T(ω2 + 4πGρ)δT = 0(∂s∂pω2 + ∂ρ∂p· 4πG ρsρns)δp+(∂s∂Tω2 − ρsρns2k2 + ∂ρ∂T· 4πG ρsρns)δT = 0. (18)In the absence of a gravitational field (G → 0) the equationsreduce to the well-known superfluid sound modes [6].In the general case, for a solution of these system of equa-tions we require the determinant to vanish,Hω4 +(4πG Hρ −(∂ρ∂pρsρns2 + ∂s∂T)k2)ω2+(ρsρns2k2 − 4πG ρsρns(∂ρ∂pρs + ∂ρ∂T))k2 = 0, (19)where H = ∂(ρ,s)∂(p,T ) is the Jacobian for changing variablesfrom (ρ, s) to (p, T ), and we assume that H > 0. Solvingfor ω2 we get2Hω2 = −(4πG Hρ −(∂ρ∂pρsρns2 + ∂s∂T)k2)±√(4πG Hρ +(∂ρ∂pρsρns2 − ∂s∂T)k2)2+ 4 ∂ρ∂Tρsρnsk2(4πG H + ∂s∂psk2). (20)In the absence of a gravitational field, we have the plus andminus sign solutions reducing to first and second sound,respectively.Consider the leading effect of the super flow in the dimen-sionless parameter ρsρn 1. At first order we get2Hω2 = −(4πG Hρ − ∂s∂Tk2)±(4πG Hρ − ∂s∂Tk2)+ ρsρn⎛⎝ ∂ρ∂ps2k2 ± ∂ρ∂ps2k2 ±2 ∂ρ∂T sk2(4πG H + ∂s∂p sk2)(4πG Hρ − ∂s∂Tk2)⎞⎠ .(21)Therefore, we have the first sound solutionHω2 = −(4πG Hρ − ∂s∂Tk2)− ρsρn⎛⎝∂ρ∂T sk2(4πG H + ∂s∂p sk2)(4πG Hρ − ∂s∂Tk2)⎞⎠ (22)and the second sound solutionHω2 = ρsρn⎛⎝∂ρ∂ps2k2 +∂ρ∂T sk2(4πG H + ∂s∂p sk2)(4πG Hρ − ∂s∂Tk2)⎞⎠ .(23)2.3 Comparison to the non-gravitational caseIt is instructive to compare special solutions to the superfluidsound equations in the presence of the gravitational interac-tion (18) to the non-gravitating case.2.3.1 The pure second sound limitIn the limit ∂ρ∂T = 0, the non-gravitational sound equations(setting G = 0 in (18)) read(∂ρ∂pω2 − k2)δp = 0(∂s∂pω2)δp +(∂s∂Tω2 − ρsρns2k2)δT = 0.(24)One has a solution where δp = 0, δT arbitrary, and thedispersion relation (4) of the pure second sound describingthe temperature wave. There is another solution, where thedispersion relation is that of a first sound mode, and the pres-sure and temperature variations are related by(∂s∂p)δp =(ρsρns2∂ρ∂p− ∂s∂T)δT . (25)When we introduce gravity, still having ∂ρ∂T = 0, we get(∂ρ∂p(ω2 + 4πGρ) − k2)δp = 0(∂s∂pω2 + ∂ρ∂p· 4πG ρsρns)δp +(∂s∂Tω2 − ρsρns2k2)δT = 0.(26)We see that the pure second sound solution (4) is not affectedby gravity. This is expected since the pure second sound ischaracterized by no variation of the density, thus gravity doesnot affect it.However, the first sound solution is changed. The disper-sion relation is the same as the Jeans dispersion relation (1),but the pressure and temperature variations are related by a1233183 Page 4 of 5 Eur. Phys. J. C (2014) 74:3183different formula,(∂s∂pω2 + ∂ρ∂p· 4πG ρsρns)δp=((∂ρ∂pρsρns2 − ∂s∂T)ω2 + ∂ρ∂p· 4πG ρsρnρs2)δT . (27)Comparing (25) and (27), we see a frequency dependenceof the linear relation between pressure and temperature (27),which vanishes as ρs → 0.2.3.2 The pure first sound limitConsider next the classical pure first sound limit, ∂s∂p = 0. Thenon-gravitational sound equations in this case are (settingG = 0 and ∂s∂p = 0 in (18)):(∂ρ∂pω2 − k2)δp +(∂ρ∂Tω2)δT = 0(∂s∂Tω2 − ρsρns2k2)δT = 0.(28)We have a solution where δT = 0, δp arbitrary, and thedispersion relation of the pure first sound. There is anothersolution, where the dispersion relation is (4), and the pressureand temperature variations are related by(∂s∂Tρnρs1s2− ∂ρ∂p)δp =(∂ρ∂T)δT . (29)When we introduce gravity, still with ∂s∂p = 0, we get(∂ρ∂p(ω2 + 4πGρ) − k2)δp + ∂ρ∂T(ω2 + 4πGρ)δT = 0(∂ρ∂p· 4πG ρsρns)δp +(∂s∂Tω2 − ρsρns2k2+ ∂ρ∂T· 4πG ρsρns)δT = 0. (30)We see that the pure first sound solution is not pure anymore.This is in contrast to the pure second sound solution, whichwas not modified by gravity.2.4 General instability conditionsIn the following we will analyze the instability conditionsarising from the roots (20).2.4.1 Identical rootsConsider the case where the two solutions (20) are identical.In this case we have the same dispersion relation for the firstand second sound and we obtain the instability conditionk2 < 4πG Hρ(∂ρ∂pρsρns2 + ∂s∂T) . (31)Dividing by H , we can recast (31) as (5). The constraint (5)is a generalization of (2) to superflows, and it is valid for boththe first and the second sounds. It reduces to (2) in the limitρs → 0. In limit T → 0, u21 = 3u22 [6], and one can see fromthe data presented in [7] that J → 0. Thus we see that theJeans length of the superfluid is larger by a factor of 2√3thanthat of a normal fluid.2.4.2 Different rootsThe conditions for instability of the two sound modes are theupper bound on the momentum (5) as well as a lower boundk2 > 4πGρ(∂ρ∂p+ ∂ρ∂T1ρs), (32)which together imply a constraint on the system independentof the gravitational field,(∂ρ∂p+ ∂ρ∂T1ρs)(∂ρ∂pρsρns2 + ∂s∂T)< H. (33)2.5 DiscussionAs the ordinary Jeans instability has been observed exper-imentally, it is natural to inquire how can we observe thegeneralized instability conditions for superfluids. It is prob-ably unlikely that current condensed matter experiments candetect the gravitational instability of superfluids. A potentialexperimental laboratory could be neutron stars. Superfluidityis expected to occur both in the inner crust and the core ofneutron stars as a consequence of the formation of Cooperpairs of neutrons that break spontaneously the U (1) Baryonsymmetry [3]. Superfluidity has been invoked in order toexplain, for instance, pulsar glitches [8]. Therefore the grav-itational instability seems relevant to the dynamics of neu-tron stars, such as its rotational and vibrational oscillations.The fluid velocities in neutron stars can be small comparedto the speed of light. For instance, the velocity at the equa-tor of the most rapidly spinning neutron stars is about 20 %of the speed of light, making the non-relativistic two fluidmodel of superfluid hydrodynamics a useful description (seee.g. [9]). It would, therefore, be interesting to work out thedetails of the gravitational instability effects in this frame-work. It would also be of interest to generalize our work torelativistic superfluid flows.Acknowledgments This work is supported in part by the I-CORE pro-gram of Planning and Budgeting Committee (grant number 1937/12),and by the US–Israel Binational Science Foundation.Open Access This article is distributed under the terms of the CreativeCommons Attribution License which permits any use, distribution, andreproduction in any medium, provided the original author(s) and thesource are credited.Funded by SCOAP3 / License Version CC BY 4.0.123Eur. Phys. J. C (2014) 74:3183 Page 5 of 5 3183References1. J.H. Jeans, The stability of a spherical nebula. Philos. Trans. R. Soc.A 199, 1 (1902)2. P. Kapitza, Viscosity of liquid helium below the λ-point. Nature 141,74 (1938)3. U. Lombardo, H.J. Schulze, Lect. Notes Phys. 578, 30 (2001).astro-ph/00122094. M.G. Alford, K. Rajagopal, F. Wilczek, QCD at finite baryon den-sity: nucleon droplets and color superconductivity. Phys. Lett. B422, 247 (1998). hep-ph/97113955. L.D. Landau, The theory of superfluidity of helium II. J. Phys. USSR5, 71 (1941)6. E.M. Lifshitz, L.D. Landau, Course of Theoretical Physics. FluidMechanics, vol. 6 (Pergamon Press, London, 1959)7. J. Brooks, R. Donnelly, The calculated thermodynamic propertiesof superfluid helium-4. J. Phys. Chem. Data 6, 1 (1977)8. P.W. Anderson, N. Itoh, Pulsar glitches and restlessness as a hardsuperfluidity phenomenon. Nature 256, 25 (1975)9. N. Chamel, Two-fluid models of superfluid neutron star cores. Mon.Not. R. Astron. Soc. 388, 737 (2008)123

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We analyze the effect of a gravitational field on the sound modes of superfluids. We derive an instability condition that generalizes the well-known Jeans instability of the sound mode in normal fluids. We discuss potential experimental implications

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Jeans Instability in Superfluids

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