Table Of ContentTrapped phonons
3
1
0 Massimo Mannarelli∗
2 LNGS-INFN
n E-mail: [email protected]
a
J
5 We analyze the effect of restricted geometries on the contribution of Nambu-Goldstone bosons
2
(phonons) to the shear viscosity, η, of a superfluid. For illustrative purpose we examine a sim-
] plifiedsystemconsistingofacircularboundaryofradiusR,confiningatwo-dimensionalrarefied
h
p gas of phonons. Considering the Maxwell-type conditions, we show that phonons that are not
- inequilibriumwiththeboundaryandthatarenotspecularlyreflectedexertashearstressonthe
p
e boundary. In this case it is possible to define an effective (ballistic) shear viscosity coefficient
h
[ η ∝ρphχR, where ρph is the density of phonons and χ is a parameter which characterizes the
typeofscatteringattheboundary. Foranopticallytrappedsuperfluidourresultscorroboratethe
1
v findingsofRefs. [1,2], whichimplythatatverylowtemperaturetheshearviscositycorrelates
4
withthesizeoftheopticaltrapanddecreaseswithdecreasingtemperature.
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XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Trappedphonons MassimoMannarelli
1. Introduction
The transport coefficients of a fluid depend on the underlying microscopic dynamics and on
theboundaryconditionsimposed. Theeffectoftheboundaryconditionsonthetransportproperties
is seldomconsidered, butmight neverthelessbe importantin particularcircumstances. Actuallya
completeknowledgeofthemicroscopicdynamicsofasystemshouldincludeadetaileddescription
oftheboundaryandoftheinteractionsoftheparticleswiththeboundary.
HereweshallfocusonasystemconsistingofNambu-Goldstonebosons(phonons)withasim-
ple boundary, that is a boundary which does not allow particle transfer. Two different geometries
are pictorially shown in Fig.1, which correspond to a spherically symmetric trap (left panel), rel-
evant for the superfluid region of a compact star, and a cylindrically symmetric trap (right panel),
relevant for optically trapped ultracold atoms. In both cases we are interested to the evaluation
of the drag force between the fluid and the boundary when the boundary rotates, with frequency
Ω, or it radially expands. However, for illustrative purposes we shall consider the simplified two-
dimensional system depicted in Fig.2, corresponding to a circular rotating boundary of radius R,
which can be thought as obtained by cutting the rotating configurations of Fig.1 with a plane or-
thogonaltotheΩ-axis.
TheeffectoftheboundaryonthefluiddynamicsisimportantinthesocalledKnudsenlayer,
which corresponds to a layer close to the boundary of width proportional to the mean free-path
(cid:96) . Any particle moving in this layer is ballistic, meaning that it has a probability of hitting the
ph
boundary equal or larger than the probability of hitting a particle. The fluid in this region cannot
bedescribedbyhydrodynamics;thecorrespondingBoltzmannequationwiththeproperboundary
conditions should instead be used. In cases where (cid:96) (cid:28)R, the Knudsen layer can be neglected,
ph
andimposingusual(no-slip)boundaryconditionstotheNavier-Stokesequationseffectivelytakes
intoaccounttheKnudsenlayer.
Figure1: Twopictorialexamplesoftrappedgeometriesofphysicalinterest. Leftpanel: Spinningcompact
star. Rightpanel: opticaltrap. Inthiscasethetrapcanbeputinrotationand/oritcanberadiallyexpanded.
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Trappedphonons MassimoMannarelli
Figure2: Schematicrepresentationofatwo-dimensionaltrappedfluid. Theboundaryrotateswithvelocity
v =ΩR. ShownistheKnudsenlayer,wheretheBoltzmannequationmustbeused. Theinnermostpartof
w
thefluidcanbedescribedbytheusualNavier-Stokesequations.
However, the finite extension of the trap does play an important role if the Knudsen number,
K =(cid:96) /R,issizable: ForK (cid:46)0.1(say)theNavier-Stokesequationarestillvalid,butthetrans-
n ph n
portcoefficientsmustbeamendedwithtermsproportionaltopowersofK ,whichdependaswell
n
on the trap geometry. For K (cid:29) 1 the system is ballistic and no hydrodynamic description can
n
be used. For 0.1(cid:46)K (cid:46)1 interparticle interactions and interactions with the boundary should be
n
consideredonanequalfooting.
2. Boundarydescription
In the ballistic regime, K (cid:29)1, the distribution function of the particles can only change by
n
interactions with the boundary. In general, the presence of a boundary changes the distribution
function of the particles because the particle which have collided with the boundary will keep
memory of it as far as they are in the Knudsen layer [3, 4, 5]. A simple boundary is equivalent
to a wall that specular reflects or diffuses the impinging particles, and in this case it is possible to
use the Maxwell-type boundary conditions, meaning that the distribution function of the particles
atthewallisgivenby
(cid:40)
χf +(1−χ)f(xxx,ξξξ(1−2(ξξξ−vvv )·nnn),t) for(ξξξ−vvv )·nnn>0
f¯= w w w , (2.1)
f(xxx,ξξξ,t) for(ξξξ−vvv )·nnn<0
w
where f is the distribution function of the particles diffused by the wall, vvv (cid:28) 1 is the wall
w w
velocity,nnnistheunitnormalvectortothewall,pointedtothefluid,andξξξ =ppp/p isthevelocityof
0
theparticle. Weshallfurthersimplifytheanalysisassumingthat f isthermal. Theaccommodation
w
coefficient, χ, describes the type of scattering at the boundary (schematically described in Fig.3);
theimpingingparticleisspecularlyreflectedwithprobability1−χ andisdiffusedwithprobability
χ. The distribution function f describes the particles that hit the wall at the time t and that are
specularlyreflected. Sinceweareconcernedwithaballisticfluid,itcorrespondstothedistribution
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Trappedphonons MassimoMannarelli
Figure3:SchematicrepresentationoftheMaxwelltheoryofasimpleboundary.Theredarrowcorresponds
tothemomentumoftheimpingingparticle; thebluearrowcorrespondstothemomentumofthescattered
particle;theblackarrowcorrespondstothemomentumtransferredtotheboundary.Specular-reflectiondoes
notproduceashearstressontheboundarybutapressureorthogonaltothesurface. Diffusiondoesproduce
ashearstress,becausethetransferredmomentumtotheboundaryisnotorthogonaltothecontactsurface.
function of the wall hit in a previous collision at the time t , that is f(xxx,ξξξ,t) = f (xxx−ξξξ(t−
0 w
t ),ξξξ,t ), where t <t. Clearly, if χ =0 phonons do not thermalize with the wall and f(xxx,ξξξ,t)
0 0 0
correspondstotheinitialdistributionfunction.
Giventheparticledistributionfunctionitispossibletodeterminetheshearstressonthewall
fromtheoff-diagonalcomponentsofthestresstensor
(cid:90) dppp
τC = (p −pv )(p −pv )f¯(xxx,ξξξ,t) , (2.2)
ij i w,i j w,j (2π)2E
where i,j are the cartesian coordinates. For the circular symmetry considered here, it is more
appropriatetheusepolarcoordinates(r,φ)andthecorrespondingstresstensorisobtainedbyτP=
U(φ)τCU(φ)T,whereU(φ)istherotationmatrix. UponpluggingEq.(2.1)inEq.(2.2),onereadily
obtains that the off-diagonal components vanish for χ =0, consistent with the fact that specular-
reflected particles do not produce shear stress on the boundary. The shear stress also vanishes if
f (xxx−ξξξ(t−t ),ξξξ,t )= f (xxx,ξξξ,t);thereasonisthatinthisconditionthesystemisstationaryand
w 0 0 w
thewallcannotbesubjecttoneitherforcenormomentofforce[4].
Therefore,fortheclosedgeometryconsidered,inordertohaveashearstressonthewallthere
mustexistanexternalagentthatperturbstheboundary. Inthepresentcaseweshallassumethatthe
boundaryisputinrotationwithafrequencyΩ. Theperturbationstartsatacertaintimet suchthat
i
t <t <t and the external agency maintains the system in rotation at the same frequency Ω. The
0 i
impinging particles have memory of the distribution function before the circle was put in rotation
and will try to catch up with the wall. This produces a shear stress (force per unit length) on the
wallgivenby
τ (t)=−Cρ ΩRχ, (2.3)
rφ ph
where C is some dimensionless number and ρ is the two-dimensional phonon density of the
ph
normalfluidcomponent. Afteracertaintime,dependingonχ,Randc ,thesystemwillequilibrate
s
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Trappedphonons MassimoMannarelli
and the phonons will “corotate" with the boundary not exerting any shear stress on it. The above
expressionofthestresstensorallowsustodefinetheballistic(effective)shearviscositycoefficient
forthenon-equilibratedsystemgivenby
η2D =Cρ χR, (2.4)
ball ph
which should be rather general, meaning that it should be the valid also for the two geometries
in Fig.1, but with a different expression of the constant C. We shall study this issue elsewhere.
This expression has been used in the study of the oscillations of compact stars in [6, 7], for the
descriptionoftheoscillationsofobjectsimmersedin4He,seee.g. [8],andforthebreathingmode
ofultracoldfermionicatoms(discussedbelow)in[1,2].
3. Applicationtoultracoldfermionicatoms
Theattractiveinteractionbetweencertaintypesoffermionicatoms(like6Li)preparedintwo
hyperfine states can be varied at will using a magnetic-field Feshbach resonance. These atoms
can be confined in a cylindrically symmetric optical trap (schematically shown in the right panel
of Fig.1) and the system can be described as a core comprising phonons and gapped fermions
and a corona of unpaired fermions, for a review see [9]. The extension of the corona decreases
with decreasing temperature and at T (cid:39)0.1T , where T is the Fermi temperature of the trapped
F F
system, thelargestpartofthesystemissuperfluid. Thecoreofthesystemceasestobesuperfluid
forT >Ttrap(cid:39)0.21T [10].
c F
Withdecreasingtemperaturetheentropyperparticlesteeplydecreasesandisnotincompatible
withthehypothesisthatitreceivesasizablecontributionsbyphonons. Presentexperimentalresults
donotallowtoinferwhetheratthereachedtemperaturethesystemisdominatedbyphonons[11],
howeverthereislittledoubtthatatsufficientlylowtemperaturephononswilldominate.
Thephonondispersionlaw(neglectingO(k/k )7 terms)canbewrittenas
F
E =c k(1+γ˜(k/k )2+δ˜(k/k )4), (3.1)
k s F F
where γ˜ (cid:39)0.18 (see [12, 13, 14] and the discussion in [2]), and a numerical estimate of δ˜ was
obtainedin[1,2]fromafitoftheexperimentalvaluesoftheshearviscositytoentropyratio. Indeed
thecontributionofbinary(hereafter4ph)and1↔2(hereafter3ph)processestotheshearviscosity
dependsonthephonondispersionlawandinRefs. [1,2]theη contributionwasevaluatedwith
3ph
thedispersionlawinEq. (3.1). WhencomparedwiththeexperimentaldataobtainedbytheDuke
group[15,16],oneobtainsδ˜ =−(0.02÷0.04).
The contribution of the ballistic shear viscosity must be included in the description of the
system because, as realized in [1, 2] and shown in Fig.4, the mean free-path of phonons diverges
at sufficiently low temperature. In this case the typical length scale appearing in Eq.(2.4) is the
smallestradialextensionoftheharmonictrapof[15,16],thatisR (cid:39)7µm.
x
Inprinciple,foranyvalueoftheKnudsennumbertheshearviscositycanbeobtainedplugging
in Eq.(2.2) the distribution function solution of the Boltzmann equation where both collisions of
phononswiththeboundaryandthecollisionsamongphononsaretakenintoaccount. Solvingthis
problemiscomplicatedevenforidealgases[3,4,5]. However,sinceweknowthebehaviorofthe
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Trappedphonons MassimoMannarelli
5
4-ph
3-ph-a
4
3-ph-b
3-ph-c
x3
R
/h
lp2
1
0 0,1 0,2 0,3
T/T
F
Figure4: Meanfree-pathofphononsinunitofR asafunctionofT/T , foropticallytrappedsuperfluid
x F
fermionicatoms.Theblackdashedlinecorrespondstothemeanfreepathassociatedtothe4phprocess.The
solidredline(named3ph-a)correspondstothe3phprocesswithδ˜ (cid:39)0.02,thedot-dashedblueline(named
3ph-b)correspondstothe3phprocesswithδ˜ (cid:39)0.03andthedottedredline(named3ph-c)correspondsto
the3phprocesswithδ˜ (cid:39)0.04. ThehorizontalblackdashedlinecorrespondstoK =1. Thehydrodynamic
n
descriptionisvalidforK (cid:28)1. Theverticaldashedgreenlineapproximatelycorrespondstothetransition
n
temperaturebetweenthenormalphaseandthesuperfluidphase[10]. Adaptedfrom[2].
relaxationtimeτ (correspondingtothetimebetweentwocollisionswiththeboundary),aswellas
b
therelaxationtimeτ (correspondingtotherelaxationtimeof3phor4phprocesses),itispossible
ph
to employ the same reasoning at the basis of the Matthiessen’s rule [17] to obtain an approximate
expression of η at any temperature. Assuming that the two above-mentioned collisions are not
correlated,onecaninterpolatebetweenthevaluesoftheshearviscositycoefficientintheballistic
andinthehydrodynamicregimes. Forthispurposeonecandefineaneffectiverelaxationtime
τ−1=τ−1+τ−1, (3.2)
eff b ph
incorporating the effects of interparticle collisions and of the collisions with the boundary. Since
the shear viscosity coefficient is proportional to the collision time, we define the total effective
shearviscosityas
(cid:16) (cid:17)−1
η = η−1+η−1 , (3.3)
eff 3ph ball
whereη isthethree-dimensionalanalogousofEq.(2.4)andη isthe3phshearviscosity[1,2].
ball 3ph
In principle, the contribution of the 4ph collisions should be considered as well, but so far it has
notbeendeterminedwiththefullexpressionofthedispersionlawgiveninEq.(3.1)andtherefore
itcannotbeconsistentlyaddedtoEq.(3.3).
InFig.5wereportseveralcontributionstotheshearviscositycoefficienttoentropyratio. The
interaction of the phonons with the boundary seems to play an important role for temperatures
T (cid:46)0.1T . Unfortunately few experimental data are available for such low temperature and it is
F
hardtosaywhethertheeffectiveshearviscosityisabletodescribethebehaviorofthesystem.
In the reported analysis a number of simplifying assumptions where used: 1) The shear vis-
cosity is dominated by the center of the trap; the density in this region is bigger and roughly con-
stant,whichassuresthatperformingatrapaveragemightnotmodifydrasticallytheresults. 2)The
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Trappedphonons MassimoMannarelli
1,2
h ballistic a=0.3 Rx 3ph
0,8 p4
s
η/
0,4
1
4π
0 0,1 0,2 0,3
T/T
F
Figure 5: Various contributions to η/s (in units of h¯/k ) as a function of T/T . The contribution of the
B F
interaction of the phonons with the boundary corresponds to the dashed black line. The contribution of
the 3ph process corresponding to the dotted red line is obtained considering δ˜ (cid:39) 0.03. The solid blue
line corresponds to η . The vertical dashed green line approximately corresponds to the phase transition
eff
temperature[10]. Thehorizontalblackdashedlinecorrespondstotheuniversalboundderivedin[18]. The
experimentalvaluesanderrorbarsweretakenfrom[19]. Adaptedfrom[2].
fermioniccontributiontoη wasneglected,bothinthesuperfluidcoreandintheouternormallayer;
this should be a valid approximation at sufficiently low T, where superfluid fermions are known
tobeexponentiallysuppressed,whiletheouterfermioniclayerofthecloudmightbetoodiluteto
provideenoughdamping. 3)Thesuperfluid-normalfluidinterfacewasmodeledasasimplebound-
ary, wherephononsarespecular-reflectedordiffusedaccordingwiththeMaxwell-typecondition;
this approximation should be fine because the Maxwell model is appropriate for the description
of an interface with no particle transfer and superfluid phonons are confined in the superfluid trap
core. However,themicroscopicdescriptionofthephononinteractionwithunpairedfermionsinan
optical trap has never been discussed and would certainly help to improve the (admittedly rough)
Maxwellmodel.
What is interesting here, is that a couple of testable predictions can be made from this study.
First, we notice that if experiments are conducted reducing/increasing the size of the trap, but
keeping T constant, then η/s at low temperatures should decrease/increase. In other words, the
F
value of η/s should correlate with the size of the gas cloud. Second, we predict that η/s should
decrease with decreasing temperature. If we naively extrapolate our results to lower temperature,
we predict that there should be a violation of the string theory proposed bound of η/s. Note,
however, that this violation happens because phonons are ballistic, while the string theory bound
concernsthehydrodynamicregime.
Both the above-mentioned predictions are independent of the detailed form of the phonon
dispersion law, in particular, they are independent of the sign of the γ term in Eq (3.1) (which
apparentlyisstillamatterofdebate).
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Trappedphonons MassimoMannarelli
4. Conclusions
Thetransportpropertiesofafluidinrestrictedgeometriesdependonthetypeofscatteringthat
takesplaceattheboundaryandonthegeometricalpropertiesoftheboundary. Wehaveevaluated
the shear viscosity coefficient for a two-dimensional ballistic phonon gas restricted by a circular
boundary. Ifthe boundaryis putin rotationand ifthe phonons arediffused bythe boundary, then
a shear stress is exerted by the phonons on the boundary in a transient period. When phonons
equilibrate with the boundary no shear stress is exerted on the boundary. These results have been
applied to the case of phonons in an optical trap, where phonons live in the superfluid core and
interactwiththenormalfluidofthecorona. Treatingtheinterfaceasawallonehasthatiftheshear
viscosity is dominated by phonons, then η/s should be proportional to the size of the system and
tothetemperature.
Acknowledgments
PartoftheworkpresentedherewasstudiedwithC.ManuelandL.Tolosin[1,2]. IthankM.Alford
for discussion and J.E. Thomas and C. Cao for providing their experimental data of the shear to
entropyratio.
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