Hydrodynamic damping in trapped Bose gases

23 Pages·0.22 MB·

by  T. Nikuni

Hydrodynamic damping in trapped Bose gases
T. Nikuni
Department of Physics, Tokyo Institute of Technology, Oh-okayama, Tokyo, 152 Japan
and Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7
A. Griffin
8 Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7
9 (February 1, 2008)
9
1

n
a Abstract
J

6

4 Griffin, Wu and Stringari have derived the hydrodynamic equations of a
v
trapped dilute Bose gas above the Bose-Einstein transition temperature. We
6
3 give the extension which includes hydrodynamic damping, following the clas-
0
sic work of Uehling and Uhlenbeck based on the Chapman-Enskogprocedure.
1
1 Our final result is a closed equation for the velocity fluctuations δv which
7 includesthehydrodynamicdampingduetotheshearviscosity η andthether-
9
mal conductivity κ. Following Kavoulakis, Pethick and Smith, we introduce
/
t
a a spatial cutoff in our linearized equations when the density is so low that
m
the hydrodynamic description breaks down. Explicit expressions are given
-
d for η and κ, which are position-dependent through dependence on the local
n fugacity when one includes the effect of quantum degeneracy of the trapped
o
gas. We also discuss a trapped Bose-condensed gas, generalizing the work of
c
: Zaremba, Griffin and Nikuni to include hydrodynamic damping due to the
v
i (non-condensate) normal fluid.
X
r PACS numbers: 03.75.Fi, 05.30.Jp, 67.40.Db
a
Typeset using REVTEX
1
I. INTRODUCTION
Recent observationofBose-Einstein condensationintrappedatomicgaseshasstimulated
interest in the collective oscillations of non-uniform Bose gases [1,2]. The hydrodynamic
equations of a trapped Bose gas (neglecting damping) have been recently derived above
[3] and below [4] the Bose-Einstein transition temperature (T ). These hydrodynamic
BEC
equations describe low frequency phenomena (ω 1/τ , where τ is the mean time between
c c
≪
collisions of atoms), when collisions are sufficiently strong to ensure local thermodynamic
equilibrium. These resultscanbeusedtoderive closedequationsforthevelocity fluctuations
in a trapped gas, whose solution gives the normal modes of oscillation. In particular, above
T , Ref. [3] gives explicit results for the frequencies of surface, monopole, and coupled
BEC
monopole-quadrupole modes. Available experimental data on the frequencies and damping
of collective oscillations in trapped gases [1,2] is mainly for the collisionless regime, rather
than the hydrodynamic regime we consider in this paper. However sufficiently high densities
which allow one to probe the hydrodynamic region at finite temperatures are expected in
the near future (see also Refs. [5,6]). Thus it is worthwhile to extend the discussion of
hydrodynamic modes in Refs. [3,4] to include damping.
Inthepresentpaper,wedothisbyfollowingthestandardChapman-Enskogprocedure,as
first generalized for quantum gases by Uehling and Uhlenbeck [7,8]. In Section II, we derive
the hydrodynamic equations of a trapped Bose gas above T which include damping due
BEC
toshearviscosity η andthermalconductivity κ. Wealsoobtainintegralequationswhichgive
expressions for the transport coefficients η and κ, and in Section III we solve for these. Our
approximation corresponds to a lowest-order polynomial approximation of the Chapman-
Enskog results such as obtained by Uehling [8]. The transport coefficients have a position
dependence when we include the effect of quantum degeneracy of the trapped Bose gas.
In Section IV, we use our results to discuss the hydrodynamic damping of the surface
normalmodes in a trappedBose gas discussed inRef. [3]. Inthe low density tailof a trapped
gas, the hydrodynamic description ceases to be valid. It is crucial to include a cutoff in the
linearized hydrodynamic equations to take this into account, as pointed out by Kavoulakis,
Petthick and Smith [6,9]. We also explicitly show that with this spatial cutoff, the damping
2
of normal modes given by our linearized hydrodynamic equations agrees with the expression
on which Ref. [6] is based.
In Section V, we extend these results to the Bose-condensed region just below T .
BEC
Zaremba, Griffin and Nikuni [4] have recently given an explicit closed derivation of two-fluid
hydrodynamic equations starting from a microscopic model of a trapped weakly-interacting
Bose-condensed gas. We generalize the results of Ref. [4] to include hydrodynamic damping.
Finally in Section VI, we give some concluding remarks.
II. CHAPMAN-ENSKOG METHOD
In Section II-IV, we limit ourselves to a non-condensed Bose gas above T , as in
BEC
Ref. [3]. The atoms are described by the semi-classical kinetic equation for the distribution
function f(r,p,t) of a Bose gas [10]
∂ p ∂f
+ ∇ ∇U(r,t) ∇ f(r,p,t) = . (1)
r p
"∂t m · − · # ∂t(cid:12)
(cid:12)coll
(cid:12)
Here U(r,t) = U (r)+2gn(r,t) includes an external potential U ((cid:12)r) as well as the Hartree-
0 0 (cid:12)
Fock (HF) self-consistent mean field 2gn(r,t), where n(r,t) is the local density. In usual
discussions [7,8], the HF field is omitted but it will play a crucial role when we discuss a
Bose-condensed gas below T (see Section V). The quantum collision integral in the right
BEC
hand side of (1) is given by [10]
∂f dp dp dp p2 p2 p2 p2
= 4πg2 2 3 4 δ(p+p p p )δ + 2 3 4
∂t(cid:12) (2π)3 (2π)3 (2π)3 2 − 3 − 4 2m 2m − 2m − 2m!
(cid:12)coll Z Z Z
(cid:12) [(1+f)(1+f )f f ff (1+f )(1+f )], (2)
(cid:12) × 2 3 4 − 2 3 4
(cid:12)
where f f(r,p,t),f f(r,p ,t). The interaction strength g = 4πh¯2a/m is determined
i i
≡ ≡
by the s-wave scattering length a.
The conservation laws are obtained by multiplying (1) by 1, p and p2 and integrating
over p. In all three cases, the integrals of the collision term in (2) vanish and one finds the
general hydrodynamic equations:
∂n
+∇ (nv) = 0, (3a)
∂t ·
∂ ∂P ∂U
mn +v ∇ v + µν +n = 0, (3b)
µ
∂t · ! ∂xν ∂xµ
∂ε
+∇ (εv)+∇ Q+D P = 0, (3c)
µν µν
∂t · ·
3
where we have defined the usual quantities:
dp
density : n(r,t) = f(r,p,t), (4)
(2π)3
Z
dp p
velocity : n(r,t)v(r,t) = f(r,p,t), (5)
(2π)3m
Z
dp p p
pressure tensor : P (r,t) = m µ v ν v f(r,p,t), (6)
µν (2π)3 m − µ m − ν
Z (cid:18) (cid:19)(cid:18) (cid:19)
dp 1
energy density : ε(r,t) = (p mv)2f(r,p,t), (7)
(2π)32m −
Z
dp 1 p
heat current : Q(r,t) = (p mv)2 v f(r,p,t), (8)
(2π)32m − m −
Z (cid:18) (cid:19)
1 ∂v ∂v
µ ν
rate-of-strain tensor: D = + . (9)
µν
2 ∂xν ∂xµ!
The first approximation for the distribution function is the local equilibrium form
f(0)(r,p,t) = exp[β(p mv)2/2m+U µ] 1 −1, (10)
{ − − − }
where the thermodynamic variables β,v and µ all depend on r and t, and U(r,t) has been
defined after (1). This expression ensures that the collision integral (2) vanishes. This gives
thelowest-order approximation inl/L, where l is themeanfree pathandL isa characteristic
wavelength. If one uses (10) to calculate the quantities from (4) to (8), one finds that Q = 0
and
1
n(r,t) = g (z(r,t)), (11)
Λ3 3/2
k T 2
P (r,t) = δ P(r,t), P(r,t) = B g (z(r,t)) = ε(r,t). (12)
µν µν Λ3 5/2 3
Here z(r,t) eβ(r,t)[µ(r,t)−U(r,t)] is the local fugacity, Λ(r,t) = [2πh¯2/mk T(r,t)]1/2 is the
B
≡
localthermaldeBrogliewavelength andg (z) = ∞ zl/ln arethewell-known Bose-Einstein
n l=1
functions. The equilibrium value of the fugacityPis given by z (r) = eβ0(µ−U(r)), with U(r) =
0
U (r)+2gn (r). Putting Q = 0 and using (12) in Eqs. (3), one obtains
0 0
∂n
+∇ (nv) = 0, (13a)
∂t ·
∂v
mn +(v ∇)v = ∇P n∇U, (13b)
"∂t · # − −
∂ε 5
+ ∇ (εv) = v ∇P. (13c)
∂t 3 · ·
4
The linearization of the equations in (13) around equilibrium leads to the equations (4-6) in
Ref. [3] if we ignore the HF field in U(r,t).
Solutions of these equations describe undamped oscillations, some example of which are
discussed in Ref. [3] (see also Section IV). In order to obtain damping of the oscillations, we
have to consider the deviation of the distribution function from the local equilibrium form
(10). One assumes a solution of the quantum Boltzmann equation (1) of the form [7,11]:
f(r,p,t) = f(0)(r,p,t)+f(0)(r,p,t)[1+f(0)(r,p,t)]ψ(r,p,t), (14)
where ψ expresses a small deviation from local equilibrium. To first order in ψ, we can
reduce the collision integral in (2) to
dp dp p2 p2 p2 p2
4πg2 2 3 dp δ(p+p p p )δ + 2 3 4
(2π)3 (2π)3 4 2 − 3 − 4 2m 2m − 2m − 2m!
Z Z Z
f(0)f(0)(1+f(0))(1+f(0))(ψ +ψ ψ ψ) Lˆ[ψ], (15)
× 2 3 4 3 4 − 2 − ≡
where ψ ψ(r,p ,t). In the left hand side of (1), we approximate f by f(0). The various
i i
≡
derivatives of v(r,t),µ(r,t),T(r,t)and U(r,t) with respect to r and t can be rewritten using
the lowest-order hydrodynamic equations given in (13). The resulting linearized equation
for ψ is (for details, see Appendix)
u ∇T mu2 5g (z) m 1
· 5/2 + D u u δ u2 f(0)(1+f(0)) = Lˆ[ψ], (16)
µν µ ν µν
( T "2kBT − 2g3/2(z)# kBT (cid:18) − 3 (cid:19))
where the thermal velocity u is defined by mu p mv and the strain tensor D is defined
µν
≡ −
in (9). The linearized collision operator Lˆ is defined by (15). This equation can be shown
to have a unique solution for ψ if we impose the constraints
dpf(0)(1+f(0))ψ = dpp f(0)(1+f(0))ψ = dpp2f(0)(1+f(0))ψ = 0, (17)
µ
Z Z Z
which mean physically that n,v and ε [see (4),(5) and (7)] are determined only by the first
term f(0) in (14). Since the left hand of (16) and also f(0) only depend on the relative
thermal velocity u, the function ψ will also depend only on u (i.e., not separately on p and
v).
It is convenient to introduce dimensionless velocity variables
m 1/2
u ξ. (18)
2k T ≡
(cid:18) B (cid:19)
5
With these dimensionless velocity variables, (16) becomes
π3 2k T 1/2 ∇T ξ 5g (z) 1
B · ξ2 5/2 +2D ξ ξ δ ξ2 f(0)(ξ)[1+f(0)(ξ)]
8a2mkB2T2  m ! T " − 2g3/2(z)# µν (cid:18) µ ν − 3 µν (cid:19)
= Lˆ′[ψ], (19)
 
with f(0)(ξ) = (z−1eξ2 1)−1 and
−
Lˆ′[ψ] dξ dξ dξ δ(ξ +ξ ξ ξ )δ ξ2 +ξ2 ξ2 ξ2
≡ 2 3 4 2 − 3 − 4 2 − 3 − 4
Z f(0)fZ(0)(1+Zf(0))(1+f(0))(ψ +ψ (cid:16)ψ ψ). (cid:17) (20)
× 2 3 4 3 4 − 2 −
For a more detailed discussion of the mathematical structure of (19) and (20), we refer to
the treatment of the analogous equations for classical gases (see for example, Ref. [12]). The
most general solution of the integral equation (19) is of the form [7]
π3 2k T 1/2 ∇T ξ 1
ψ = B · A(ξ)+2D ξ ξ δ ξ2 B(ξ) . (21)
8a2mkB2T2  m ! T µν (cid:18) µ ν − 3 µν (cid:19) 
 
The functions A(ξ) and B(ξ) obey the following integral equations:
5g (z)
Lˆ′[ξA(ξ)] = ξ ξ2 5/2 f(0)(1+f(0)), (22a)
" − 2g3/2(z)#
1 1
Lˆ′ ξ ξ δ ξ2 B(ξ) = ξ ξ δ ξ2 f(0)(1+f(0)). (22b)
µ ν µν µ ν µν
− 3 − 3
(cid:20)(cid:18) (cid:19) (cid:21) (cid:18) (cid:19)
For (21) to satisfy the constraints given in (17), we must also have
dξξ2A(ξ)f(0)(1+f(0)) = 0. (23)
Z
Using a solution of the form(21) in conjunction with (14), one can calculate the heat current
density Q in (8) and the pressure tensor P in (6). One finds these have the form
µν
Q = κ∇T, (24a)
−
1
P = δ P 2η D (TrD)δ , (24b)
µν µν µν µν
− − 3
(cid:20) (cid:21)
where the last term in (24b) involves the non-equilibrium stress tensor. The thermal con-
ductivity κ and the shear viscosity coefficient η are given in terms of the functions A(ξ) and
B(ξ):
6
1/2
k 2k T
κ = B B dξξ4A(ξ)f(0)(1+f(0)), (25a)
−48a2 m !
Z
1/2
m 2k T
η = B dξξ4B(ξ)f(0)(1+f(0)). (25b)
−120a2 m !
Z
Introducing (24) into the general hydrodynamic equations (3), one obtains the following
hydrodynamic equations with the effect of shear viscosity and heat conduction included:
∂n
+∇ (nv) = 0, (26a)
∂t ·
∂ ∂P ∂U ∂ 1
mn +v ∇ v + +n = 2η D (TrD)δ , (26b)
µ µν µν
∂t · ! ∂xµ ∂xµ ∂xν (cid:26) (cid:20) − 3 (cid:21)(cid:27)
∂ε 1 2
+∇ (εv)+(∇ v)P = ∇ (κ∇T)+2η D (TrD)δ . (26c)
µν µν
∂t · · · − 3
(cid:20) (cid:21)
We recall that n,v and ε are still given by (4)-(7) with f = f(0), which means that the
expressions in (11) and (12) for n,ε and P are valid. The form of these equations can be
shown to agree with those originally obtained by Uehling and Uhlenbeck [7]. We note that η
and κ are position-dependent, but only through their dependence on the equilibrium value
of the fugacity z = z (r). One slight generalization we have made over the derivation in
0
Ref. [7] is that we have included the Hartree-Fock mean field.
III. THE TRANSPORT COEFFICIENTS
A. The thermal conductivity
To find the thermal conductivity as given by (25a), we can introduce a simple ansatz for
the form of the function A(ξ) [8,12]:
5g (z)
A(ξ) = A ξ2 5/2 . (27)
" − 2g3/2(z)#
The constant A is determined by multiplying (22a) by ξ[ξ2 5g (z)/2g (z)] and inte-
5/2 3/2
−
grating over ξ:
2
5g (z)
A = dξξ2 ξ2 5/2 f(0)(1+f(0))
Z " − 2g3/2(z)#
−1
5g (z) 5g (z)
dξ ξ2 5/2 ξ Lˆ′ ξ2 5/2 ξ
×(Z − 2g3/2(z)! · " − 2g3/2(z)! #)
7
15π3/2 7 5g2 (z)
5/2
= g (z) , (28)
7/2
4IA "2 − 2g3/2(z)#
where the integral I is defined by
A
5g (z) 5g (z)
I dξ ξ2 5/2 ξ Lˆ′ ξ2 5/2 ξ = dξξ2ξ Lˆ′[ξ2ξ]. (29)
A
≡ Z " − 2g3/2(z)# · " − 2g3/2(z)! # Z ·
In order to evaluate the integral in (29), it is convenient to introduce the change of variables
1 1
′ ′
ξ = (ξ +ξ ), ξ = (ξ ξ ),
√2 0 2 √2 0 −
1 1
′′ ′′
ξ = (ξ +ξ ), ξ = (ξ ξ ). (30)
3 √2 0 4 √2 0 −
′ ′ ′ ′ ′ ′ ′′ ′′ ′′ ′′ ′′ ′′ ′ ′′
Then we introduce transformations from ξ ξ ξ to ξ θ φ and ξ ξ ξ to ξ θ φ , where θ ,θ
x y z x y z
and φ′,φ′′ are the polar and azimuthal angles with respect to the vector ξ . One obtains the
0
following expression for I ,
A
I = 4√2π3I′ ,
A − A
∞ ∞ 1 1
I′ (z) = dξ dξ′ dy′ dy′′ξ4ξ′7F(ξ ,ξ′,y′,y′′;z)[y′2 +y′′2 2y′2y′′2], (31)
A 0 0 0 −
Z0 Z0 Z0 Z0
′ ′ ′′ ′′
where y = cosθ ,y = cosθ . Here the function F is defined by
z2e−(ξ02+ξ′2)
F f(0)f(0)(1+f(0))(1+f(0)) = , (32)
≡ 2 3 4 (1 ze−ξ2)(1 ze−ξ22)(1 ze−ξ32)(1 ze−ξ42)
− − − −
with
1 1
ξ2 = (ξ2 +2ξ ξ′y′+ξ′2), ξ2 = (ξ2 2ξ ξ′y′+ξ′2),
2 0 0 2 2 0 − 0
1 1
ξ2 = (ξ2 +2ξ ξ′y′′ +ξ′2), ξ2 = (ξ2 2ξ ξ′y′′+ξ′2). (33)
3 2 0 0 4 2 0 − 0
Inserting the expression in (27) into (25a) and carrying out the integration, we obtain
the following expression for the thermal conductivity κ:
75k mk T 1/2 π1/2 7 5g2 (z) 2
B B 5/2
κ = g (z) , (34)
−64a2m π ! 16IA′ (z) "2 7/2 − 2g3/2(z)#
′
where the function I (z) is defined in (31).
A
8
B. The shear viscosity
In evaluating the shear viscosity in (25b), the simplest consistent approximation [8,12] is
touseB(ξ) B. TheconstantB canbedetermined bymultiplying (22b)by(ξ ξ δ ξ2/3)
µ ν µν
≡ −
and integrating over ξ,
1 2 1 1 −1
B = dξ ξ ξ δ ξ2 f(0)(1+f(0)) dξ ξ ξ δ ξ2 Lˆ′ ξ ξ δ ξ2
µ ν µν µ ν µν µ ν µν
(Z (cid:18) − 3 (cid:19) )(cid:26)Z (cid:18) − 3 (cid:19) (cid:20) − 3 (cid:21)(cid:27)
5π3/2g (z)
5/2
= . (35)
2I
B
The function I is defined by
B
1 1
I = dξ ξ ξ δ ξ2 Lˆ′ ξ ξ δ ξ2 = dξ(ξ ξ )Lˆ′[ξ ξ ]
B µ ν µν µ ν µν µ ν µ ν
− 3 − 3
Z (cid:18) (cid:19) (cid:20) (cid:21) Z
2√2π3I′ , (36)
≡ − B
where
∞ ∞ 1 1
I′ (z) = dξ dξ′ dy′ dy′′F(ξ ,ξ′,y′,y′′;z)ξ2ξ′7(1+y′2 +y′′2 3y′2y′′2). (37)
B 0 0 0 −
Z0 Z0 Z0 Z0
This involves the same function F as defined in (32). Using the ansatz B(ξ) B in
≡
conjunction with (35) in (25b), we can carry out the integral. Our final the expression for
the viscosity η is
5 1 mk T 1/2 π1/2
η = B g2 (z). (38)
16a2 π ! 8I′ (z) 5/2
B
C. High-temperature limit
The formulas in (34) and (38) give κ and η as a function of the fugacity z. We note
′ ′
that the integrals I and I also depend on z. These four-dimensional integrals in (31) and
A B
(37) can be evaluated numerically. In Figs. 1 and 2, we plot both κ and η as a function of
the fugacity z. These graphs are valid for any trapping potential U (r) since the latter only
0
enters into the equilibrium fugacity z = exp[β (µ U (r) 2gn (r))].
0 0 0 0
− −
In the high-temperature limit, where the fugacity z = eβ(µ−U(r)) is small, the local dis-
tribution function f(0) can be expanded in terms of z. To third order in z, the function F
in (32) reduces to
9
F = z2e−(ξ02+ξ′2) +2z3e−23(ξ02+ξ′2)[cosh(ξ ξ′y′)+cosh(ξ ξ′y′′)]+O(z4). (39)
0 0
The integrals in (31) and (37) can be evaluated analytically,
9 3
I′ (z) = 2I′ (z) = π1/2z2 1+z . (40)
B A  16s2
 
We thus obtain the following explicit expressions for the transport coefficients to first order
in the fugacity z:
1 75 k k T 1/2 7√2 9 3
B B
κ = 1+z
8 (cid:18)64(cid:19) a2 πm !   16 − 16s2
1 75 k k T 1/2  
B B (1 0.07n Λ3), (41a)
≈ 8 (cid:18)64(cid:19) a2 πm ! − 0
1/2
1 5 m k T 1 9 3
B
η = 1+z
8 (cid:18)16(cid:19) a2 πm !  2√2 − 16s2
1 5 m k T 1/2  
B 1 0.335n Λ3 . (41b)
0
≈ 8 (cid:18)16(cid:19) a2 πm ! (cid:16) − (cid:17)
The local equilibrium density n (r) in the high-temperature limit is the classical result
0
n = Λ−3eβ(µ−U(r)). The terms first order in z in (41a) and (41b) give the first order
0
corrections to the classical results due to Bose statistics. If we ignore the HF mean field
2gn (r), these transport coefficients reduce to the expressions first obtained for a uniform
0
Bose gas by Uehling [8]. The Bose quantum corrections to the classical results in both η
and κ depend on the local fugacity z and, through this, on position in a trapped gas.
The density-independent terms inEqs. (41a)and(41b) are8times smaller thanthe well-
known Chapman-Enskog expressions for classical hard spheres. This is due to the difference
(see also Ref. [8]) in the quantum binary scattering cross-section for Bosons when correctly
calculated using symmetrized wavefunctions (σ = 8πa2, instead of πa2).
IV. HYDRODYNAMIC DAMPING OF NORMAL MODES
The linearized version of the hydrodynamic equations in (26) are
∂δn
+∇ (n δv) = 0, (42a)
0
∂t ·
∂δv ∂δP ∂U ∂δn ∂ 1
µ
mn = δn 2gn + 2η D (TrD)δ , (42b)
0 0 µν µν
∂t −∂x − ∂x − ∂x ∂x − 3
µ µ µ ν (cid:26) (cid:20) (cid:21)(cid:27)
∂δP 5 2 2
= ∇ (P δv)+ δv ∇P + ∇ (κ∇T), (42c)
0 0
∂t −3 · 3 · 3 ·
10