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Quantum vortex waves in superfluid helium
PHYSICA
Physica B 182 (1992) 278 286 North-t Iolland
Quantum vortex waves in superfluid helium A Hamiltonian approach R. O w c z a r c k
and T. Slupski
In.s'tilttlc o/ l"undanu'tmd l'echnoh*~,,lcal Re,~(,ar(h, I'olish Academy v/M'iem'c~. ,Swtct,*krzv*ka 21. O0-114u g)u*w*. t'oland
Received 13 September Iqt)l
In
this p a p e r v~c construct
il
c a n o n i c a l H a n l i l t o l l i a n svMclll +.)l Cqtlations
Lu,~l()fe|'l]il]g
the C];AsMC:.II molioll ol ,i \ o l t c x
lilamcnt in an incompressible fluid in a three-dimensional conliguration. Subsequently wc quantize the system, dcri~c the frequency spectrum of energy and compute some thcrmodynamical palalllClCFS O| the qllalltUlll excitations, such as spccilic heal and the m e a n - s q u a r e d i s p l a c e m e n t of the z e r o - p o i n t motion.
!. Introduction
:-+(x(z),y(:),zj.
Thc theory concerns three-dimensional vof tices in s u p e r f l u i d h e l i u m He 4, which arc e x t c n d ed q u a n t u m o b j e c t s , o b s e r v e d in n a t u r e . T h e i r m o t i o n can bc t r e a t e d classically [3, 6, 1()], with the q u a n t i z e d c i r c u l a t i o n as the only r e m n a n t of t h e i r q u a n t u m m e c h a n i c a l origin. In the classical d e s c r i p t i o n s o m e r c n o r m a l i z a t i o n is a s s u m e d : the v o r t e x c o r e o f finite d i a m e t e r (I ,~,) induces a linite v e l o c i t y and a finite e n e r g y in the Blot S a v a r t f o r m u l a . We c o n s i d e r also the local selfi n d u c t i o n a p p r o x i m a t i o n [10], w h e r e the e n e r g y is p r o p o r t i o n a l to the arc length of a w~rtcx curve. In the f r a m e of the q u a n t u m t h e o r y the localiz a t i o n o f the c o r e in a w m e x lattice is analogous to the l o c a l i z a t i o n of p o i n t masses in a crystal lattice, a n d it is s u b j e c t e d to q u a n t u m and thermal fluctuations. In rcf. [3] F c t t c r i n v e s t i g a t e d a H a m i l t o n i a n f o r m a l i s m for slightly d e f o r m e d r e c t i l i n e a r vortices. T h e v o r t i c e s have b e e n p a r a m e t e r i z e d m the f o l l o w i n g way:
w h e r e x ( z ) and y ( c ) arc scalar fields d e p e n d i n g on the 2-\r'ariable. Such a system is a n a l o g o u s to the c a n o n i c a l s y s t e m of e q u a t i o n s for point \ o r rices in two d i m e n s i o n s , w h e r c x and v arc c o n j u g a t e v a r i a b l c s of a v o r t e x point. T h e system has b e e n q u a n t i z e d by i n t c r p r c t i n g these varia b l e s as q u a n t u m m e c h a n i c a l o p e r a t o r s s u b j e c t to the c a n o n i c a l c o m m u t a t i o n relations. A c c o r d ing to F e t t e r [31 this c a n o n i c a l a p p r o a c h was c s s e n t i a l l y t w o - d i m e n s i o n a l and correct only for r e c t i l i n e a r vortices, b e c a u s c "'only a 2I) s y s t e m allows for the classical H a m i l t o n i a n for+ mulation.'" H o w c v e r , r e c e n t l y , following s o m e ideas of A r n o l d [11 and M a r s d e n and W e i n s t e i n [S I. thc g e n e r a l s y m p l e c t i c s t r u c t u r e can be d e r i v e d for a n u m b e r of t h r e e - d i m e n s i o n a l w)rtex systems, in p a r t i c u l a r for c u r v e d w~rtcx filaments. Extensive b a c k g r o u n d m a t e r i a l can be f o u n d in 181 and in the b i b l i o g r a p h y listed there. T h e s y m p l e c t i c s t r u c t u r c i n t r o d u c e d in [S] is non-canonical, t h e r c f o r c o u r ailn was to lind c a n o n i c a l scalar lields p ( s ) and q(s). which g e n e r a l i z c the choice of .r(z) and y ( z ) m the r e c t i l i n e a r case. Wc a g r e e that the c a n o n i c a l q u a n t i z a t i o n d e p e n d i n g on the a r b i t r a r y choice
('orresl~ondence IO: R. Owczarek, Institute of Fundamental Techn. Rcscarch. Polish Academy of Sciences, Swietokrzx ska 21, (l{1-049 Warsaw, Pohmd.
0t)21-4526/~*2/$05.1)0 © 1992- Elsevier Science Pulqishcrs H.V. All rights rcser\cd
R. Owczarek, T. Slupski / Quantum vortex waves in superfluid helium
of canonical coordinates must be considered heuristic. A n o t h e r approach to the problem of quantization is proposed in ref. [9], based on more natural "complex polarization" of the vortex phase space, but the canonical complex coordinates z ( s ) = q ( s ) + ip(s) can still not be found explicitly. The problem concerning the rigorous basis for the quantization of the Hamiltonian seems still to be open. In this p a p e r we quantize only the arc-length Hamiltonian of the local self-inducted approximation. Therefore we can compare our results with that part of Fetter's paper [3] which concerns a single, independent vortex line.
K is a circulation constant, which for superfluid helium is a quantum analogon to h / m (m = mass of a helium atom). Formula (2.1) diverges logarithmically as r~ approaches the vortex line. If we assume that the filament has a core of small non-zero radius (in superfluid helium a ~ 1 A), then the integration can be confined to I r, - rl > a. Some argumentation (see ref. [10]) shows that introducing an u p p e r cut-off p a r a m e t e r I r ~ r[ < R, where R is the mean radius of curvature of the vortex, we obtain the approximated value of the velocity on the vortex line v ~- a K ( t
Let us consider the evolution of a curved vortex filament in ~3. We consider two cases: a vortex loop and a vortex helical spiral. A parameterization of a vortex loop is given by a smooth function on a unit circle 51 into [R3, or equivalently, by a periodic, smooth function on an interval (0, 2rr} into N3. A parameterization of a helical spiral is defined also on 5~, but the image is in the cylinder IR2 x 5l. Equivalently it is given by a smooth function on ~ into N3
r = r(s) = (x(s), y(s), z(s)) with periodicity conditions
49l--
(2.2)
K ln(R/a) 4~r
T h e velocity in (2.2) is called "the locally selfinduced approximation" as it depends only on the local quantities at the point rl, and neglects the influence of the other parts of the vortex line. H o w e v e r , it is estimated in ref. [10] that the local self-induction approximation is accurate to about 10% c o m p a r e d with (2.1). The equation of motion (2.2) will be shown to be conservative with the following Hamiltonian:
H~,f = pK~ f [drl,
x(s + 2"rr) = x(s) + x 0 ,
(2.3)
,6
y(s + 2'rr) = y(s) + y,, ,
It is proportional to the arc-length of the curve
z(s + 2rr) = z(s) + z,,. is a regular helical spiral, the only curve in ~3 with torsion and constant curresult of classical hydrothe velocity in a point r 1 filament q¢ is
t< f d r x ( r l - r ) v(G) = ~ a ]rt~-r] ~
x n),
where t is the tangent versor, n is the normal versor, K is the curvature in the point rl, a is a "cut-off" p a r a m e t e r equal to
2. H a m i l t o n i a n s on vortex filaments
A typical example defined in (3.1) constant, non-zero vature. A fundamental dynamics says that induced by a vortex
279
'
(2.1)
The B i o t - S a v a r t formula of motion (2.1) will be shown to be related to the following Hamiltonian: H = ~
Jr, - r2l 1 dr 1 dr 2
(2.4)
(introducing the cut-off Irl -- rel > a). We shall use the Hamiltonian (symplectic) structure for the vortex filaments, which has been c o m p u t e d from the general Poisson brac-
28(I
R. O w c z a r e k , T. Slupski
, Q u a n t u m vortex waves' in suFerfluid helium
kets of hydrodynamics by Marsden and Weinstein in [8]. According to ref. [8], the Poisson bracket of the functionals H and F on the space of vortex filaments takes the form
{H, F} =OK
f SH
8F
(2.5)
8 r x 8rr dr
(
3.
The
stationary
solutions
We shall consider a 2-parameter family of solutions of the equation pK Or~ hi x t-
pK(~ K n
.
For simplification let us temporarily assume that pK 1, (~ - 1, hence
and the appropriate system of Hamiltonian equations is
i~r/#t
ar pK at
For A ~ [~. R > () wc define the time-dependent curves
--
x t-
8H , 8r
(2.6)
where × denotes the vector product and 8 H / S r is a functional derivative. We identify the functional derivative with a vector field v which is orthogonal to the curve Y,:
v-
8H/gr.
v" t-
(),
such that
d/de(H(r
x
t-
Kn
r(O, t) = ( R cos 0, R sin 0, RAO + ct).
(2.7)
t
n,=(-cos0,
sin0.()),
b,,=a(l+A:)
i "(sin0.
We also find
8H~,.,/Sr = -puog i ~ t / a s -
l,
This yields immediately (2.2). For the Hamiltonian of the Biot-Savart type (2.4), a slightly longer calculation yields [)K e f d r x (r, - r) 8H/ar(r, ) = -~ }r, - rl ~
K=(R(I
+A2)) ~
(3.2) cos0,1/A).
T-AK.
2rrRV'l ¢ a : .
where K denotes the curvature. 7" the torsion and L the arc-length of the vortex line. From eq. (3.0), wc find c=(RVI+A2)
× t(r,
sinO, cosO, a ) ,
~:(
for each smooth vector field w. For the arc-length Hamiltonian (2.3), we call easily compute, using the Frenet equations (4.3),
pKoeKn .
(3.1)
For A = 0 wc have a vortex ring, whereas for other a we have helical spirals. Now, we compute the Frenct triad (the tangent, normal and binormal versor): t,,-(l+a:)
+ ew))~ =,1 -- J v . w d s ,
(3.(l)
.
~
(3.3)
) .
(
This proves the equivalence of the Biot-Savart formula (2.1) with the Hamiltonian equation (2.6) for the Hamiltonian (2.4). Let us notice that Fetters remark [3, p. 144]: "for a general 3D configuration eqs. (2.1) and (2.4) are not directly related" is too pessimistic, as we have proven above.
and
ce~-
Tt,, + Kb,,.
(3.4)
Our curve moves with constant velocity ('e~, but in the moving frame we can see it as a stationary vortex line. In the moving frame we must replace the Hamiltonian H o with the effective Hamiltonian H , - H~, where
R. Owczarek, T. Slupski / Quantum vortex waves in superfluid helium f H I ( ~ ) = ~1 I (ce3 x r) dr
(3.5)
281
we obtain r ' = (1 + K Q ) t o + ( Q ' - T P ) n o + ( P ' + T Q ) b o .
and we may calculate O(r-
(4.4)
ce3)/at = b H o / ~ r -
(3.6)
~H1/~r .
For r defined in (3.1) the curve r - c e 3 is the stationary solution of (3.6). Note that the curve r moves with velocity c, which is inversely proportional to R.
We also have ar/Ot = 0" t o + o Q / o t , n o + 8P/Ot. b o , therefore n o . ( O r / a t x r ' ) = (1 - K Q ) " 8 Q / o t , b o. (Or/Ot x r ' ) = - ( 1 -
4. Local canonical coordinates
In this section, we again simplify formulae (3.0) etc., taking the constant pK-= 1. Now we search for the local coordinates p, q, which are periodic scalar functions on E, with period equal to L. For given p = p(s) and q = q(s) the parameterization of the curve ~ ought to be
KQ)" aP/ot.
But from (2.6) we obtain Or/at x Or~as = 8 H / ~ r . It' I , and from (4.2), ~r = ~Qn
o + ~Pb o ,
r(s) : ro(S ) + rl(s, p(s), q ( s ) ) ,
which allows us to compute the functional derivatives
where r 0 is an initial curve with normal parameterization, i.e.
8 H / O Q ds = 8 H / ~ r noldr I , ~ H / ~ P ds = ~ H / ~ r boldr I ,
IOro/OS [ = 1
and
ri(s,O,O)=O.
[dr I = [r'[ d s .
We shall consider that the p a r a m e t e r s C (0, L ) , where L is the arc-length of the curve r0, rem e m b e r i n g the periodic conditions for p(s) and q(s). We try to obtain the canonical equations
Finally, we obtain (1 -
KQ ) oQ/at = ~H/~P,
(1 - K Q ) oP/Ot = - ~ H / ~ Q . aq/Ot = ~ H / b p ,
Op/Ot = - ~ H / S q
.
(4.1)
For the m o m e n t we introduce another pair of scalar functions Q(s, t), P(s, t), and define
The system of equations (4.1) becomes canonical after the following transformation of variables: q=Q-
r(s, t) : ro(s ) + Q(s, t)n o + P(s, t)bo ,
where to, no, b o form the Frenet triad for the curve r o. Using the Frenet formula (t' denotes the derivative Ot/Os, etc.), t' =
Kn
,
n' =
-Kt
+
Tb
,
b' = -
½KQ 2,
p=P.
(4.5)
(4.2)
Tn
,
(4.3)
The transformation is invertible for ] q l < 2 / K and IQ] < 1 / K . Hence we have proven the following lemma. L e m m a 1. In the vicinity (in c~2-topology) o f the curve r o with a well-defined Frenet triad there exist canonical variables f o r the Hamiltonian (symplec-
R. Owczarek, T, Slupski
282
Quantum vorte.~ wm'e,s in supep:fluid helium
tic') structure o[ the vortex motion. The variables p, q are defined by the j ~ r m u l a e (4.2) and (4.5).
r e m e m b e r i n g that \/1
The assumption about the Frenet triad (i.e. K > ( ) ) is not essential for the sections of the curve with vanishing curvature; we can construct even simpler canonical variables.
2Kq + Aq ~
~(ce~ × r ) - r ' -
Assume that ~; is a straight line. the z-axis in ~. Define r(s, t)
se~ + q(s, t)e I + p(s. t)e~ .
An analogous computation as above would show that q, p are canonical. These variables are well known in the two-dimensional vortex theory (where they do not depend on s) and they were introduced as s-dependent deformations by Fetter in ref. [31.
K,~ ~ !(
-: l
Kq+
Kq ~ ~ A,/: + O()ql~)).
- ~('/'t,, + Kb,,) × (
Rn,,). r'
i(.F-'p: ~- T'-q ~+ T p ' q
Tq'p)
o(I,/1' + I.1 * Iq'l '÷ I/,'l') • The first order terms are identical in H~j and H~, so they disappear m H, H~, which implies that ( 0 , 0 ) is the point of equilibrium. The second order term of H , - H~ may be rewritten as [
H=
! ] {(q')::+(P'):-
K-'q~-
ql
7"( p ' q
q /, ))j dIs .
(5.1)
5. Diagonalization of the Hamiltonian For the linearization of the Hamiltonian equations close to the stationary solution (3.1) wc expand the effective Hamiltonian H o - H~ up to second order terms with respect to the canonical variables p(s), q(s), which are considered as the displacements from equilibrium in the moving frame. We use the formulae (3.1), (3.4), (4.2), (4.3) computing k
q(s. t) -- q exp(i(wt + ks)). p(s, t) = p exp(i(wt
ks)).
and obtain the dispersion relation for w = w ( k ) . but the same can be done in the diagonalization procedure of section 6.
l.
x r)- r' ds 0
0
and obtain
[r'l=X/(l
In thc classical approach we should search the solutions in the form of plane waves:
KO)~+(O '
TP): + ( P ' + T Q ) ~ .
From eqs. (4.5) we can put q in place of Q and p in place of P in all the quadratic terms except (1-KQ)2=I-2Kq, which is linear in q. Hence ]
Ir'l = l - Kq + , {
K'-q~ + ( q
t
Tp) ~
+ ( p ' + Tq) ~}
+ O([q[" + Ipl 3 + Iq'l ~ + I p ' [ " ) ,
6. Quantization and diagonalization of the Hamiltonian We assume that the excitations of a vortex filament may be quantized by interpreting q and p as quantummechanical operators that obey Heisenberg's equations of motion ih a q / a t = [q(s),
HI,
ih a p / a t = [p(s). H I . (~.l)
The operators q and p satisfy the canonical c o m m u t a t i o n relation [q(s), p ( s ' ) ] = i h S ( s
s').
((~.2)
283
R. Owczarek, T. Slupski / Quantum vortex waves in superfluid helium
q(s) = L
2
I/2
(6.9)
cok = ~ / k 2 ( k2 - K 2 ) - T k .
We can expand them in a Fourier series: qk e x p ( + i k s ) ,
(6.3)
The operators a k and a~, obey the boson canonical c o m m u t a t i o n relations
p ( s ) = L - ' / 2 ~ , Pk e x p ( - i k s ) , [a k, a~] = 6kl,
where k = 2 v n / L , n = 0 , +1, +2 . . . . . Then, the coefficients p and q obey the canonical commutation relation
For n = 1 we have k~ = 2 " r r / L = V ~
[ qk, Pt]
= ihfkt
~ ~] (k 2
= -
+ T 2.
,
and the quantized Hamiltonian (5.1) becomes H
[a k, a,] = O.
2
,
2
H e n c e c o ( k 1 ) = 0 for each T and K. All frequencies co are real and this is a necessary condition for the stability of the stationary solutions.
,
- K )qkqk + k PkPk
k
- iTk(qkPk
+ Pkqk),
(6.4)
which may be diagonalized by a linear transformation of variables: a k = Ckp k - - i q _ k / 2 h c k ,
(6.5) r Ck
k2
~/2,n-(k 2 _ K 2) "
We see that the Hamiltonian does not depend on P0, hence qo is the constant of motion and we may cancel 0-terms in the Fourier series. In the case of a vortex ring ( T = 0, K > 0), the Hamiltonian does not depend on ql and pl is the constant of motion. H e n c e for a vortex ring we define k m i n = k 2 and calculate kmi . = k 2 = 4"rr/L = 2K . For non-zero torsion T S 0 , we have and obtain kmi n = k! = 2 ~ r / L = "~/T 2 + K 2 .
(6.6) kmi n = k 1
7. S p e c i f i c h e a t
As in ref. [3], the Hamiltonian equation (6.1) may be used to compute the thermal energy of vibration of a single vortex filament with constant K and constant torsion T. We consider two special cases: K = 0, which is the case of a single straight line and K > 0 , T = 0, which is the case of a vortex ring. (It is possible to find also a formula for the case T # 0, but with rather unpleasant integrals, difficult to compute or estimate.) Let us return to the Hamiltonian (2.3), with the coefficient a = K/4"rr l n ( R / a ) . Then formula (6.9) reads to(k) = o~(Ikl(k ~ - K 2 ) ~/2 - T k ) .
It was shown that stable vibration is possible for Ik I/> kmin and then co(k) >10. At the t e m p e r a t u r e T = ( k B [ 3 ) -1 the mean thermal energy may be found assuming that excitations fulfill B o s e - E i n s t e i n statistics. Hence
(6.7) E=
Substitution of (6.5) into (6.4) yields H=
~ ~
hcok(aka ~ + a k a k ) ,
k#O
where the dispersion relation reads
(6.8)
(7.1)
~
Ikl>kmin
( L
hco k exp(h
k)--l+2
1)
"
(7.2)
The second term in this equation represents the energy of zero-point motion and will be omitted, because it does not affect the specific heat. The sum over k may be approximated by an integral:
284
R. O w c z a r e k .
/" m i n
E = ~
T. .~Tupski
' QUaHItCtll v o r t e t
wave~ m .~u/)edtutd helium
implies
':
+
. /'mm 1.".
____
17
In the case of straight vortex lines, wc have co(k) = a k ~ and hence
L f E
.
exp(h/3a(k:
Ix
Lhc,__
]
W
•
x d.t
exp(fi/iak~')- 1
?
.... L ( k , ~ l ) ;
[
L((3)(k ~ "
1
7')':(2hK) '-~ In(R:) B (12
,dE 3(3) dT - 5 (
I
2 kt~(kBT)J
2
/
I
exp(fdBalkl(k:
Kz) ' z)
1
As in Fetter [3], we use the' Hamiltonian lorrealism to ealculatc the average displacement of the vortex core:
( u~(s)).,
Ikl(k: - ~~)' ~ dk
A n u p p e r estimate of the energy can be derived in the following way: for x = ( k 2 ( k ~ - K~-)) ~ ~ w e have k = 2 l"2K(l + ( 1 + 4 x a K 4)1 _~)t 2 and calculate that dk < dx for k > K, and hence
K
/
L
'
J ds(u~(s))
K:)' -')--1 (7.6)
w
(7.8)
/
exp(h/3~lkl(k:
Lf,,~ f
)]'
l
K
_ Lho,_.~ j
'
8. Zero-point motion
K:I' +
~
I 2
In( R:_~)
(7.5)
L (f
E<--
1
For a m a c r o s c o p i c w)rtex ring (1 K ~ I A). the lower and u p p e r estimales are practically equal as the series tend to ( ( 3 / 2 ) . The c o m m o n value is identical with the result for rectilinear w m i c e s (7.4). The same concerns the specific heat (7.5).
in the case of a vortex ring c o ( k ) = ~ l k ] ( k : K2) 1'2 and kmm = K. H e n c e
E:
'
V. [exp(-:~h / & ~ K : (7.4)
~-(2hK)
ln--t l 2
x
:(2ilK)
U-
and the specific heat per unit length b e c o m e s L
[Ke))
exp(fi/&~(x e +
hak ~ dk
~ =
! K ~)) - 1
dx
+o(lql ~ +!pl'~
(~.1) wherc Ya can be calculated from (6.5 1
ke
(k:
as
K2) )
(8.2)
exp(h/3ax-') - 1 I 2
,
,
a 2
(7.7)
and the ensemble average ( a k a ~ + a ~ a a ) is equal to the thermal factor coth( !/3hco,). Hence
T o obtain a lower estimate, we notice that for K > 0, the estimation
{ u-'(s)),, = f l ( p K L )
i N~ c o t h ( f i h w k / 2 ) y a .
[k[(k e - K : ) I : < k z - { K -~
In the case of a vortex ring, co, depends only on
(S.3)
285
R. Owczarek, T. Slupski / Quantum vortex waves in superfluid helium
k 2, therefore we take the factor 2 and sum only for k > 0 . As in eq. (6.6) we have the lower limit k > kmi n = 2K, and according to ref. [3] we introduce also an upper limit
= (~OK/3a)-, J (k-2 + (k 2 _ K 2) 1) dk 2K
-- (avpK/3a)-'K-' ½(1 + In 3)
--rr 2(pK/3a) ' ~ L ( l + l n 3 ) .
(8.8)
(8.4)
kma x = 2'rr/Ami n = 2~r/a ,
Finally we obtain determined by the condition for the shortest wavelength allowed Amin, which has to be equal to the radius a of the vortex core. At zero temperature the thermal factor c o t h ( / 3 h % / 2 ) reduces to 1, and the average zero-displacement is simply
(u2(s) ) av ~ 4 L( OK2/3~ l n ( L / 2 % ) ) - 1 x ¼(1+ I n 3 ) .
(8.9)
9. Discussion
kmax
~ ~- 2 h ( p K L ) - ' ~
(8.5)
"y~ ,
Our result for curved vortices
kmin
dT-
which can be approximated by the integral
2
kB(kBT),/:(2h,,) ,/2
krnax
< u2 ( s ) ) a v ~ - 2 h ( p K L )
'~
Lf
y(k) dk
is similar to Fetter's result
kmin = h(pK~r)-X((k
2 --
K2) 1/2
L_ l dE 3 /3\ 1/~ 1/2 d T - 2 ~(-2) k B ( k B T ) "(2hK)-
k=kma - K arctg((k-2 - K 2) 1/2/2K))]k=ZK
x •
(8.6) For kma x >~ K the arctg term is smaller than ½7r, and can be neglected, as well as all the terms of order K. Hence (
u 2 (s))av
~-
h( pK'lT) -'. kma x =- 2h/pKa
,
for T = 0.
(8.7)
Our result for vortex rings at zero temperature is the same as Fetter's case of the straight vortex lines. At the finite non-zero temperature, the coth(x) can be approximated by x ~, and the upper limit kma x c a n be taken +oc (the integral above kma x is small), hence
(U2(S)>
h(~rrpK) -1 ff (2/3hbgk) 1 kmin
l Tk d k
[In( hK ] ] X k \a2kuT/J
,/2
for the " p e r t u r b e d " 2D case of rectilinear vortices. The specific heat associated with the vortex waves varies approximately a s T 1/2 at low temperatures. Fetter's additional logarithmic factor is negligible, because the logarithmic variation is much smaller than T 1/2. It seems that it is connected with the non-local influence of the B i o t Savart interaction, because it has not been obtained in the local approximation. The same concerns eq. (8.9) for the average zerodisplacement. The formulae apply also to the macroscopic vortex ring of radius R >> 1 ~ and to the fully 3D helical spirals. Therefore the validity of the theory for non-interacting curved filaments in three dimensions is now proved. We suppose that the case of interacting vortex filaments (in particular vortex lattices) can be
286
R. O w c z a r e k , 7". Slupski / Quantum vortex waves in superltuid helium
investigated in a similar way for the Biot-Savart Hamiltonian. Also some perturbation calculations can be developed if one considers the Biot-Savart Hamiltonian as a small perturbation of the arc-length Hamiltonian.
References [I 1 V. Arnold, Usp, Mat. Nauk 24 (1969) 225. [2] V. Arnold. PWN (19810.
[31 A.L. Fetter. Phys. Rcv. 162 11967) 143. 14l R. Hasimoto, J. Fluid Mech. 51 (1972)477. 15} V. kebedev and I. Khalatniko~, Pis'ma Zh. Eksp. T e o r Fis. 28 11978) 89. [61 V. Lebcdev and I, Khalatnikov, Hamiltoniatl Ilydr~-dynamics equations in the presence of solitons, Zh, Eksp. Teor. Fis. 75 (1978) 2312. [7] J. Marsden, T. Ratiu and A. Weinstein, Trans. Am. Math. Soc. 281 11984) 147. 181 J. Marsden and A. Weinstein. Physica D 7 (1983) 31~5. [9] V. Penna and M. Spera, J. Math. Phys. 31~ (1989) 2778 [I(I] K.W. Schwarz, Phys. Re,,'. B 31 (1985) 5782. [11] T. Slupski, Arch. Mech. 36 11984) 715.
Physica B 182 (1992) 278 286 North-t Iolland
Quantum vortex waves in superfluid helium A Hamiltonian approach R. O w c z a r c k
and T. Slupski
In.s'tilttlc o/ l"undanu'tmd l'echnoh*~,,lcal Re,~(,ar(h, I'olish Academy v/M'iem'c~. ,Swtct,*krzv*ka 21. O0-114u g)u*w*. t'oland
Received 13 September Iqt)l
In
this p a p e r v~c construct
il
c a n o n i c a l H a n l i l t o l l i a n svMclll +.)l Cqtlations
Lu,~l()fe|'l]il]g
the C];AsMC:.II molioll ol ,i \ o l t c x
lilamcnt in an incompressible fluid in a three-dimensional conliguration. Subsequently wc quantize the system, dcri~c the frequency spectrum of energy and compute some thcrmodynamical palalllClCFS O| the qllalltUlll excitations, such as spccilic heal and the m e a n - s q u a r e d i s p l a c e m e n t of the z e r o - p o i n t motion.
!. Introduction
:-+(x(z),y(:),zj.
Thc theory concerns three-dimensional vof tices in s u p e r f l u i d h e l i u m He 4, which arc e x t c n d ed q u a n t u m o b j e c t s , o b s e r v e d in n a t u r e . T h e i r m o t i o n can bc t r e a t e d classically [3, 6, 1()], with the q u a n t i z e d c i r c u l a t i o n as the only r e m n a n t of t h e i r q u a n t u m m e c h a n i c a l origin. In the classical d e s c r i p t i o n s o m e r c n o r m a l i z a t i o n is a s s u m e d : the v o r t e x c o r e o f finite d i a m e t e r (I ,~,) induces a linite v e l o c i t y and a finite e n e r g y in the Blot S a v a r t f o r m u l a . We c o n s i d e r also the local selfi n d u c t i o n a p p r o x i m a t i o n [10], w h e r e the e n e r g y is p r o p o r t i o n a l to the arc length of a w~rtcx curve. In the f r a m e of the q u a n t u m t h e o r y the localiz a t i o n o f the c o r e in a w m e x lattice is analogous to the l o c a l i z a t i o n of p o i n t masses in a crystal lattice, a n d it is s u b j e c t e d to q u a n t u m and thermal fluctuations. In rcf. [3] F c t t c r i n v e s t i g a t e d a H a m i l t o n i a n f o r m a l i s m for slightly d e f o r m e d r e c t i l i n e a r vortices. T h e v o r t i c e s have b e e n p a r a m e t e r i z e d m the f o l l o w i n g way:
w h e r e x ( z ) and y ( c ) arc scalar fields d e p e n d i n g on the 2-\r'ariable. Such a system is a n a l o g o u s to the c a n o n i c a l s y s t e m of e q u a t i o n s for point \ o r rices in two d i m e n s i o n s , w h e r c x and v arc c o n j u g a t e v a r i a b l c s of a v o r t e x point. T h e system has b e e n q u a n t i z e d by i n t c r p r c t i n g these varia b l e s as q u a n t u m m e c h a n i c a l o p e r a t o r s s u b j e c t to the c a n o n i c a l c o m m u t a t i o n relations. A c c o r d ing to F e t t e r [31 this c a n o n i c a l a p p r o a c h was c s s e n t i a l l y t w o - d i m e n s i o n a l and correct only for r e c t i l i n e a r vortices, b e c a u s c "'only a 2I) s y s t e m allows for the classical H a m i l t o n i a n for+ mulation.'" H o w c v e r , r e c e n t l y , following s o m e ideas of A r n o l d [11 and M a r s d e n and W e i n s t e i n [S I. thc g e n e r a l s y m p l e c t i c s t r u c t u r e can be d e r i v e d for a n u m b e r of t h r e e - d i m e n s i o n a l w)rtex systems, in p a r t i c u l a r for c u r v e d w~rtcx filaments. Extensive b a c k g r o u n d m a t e r i a l can be f o u n d in 181 and in the b i b l i o g r a p h y listed there. T h e s y m p l e c t i c s t r u c t u r c i n t r o d u c e d in [S] is non-canonical, t h e r c f o r c o u r ailn was to lind c a n o n i c a l scalar lields p ( s ) and q(s). which g e n e r a l i z c the choice of .r(z) and y ( z ) m the r e c t i l i n e a r case. Wc a g r e e that the c a n o n i c a l q u a n t i z a t i o n d e p e n d i n g on the a r b i t r a r y choice
('orresl~ondence IO: R. Owczarek, Institute of Fundamental Techn. Rcscarch. Polish Academy of Sciences, Swietokrzx ska 21, (l{1-049 Warsaw, Pohmd.
0t)21-4526/~*2/$05.1)0 © 1992- Elsevier Science Pulqishcrs H.V. All rights rcser\cd
R. Owczarek, T. Slupski / Quantum vortex waves in superfluid helium
of canonical coordinates must be considered heuristic. A n o t h e r approach to the problem of quantization is proposed in ref. [9], based on more natural "complex polarization" of the vortex phase space, but the canonical complex coordinates z ( s ) = q ( s ) + ip(s) can still not be found explicitly. The problem concerning the rigorous basis for the quantization of the Hamiltonian seems still to be open. In this p a p e r we quantize only the arc-length Hamiltonian of the local self-inducted approximation. Therefore we can compare our results with that part of Fetter's paper [3] which concerns a single, independent vortex line.
K is a circulation constant, which for superfluid helium is a quantum analogon to h / m (m = mass of a helium atom). Formula (2.1) diverges logarithmically as r~ approaches the vortex line. If we assume that the filament has a core of small non-zero radius (in superfluid helium a ~ 1 A), then the integration can be confined to I r, - rl > a. Some argumentation (see ref. [10]) shows that introducing an u p p e r cut-off p a r a m e t e r I r ~ r[ < R, where R is the mean radius of curvature of the vortex, we obtain the approximated value of the velocity on the vortex line v ~- a K ( t
Let us consider the evolution of a curved vortex filament in ~3. We consider two cases: a vortex loop and a vortex helical spiral. A parameterization of a vortex loop is given by a smooth function on a unit circle 51 into [R3, or equivalently, by a periodic, smooth function on an interval (0, 2rr} into N3. A parameterization of a helical spiral is defined also on 5~, but the image is in the cylinder IR2 x 5l. Equivalently it is given by a smooth function on ~ into N3
r = r(s) = (x(s), y(s), z(s)) with periodicity conditions
49l--
(2.2)
K ln(R/a) 4~r
T h e velocity in (2.2) is called "the locally selfinduced approximation" as it depends only on the local quantities at the point rl, and neglects the influence of the other parts of the vortex line. H o w e v e r , it is estimated in ref. [10] that the local self-induction approximation is accurate to about 10% c o m p a r e d with (2.1). The equation of motion (2.2) will be shown to be conservative with the following Hamiltonian:
H~,f = pK~ f [drl,
x(s + 2"rr) = x(s) + x 0 ,
(2.3)
,6
y(s + 2'rr) = y(s) + y,, ,
It is proportional to the arc-length of the curve
z(s + 2rr) = z(s) + z,,. is a regular helical spiral, the only curve in ~3 with torsion and constant curresult of classical hydrothe velocity in a point r 1 filament q¢ is
t< f d r x ( r l - r ) v(G) = ~ a ]rt~-r] ~
x n),
where t is the tangent versor, n is the normal versor, K is the curvature in the point rl, a is a "cut-off" p a r a m e t e r equal to
2. H a m i l t o n i a n s on vortex filaments
A typical example defined in (3.1) constant, non-zero vature. A fundamental dynamics says that induced by a vortex
279
'
(2.1)
The B i o t - S a v a r t formula of motion (2.1) will be shown to be related to the following Hamiltonian: H = ~
Jr, - r2l 1 dr 1 dr 2
(2.4)
(introducing the cut-off Irl -- rel > a). We shall use the Hamiltonian (symplectic) structure for the vortex filaments, which has been c o m p u t e d from the general Poisson brac-
28(I
R. O w c z a r e k , T. Slupski
, Q u a n t u m vortex waves' in suFerfluid helium
kets of hydrodynamics by Marsden and Weinstein in [8]. According to ref. [8], the Poisson bracket of the functionals H and F on the space of vortex filaments takes the form
{H, F} =OK
f SH
8F
(2.5)
8 r x 8rr dr
(
3.
The
stationary
solutions
We shall consider a 2-parameter family of solutions of the equation pK Or~ hi x t-
pK(~ K n
.
For simplification let us temporarily assume that pK 1, (~ - 1, hence
and the appropriate system of Hamiltonian equations is
i~r/#t
ar pK at
For A ~ [~. R > () wc define the time-dependent curves
--
x t-
8H , 8r
(2.6)
where × denotes the vector product and 8 H / S r is a functional derivative. We identify the functional derivative with a vector field v which is orthogonal to the curve Y,:
v-
8H/gr.
v" t-
(),
such that
d/de(H(r
x
t-
Kn
r(O, t) = ( R cos 0, R sin 0, RAO + ct).
(2.7)
t
n,=(-cos0,
sin0.()),
b,,=a(l+A:)
i "(sin0.
We also find
8H~,.,/Sr = -puog i ~ t / a s -
l,
This yields immediately (2.2). For the Hamiltonian of the Biot-Savart type (2.4), a slightly longer calculation yields [)K e f d r x (r, - r) 8H/ar(r, ) = -~ }r, - rl ~
K=(R(I
+A2)) ~
(3.2) cos0,1/A).
T-AK.
2rrRV'l ¢ a : .
where K denotes the curvature. 7" the torsion and L the arc-length of the vortex line. From eq. (3.0), wc find c=(RVI+A2)
× t(r,
sinO, cosO, a ) ,
~:(
for each smooth vector field w. For the arc-length Hamiltonian (2.3), we call easily compute, using the Frenet equations (4.3),
pKoeKn .
(3.1)
For A = 0 wc have a vortex ring, whereas for other a we have helical spirals. Now, we compute the Frenct triad (the tangent, normal and binormal versor): t,,-(l+a:)
+ ew))~ =,1 -- J v . w d s ,
(3.(l)
.
~
(3.3)
) .
(
This proves the equivalence of the Biot-Savart formula (2.1) with the Hamiltonian equation (2.6) for the Hamiltonian (2.4). Let us notice that Fetters remark [3, p. 144]: "for a general 3D configuration eqs. (2.1) and (2.4) are not directly related" is too pessimistic, as we have proven above.
and
ce~-
Tt,, + Kb,,.
(3.4)
Our curve moves with constant velocity ('e~, but in the moving frame we can see it as a stationary vortex line. In the moving frame we must replace the Hamiltonian H o with the effective Hamiltonian H , - H~, where
R. Owczarek, T. Slupski / Quantum vortex waves in superfluid helium f H I ( ~ ) = ~1 I (ce3 x r) dr
(3.5)
281
we obtain r ' = (1 + K Q ) t o + ( Q ' - T P ) n o + ( P ' + T Q ) b o .
and we may calculate O(r-
(4.4)
ce3)/at = b H o / ~ r -
(3.6)
~H1/~r .
For r defined in (3.1) the curve r - c e 3 is the stationary solution of (3.6). Note that the curve r moves with velocity c, which is inversely proportional to R.
We also have ar/Ot = 0" t o + o Q / o t , n o + 8P/Ot. b o , therefore n o . ( O r / a t x r ' ) = (1 - K Q ) " 8 Q / o t , b o. (Or/Ot x r ' ) = - ( 1 -
4. Local canonical coordinates
In this section, we again simplify formulae (3.0) etc., taking the constant pK-= 1. Now we search for the local coordinates p, q, which are periodic scalar functions on E, with period equal to L. For given p = p(s) and q = q(s) the parameterization of the curve ~ ought to be
KQ)" aP/ot.
But from (2.6) we obtain Or/at x Or~as = 8 H / ~ r . It' I , and from (4.2), ~r = ~Qn
o + ~Pb o ,
r(s) : ro(S ) + rl(s, p(s), q ( s ) ) ,
which allows us to compute the functional derivatives
where r 0 is an initial curve with normal parameterization, i.e.
8 H / O Q ds = 8 H / ~ r noldr I , ~ H / ~ P ds = ~ H / ~ r boldr I ,
IOro/OS [ = 1
and
ri(s,O,O)=O.
[dr I = [r'[ d s .
We shall consider that the p a r a m e t e r s C (0, L ) , where L is the arc-length of the curve r0, rem e m b e r i n g the periodic conditions for p(s) and q(s). We try to obtain the canonical equations
Finally, we obtain (1 -
KQ ) oQ/at = ~H/~P,
(1 - K Q ) oP/Ot = - ~ H / ~ Q . aq/Ot = ~ H / b p ,
Op/Ot = - ~ H / S q
.
(4.1)
For the m o m e n t we introduce another pair of scalar functions Q(s, t), P(s, t), and define
The system of equations (4.1) becomes canonical after the following transformation of variables: q=Q-
r(s, t) : ro(s ) + Q(s, t)n o + P(s, t)bo ,
where to, no, b o form the Frenet triad for the curve r o. Using the Frenet formula (t' denotes the derivative Ot/Os, etc.), t' =
Kn
,
n' =
-Kt
+
Tb
,
b' = -
½KQ 2,
p=P.
(4.5)
(4.2)
Tn
,
(4.3)
The transformation is invertible for ] q l < 2 / K and IQ] < 1 / K . Hence we have proven the following lemma. L e m m a 1. In the vicinity (in c~2-topology) o f the curve r o with a well-defined Frenet triad there exist canonical variables f o r the Hamiltonian (symplec-
R. Owczarek, T, Slupski
282
Quantum vorte.~ wm'e,s in supep:fluid helium
tic') structure o[ the vortex motion. The variables p, q are defined by the j ~ r m u l a e (4.2) and (4.5).
r e m e m b e r i n g that \/1
The assumption about the Frenet triad (i.e. K > ( ) ) is not essential for the sections of the curve with vanishing curvature; we can construct even simpler canonical variables.
2Kq + Aq ~
~(ce~ × r ) - r ' -
Assume that ~; is a straight line. the z-axis in ~. Define r(s, t)
se~ + q(s, t)e I + p(s. t)e~ .
An analogous computation as above would show that q, p are canonical. These variables are well known in the two-dimensional vortex theory (where they do not depend on s) and they were introduced as s-dependent deformations by Fetter in ref. [31.
K,~ ~ !(
-: l
Kq+
Kq ~ ~ A,/: + O()ql~)).
- ~('/'t,, + Kb,,) × (
Rn,,). r'
i(.F-'p: ~- T'-q ~+ T p ' q
Tq'p)
o(I,/1' + I.1 * Iq'l '÷ I/,'l') • The first order terms are identical in H~j and H~, so they disappear m H, H~, which implies that ( 0 , 0 ) is the point of equilibrium. The second order term of H , - H~ may be rewritten as [
H=
! ] {(q')::+(P'):-
K-'q~-
ql
7"( p ' q
q /, ))j dIs .
(5.1)
5. Diagonalization of the Hamiltonian For the linearization of the Hamiltonian equations close to the stationary solution (3.1) wc expand the effective Hamiltonian H o - H~ up to second order terms with respect to the canonical variables p(s), q(s), which are considered as the displacements from equilibrium in the moving frame. We use the formulae (3.1), (3.4), (4.2), (4.3) computing k
q(s. t) -- q exp(i(wt + ks)). p(s, t) = p exp(i(wt
ks)).
and obtain the dispersion relation for w = w ( k ) . but the same can be done in the diagonalization procedure of section 6.
l.
x r)- r' ds 0
0
and obtain
[r'l=X/(l
In thc classical approach we should search the solutions in the form of plane waves:
KO)~+(O '
TP): + ( P ' + T Q ) ~ .
From eqs. (4.5) we can put q in place of Q and p in place of P in all the quadratic terms except (1-KQ)2=I-2Kq, which is linear in q. Hence ]
Ir'l = l - Kq + , {
K'-q~ + ( q
t
Tp) ~
+ ( p ' + Tq) ~}
+ O([q[" + Ipl 3 + Iq'l ~ + I p ' [ " ) ,
6. Quantization and diagonalization of the Hamiltonian We assume that the excitations of a vortex filament may be quantized by interpreting q and p as quantummechanical operators that obey Heisenberg's equations of motion ih a q / a t = [q(s),
HI,
ih a p / a t = [p(s). H I . (~.l)
The operators q and p satisfy the canonical c o m m u t a t i o n relation [q(s), p ( s ' ) ] = i h S ( s
s').
((~.2)
283
R. Owczarek, T. Slupski / Quantum vortex waves in superfluid helium
q(s) = L
2
I/2
(6.9)
cok = ~ / k 2 ( k2 - K 2 ) - T k .
We can expand them in a Fourier series: qk e x p ( + i k s ) ,
(6.3)
The operators a k and a~, obey the boson canonical c o m m u t a t i o n relations
p ( s ) = L - ' / 2 ~ , Pk e x p ( - i k s ) , [a k, a~] = 6kl,
where k = 2 v n / L , n = 0 , +1, +2 . . . . . Then, the coefficients p and q obey the canonical commutation relation
For n = 1 we have k~ = 2 " r r / L = V ~
[ qk, Pt]
= ihfkt
~ ~] (k 2
= -
+ T 2.
,
and the quantized Hamiltonian (5.1) becomes H
[a k, a,] = O.
2
,
2
H e n c e c o ( k 1 ) = 0 for each T and K. All frequencies co are real and this is a necessary condition for the stability of the stationary solutions.
,
- K )qkqk + k PkPk
k
- iTk(qkPk
+ Pkqk),
(6.4)
which may be diagonalized by a linear transformation of variables: a k = Ckp k - - i q _ k / 2 h c k ,
(6.5) r Ck
k2
~/2,n-(k 2 _ K 2) "
We see that the Hamiltonian does not depend on P0, hence qo is the constant of motion and we may cancel 0-terms in the Fourier series. In the case of a vortex ring ( T = 0, K > 0), the Hamiltonian does not depend on ql and pl is the constant of motion. H e n c e for a vortex ring we define k m i n = k 2 and calculate kmi . = k 2 = 4"rr/L = 2K . For non-zero torsion T S 0 , we have and obtain kmi n = k! = 2 ~ r / L = "~/T 2 + K 2 .
(6.6) kmi n = k 1
7. S p e c i f i c h e a t
As in ref. [3], the Hamiltonian equation (6.1) may be used to compute the thermal energy of vibration of a single vortex filament with constant K and constant torsion T. We consider two special cases: K = 0, which is the case of a single straight line and K > 0 , T = 0, which is the case of a vortex ring. (It is possible to find also a formula for the case T # 0, but with rather unpleasant integrals, difficult to compute or estimate.) Let us return to the Hamiltonian (2.3), with the coefficient a = K/4"rr l n ( R / a ) . Then formula (6.9) reads to(k) = o~(Ikl(k ~ - K 2 ) ~/2 - T k ) .
It was shown that stable vibration is possible for Ik I/> kmin and then co(k) >10. At the t e m p e r a t u r e T = ( k B [ 3 ) -1 the mean thermal energy may be found assuming that excitations fulfill B o s e - E i n s t e i n statistics. Hence
(6.7) E=
Substitution of (6.5) into (6.4) yields H=
~ ~
hcok(aka ~ + a k a k ) ,
k#O
where the dispersion relation reads
(6.8)
(7.1)
~
Ikl>kmin
( L
hco k exp(h
k)--l+2
1)
"
(7.2)
The second term in this equation represents the energy of zero-point motion and will be omitted, because it does not affect the specific heat. The sum over k may be approximated by an integral:
284
R. O w c z a r e k .
/" m i n
E = ~
T. .~Tupski
' QUaHItCtll v o r t e t
wave~ m .~u/)edtutd helium
implies
':
+
. /'mm 1.".
____
17
In the case of straight vortex lines, wc have co(k) = a k ~ and hence
L f E
.
exp(h/3a(k:
Ix
Lhc,__
]
W
•
x d.t
exp(fi/iak~')- 1
?
.... L ( k , ~ l ) ;
[
L((3)(k ~ "
1
7')':(2hK) '-~ In(R:) B (12
,dE 3(3) dT - 5 (
I
2 kt~(kBT)J
2
/
I
exp(fdBalkl(k:
Kz) ' z)
1
As in Fetter [3], we use the' Hamiltonian lorrealism to ealculatc the average displacement of the vortex core:
( u~(s)).,
Ikl(k: - ~~)' ~ dk
A n u p p e r estimate of the energy can be derived in the following way: for x = ( k 2 ( k ~ - K~-)) ~ ~ w e have k = 2 l"2K(l + ( 1 + 4 x a K 4)1 _~)t 2 and calculate that dk < dx for k > K, and hence
K
/
L
'
J ds(u~(s))
K:)' -')--1 (7.6)
w
(7.8)
/
exp(h/3~lkl(k:
Lf,,~ f
)]'
l
K
_ Lho,_.~ j
'
8. Zero-point motion
K:I' +
~
I 2
In( R:_~)
(7.5)
L (f
E<--
1
For a m a c r o s c o p i c w)rtex ring (1 K ~ I A). the lower and u p p e r estimales are practically equal as the series tend to ( ( 3 / 2 ) . The c o m m o n value is identical with the result for rectilinear w m i c e s (7.4). The same concerns the specific heat (7.5).
in the case of a vortex ring c o ( k ) = ~ l k ] ( k : K2) 1'2 and kmm = K. H e n c e
E:
'
V. [exp(-:~h / & ~ K : (7.4)
~-(2hK)
ln--t l 2
x
:(2ilK)
U-
and the specific heat per unit length b e c o m e s L
[Ke))
exp(fi/&~(x e +
hak ~ dk
~ =
! K ~)) - 1
dx
+o(lql ~ +!pl'~
(~.1) wherc Ya can be calculated from (6.5 1
ke
(k:
as
K2) )
(8.2)
exp(h/3ax-') - 1 I 2
,
,
a 2
(7.7)
and the ensemble average ( a k a ~ + a ~ a a ) is equal to the thermal factor coth( !/3hco,). Hence
T o obtain a lower estimate, we notice that for K > 0, the estimation
{ u-'(s)),, = f l ( p K L )
i N~ c o t h ( f i h w k / 2 ) y a .
[k[(k e - K : ) I : < k z - { K -~
In the case of a vortex ring, co, depends only on
(S.3)
285
R. Owczarek, T. Slupski / Quantum vortex waves in superfluid helium
k 2, therefore we take the factor 2 and sum only for k > 0 . As in eq. (6.6) we have the lower limit k > kmi n = 2K, and according to ref. [3] we introduce also an upper limit
= (~OK/3a)-, J (k-2 + (k 2 _ K 2) 1) dk 2K
-- (avpK/3a)-'K-' ½(1 + In 3)
--rr 2(pK/3a) ' ~ L ( l + l n 3 ) .
(8.8)
(8.4)
kma x = 2'rr/Ami n = 2~r/a ,
Finally we obtain determined by the condition for the shortest wavelength allowed Amin, which has to be equal to the radius a of the vortex core. At zero temperature the thermal factor c o t h ( / 3 h % / 2 ) reduces to 1, and the average zero-displacement is simply
(u2(s) ) av ~ 4 L( OK2/3~ l n ( L / 2 % ) ) - 1 x ¼(1+ I n 3 ) .
(8.9)
9. Discussion
kmax
(8.5)
"y~ ,
Our result for curved vortices
kmin
dT-
which can be approximated by the integral
2
kB(kBT),/:(2h,,) ,/2
krnax
< u2 ( s ) ) a v ~ - 2 h ( p K L )
'~
Lf
y(k) dk
is similar to Fetter's result
kmin = h(pK~r)-X((k
2 --
K2) 1/2
L_ l dE 3 /3\ 1/~ 1/2 d T - 2 ~(-2) k B ( k B T ) "(2hK)-
k=kma - K arctg((k-2 - K 2) 1/2/2K))]k=ZK
x •
(8.6) For kma x >~ K the arctg term is smaller than ½7r, and can be neglected, as well as all the terms of order K. Hence (
u 2 (s))av
~-
h( pK'lT) -'. kma x =- 2h/pKa
,
for T = 0.
(8.7)
Our result for vortex rings at zero temperature is the same as Fetter's case of the straight vortex lines. At the finite non-zero temperature, the coth(x) can be approximated by x ~, and the upper limit kma x c a n be taken +oc (the integral above kma x is small), hence
(U2(S)>
h(~rrpK) -1 ff (2/3hbgk) 1 kmin
l Tk d k
[In( hK ] ] X k \a2kuT/J
,/2
for the " p e r t u r b e d " 2D case of rectilinear vortices. The specific heat associated with the vortex waves varies approximately a s T 1/2 at low temperatures. Fetter's additional logarithmic factor is negligible, because the logarithmic variation is much smaller than T 1/2. It seems that it is connected with the non-local influence of the B i o t Savart interaction, because it has not been obtained in the local approximation. The same concerns eq. (8.9) for the average zerodisplacement. The formulae apply also to the macroscopic vortex ring of radius R >> 1 ~ and to the fully 3D helical spirals. Therefore the validity of the theory for non-interacting curved filaments in three dimensions is now proved. We suppose that the case of interacting vortex filaments (in particular vortex lattices) can be
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R. O w c z a r e k , 7". Slupski / Quantum vortex waves in superltuid helium
investigated in a similar way for the Biot-Savart Hamiltonian. Also some perturbation calculations can be developed if one considers the Biot-Savart Hamiltonian as a small perturbation of the arc-length Hamiltonian.
References [I 1 V. Arnold, Usp, Mat. Nauk 24 (1969) 225. [2] V. Arnold. PWN (19810.
[31 A.L. Fetter. Phys. Rcv. 162 11967) 143. 14l R. Hasimoto, J. Fluid Mech. 51 (1972)477. 15} V. kebedev and I. Khalatniko~, Pis'ma Zh. Eksp. T e o r Fis. 28 11978) 89. [61 V. Lebcdev and I, Khalatnikov, Hamiltoniatl Ilydr~-dynamics equations in the presence of solitons, Zh, Eksp. Teor. Fis. 75 (1978) 2312. [7] J. Marsden, T. Ratiu and A. Weinstein, Trans. Am. Math. Soc. 281 11984) 147. 181 J. Marsden and A. Weinstein. Physica D 7 (1983) 31~5. [9] V. Penna and M. Spera, J. Math. Phys. 31~ (1989) 2778 [I(I] K.W. Schwarz, Phys. Re,,'. B 31 (1985) 5782. [11] T. Slupski, Arch. Mech. 36 11984) 715.