Writing

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~ eimv, x/h (

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0/ (x)

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(13.40)

(0/)

= r eicf> in general, we can obtain the superfluid velocity as

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Ii vs = - vep m

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1 Proof. e-'G 'e'G= '-i[G, ']- 2T[G,[G, ']]

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third term.

(13.41)

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+ ... The series terminates after the

SUPERFLUIDS

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Therefore the phase of (tf; > is the superfluid velocity potential. It follows from (13.40) that the flow the superfluid is irrotational (see, however, Section 13.6):

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V' X Vs =

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(13.42)

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Superfluidity:

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As an Independent Degree of Freedom

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In absolute thermodynamic equilibrium Vs is not independent of the total momentum P of the system. In fact, as we show below,

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Nm(vs >

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(Absolute equilibrium)

(13.43)

which means that there is no relative motion between the centers of mass of normal fluid and superfluid. In nonequilibrium situations, (13.43) need not be true, and the question is one of how long it will take for equilibrium relation (13.43) to be established. We argue that it will take a macroscopically long time. In fact, to the extend that the energy levels (13.36) are valid, the absolute equilibrium can never be established. The reason is as follows. To establish the relation (13.43) momentum must be transferred from the gas of excitations to somewhere else. According to (13.36), however, the excitations are stable, and hence there is no mechanism for momentum transfer. In a more accurate treatment, the elementary excitations should interact with one another and have finite lifetimes. Only in such an accurate treatment can we discuss the approach to absolute equilibrium. We assume that it is a good approximation to take the lifetime of an elementary excitation to be infinite. The kinetic theory which results from this assumption is analogous to the zero-order approximation in the classical kinetic theory of gases and leads to nonviscous hydrodynamics. Specifically, the assumption is that P and Vs are independent variables. Such a treatment does not correspond to the situation of absolute equilibrium; it corresponds instead to situations of quasi-equilibrium. By this assumption, the liquid is endowed with a new degree of freedom, namely, the relative motion between the normal fluid and the superfluid. This new degree of freedom is the essence of the transport phenomena in He II known collectively as superfluidity. The new degree of freedom Vs leads to a two-fluid hydrodynamics, in which the superfluid moves independently of the normal fluid. * To complete the discussion, we supply a proof of (13.43). Let us calculate the partition function of the liquid with the energy levels (13.36), subject to the condition that

Jikn k

Nmvs

(13.44)

where P is a given vector. The partition function will be a function of P and also

'See K. Huang in Studies in Statistical Mechanics, Vol. II, J. De Boer and G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1964).

SPECIAL TOPICS IN STATISTICAL MECHANICS

of N, v, and T, but we leave the latter variables understood. Thus Q(P) = e- PEo where the sum

v,,{n)

exp { - f3 [~Nmv; +

h( W k + k v.)n k ] } (13.4S

extends over all sets {n q} and all values of

~(w)

that satish

(13.461

(13.44). Define the following generating function:

e Pw ' P Q(P)

where the sum over P extends over all vectors of the form

P=-L

27TOh

(13.471

where L = Vl/ 3 and 0 is a vector whose components take on the value~ 0, 1, 2, .... One easily finds

~(w)

= Ve- PEo +(1/2)Npmw' f - - e- N[Ph'k'/2m+g(k)]

(27T )3

(13.48)

d 3k

where

g(k) = vf--Iog [1 - e-Ph(wq+k' q )]

(27T )3

As N

~ 00

we can evaluate

= - -

~(w)

by the method of saddle-point integration:

d 3k

1 - log ~(w)

f3Eo

+ lf3mw 2 - vf --log (1 - e- PhWk ) (13.49)

(27T)3

We now recover Q(P) through

Q(p) = (27Ti)3 dt 1 dt 2 dt 3 t~1+lt~2+1t33+1

~(w)

(13.50)

where tj = exp(27Tf3wjL), n j = (L/27T)Ij, and the contours of integration are circles about tj = 0. We can also write Q(P) The result is

1 1 -logQ(P) = -logQ - - f3(P)2 N N 0 2m N

(13.52)

( - if3 )3 'TT 1 2'TT 12'TT P 12 dW l dW2 dW 3 e- P'W ~(w) v 0 0 0

(13.51)

where Qo is the partition function for Vs = 0. Thus, (n k >is still given by (13.19). Taking the ensemble average of (13.44) and noting that Lkn k = 0, we obtain

P = Nm(vs >

SUPERFLUIDS

13.6 SUPERFLUID FLOW

Anderson's Equations Consider pure superfluid flow in a hydrodynamic regime, in which the phase cP, and therefore the superfluid velocity vs ' varies slowly on an atomic scale. We shall perform coarse graining, by imagining the superfluid (i.e., liquid He 4 in its ground state), being divided into cells containing a large number of atoms, but which are small on a macroscopic scale. We assume that both the phase cp and the particle number N vary only slightly from cell to cell, and average their values over a cell. These average values will be used as macroscopic thermodynamic variables. First let us construct a state of definite phase cp by superposing states of definite particle number N. As a model we choose

ICP) =