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Superfluidity in the new quantum statistical approach
PHYSlCA ELSEVIER
Physica A 246 (1997) 275-287
Superfluidity in the new quantum statistical approach V.A. Golovko Moscow Evening Metallurgical Institute, Lefortovsky Val 26, Moscow 111250, Russia
Received 23 April 1996; received in revised form 24 February 1997
Abstract This paper is a further development of the approach in quantum statistical mechanics proposed by the author. The hierarchy of equations for reduced density matrices obtained previously is extended to the case corresponding to the Bose condensation. The relevant state of the system with a condensate can be superfluid as well as nonsuperfluid. Special attention is given to the thermodynamics of superfluid systems. According to the results of the paper superfluidity is the state of a fluid whose symmetry is spontaneously broken because of a stationary flow. The state corresponds to thermodynamic equilibrium while the magnitude of the flow depends upon the temperature and is determined by thermodynamic considerations. The physical origin of superfluidity, peculiarities of the phenomenon in closed volumes and the critical velocity are discussed as well. PACS: 67.40; 64.60; 05.30 Keywords: Superfluidity; Bose condensation; Quantum statistical thermodynamics; Reduced density matrices
1. Introduction In a previous paper [1] (hereafter referred to as I) it was shown that properties o f an equilibrium quantum system can be treated on the basis o f a hierarchy o f equations for reduced density matrices. However, in the event o f a Bose system the examples considered in 1 show that the equations obtained have no solution at low temperatures, which points out that special investigation is needed in order to extend the approach o f I to that temperature region while bearing in mind that a correct physical theory o f Bose systems must necessarily lead to such a phenomenon as superfluidity. The present paper, which is a continuation o f the work begun in I, deals with the low-temperature behaviour o f a Bose system and especially with superfluidity. One may hope that this will also afford a deeper insight into the phenomenon. 0378-4371/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PHS0378-4371(97)00336-1
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In the present paper we consider the same systems of particles as in I and use the same notation. For the sake of convenience, when referring to an equation of I, we shall place I in front; so, for example, Eq. (1.2.7) will stand for Eq. (2.7) of I.
2. Basic equations The starting point of I is the hypothesis that at thermodynamic equilibrium reduced density matrices (we imply the time-independent term in Eq. (I.2.7)) are of the form (see Eq. (I.2.11))
Rs(xs, x~) ---- Z ns(e~s))~v(Xs)~h*(x',)"
(2.1)
In I it was assumed that the functions ns(~ s)) were smooth and even analytic. However, the smoothness of ns(e~s)) is not obligatory in the hypothesis expressed by Eq. (2.1). Let us suppose now that n~(e~s)) is mainly a smooth function of ~,s) except for a value v--v0 at which its magnitude is larger than that required by the smoothness. Then Eq. (2.1) can be rewritten as
Rs(xs, X~) =Rs(~')(Xs, Xs)+Rs ' (') (xs, x,) '
(2.2)
with (¢)
t
*
t
,% (x,,x,)=~os(x,)q~s (Xs),
(n)
t
Rs (xs' xs) = Z
n.~(e~s))lPv(Xs)Or*(x,),'
(2.3)
Y
where the superscript (c) means the condensate fraction and (n) the normal one (we resort to the terminology used in consideration of the Bose condensation of an ideal gas). In R} n) of Eq. (2.3) we have retained the term with v = v0 upon smoothing down n~(e~s)). Because of this the term RI c) appears in which the function g0s(Xs) differs from ~h~.0(Xs)in the normalization only: Ov0(Xs) is normalized to unity as in I while the normalization of q)s(Xs) includes the unknown quantity ns(e~i~,)). Since the functions ns(z) in Eq. (2.3) are smooth, Eqs. (I.2.16)-(I.2.24) remain valid for R~m(Xs,X~). As to the function (ps(Xs), according to Eq. (I.2.8) it obeys the equation ~2qgs(Xs)+ [e(s) - U,(xs)](ps(X,)= 0,
2m
(2.4)
j--1
in which e(s) is written for e~). When deducing Eqs. (I.2.14) and (I.3.1) no condition on n~(e~s)) was implied, therefore, they can be used as they stand. Analogously with Eq. (2.2), we write for diagonal elements ps(xs) = pSC)(xs) + p~')(x,), with Eq. (I.3.3) for pj')(x~).
p~C)(x~.) = kOs(xs)l =
(2.5)
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Thus, as in I we have again obtained a hierarchy of equations for diagonal elements of reduced density matrices, which contains p,(x,), Us(x,), vs(xs, ms,Z), and now q~(X,.) in addition. The equations are (2.5), (I.2.22), 0.3.1) and (2.4). In order to find n~.(z),e(.~) and the normalization of ~p.~(xs) we turn to the interrelation (I.4.1). By virtue of its linearity it can be satisfied by RI c) and RI ") separately. [t can be shown [2] that we again have Eqs. (I.4.15)-(I.4.17) for n~(z). Note that even in a condensed state n~.(z)is not of the form of a Bose distribution (cf. Appendix H of I). Now we turn to Eq. (I.4.1) as applied to R I"). Upon using the expression for R~,:/ of Eq. (2.3) and assuming that s <{ N we obtain ~ps_l(Xs
1)(/9 s. l ( X s, , _ l )
N1
/ ' ~p,.(x,_l,r~)q~,• (x,_l,r,.)dr, . t
(2.6)
v
Let us find q~l(rl) first. In the present paper we restrict ourselves to spatially homogeneous boundless systems, in which case we can put UI = 0 (see Eq. (1.5.1)). Then Eq. (2.4) at s = 1 becomes 2~1 V2 ~ol (r) + ~;(j)q01(r) = 0.
(2.7)
The solution of this equation, which gives p~l")= constant, is q~l(r)=x/p-Texp
( i~p0r )
~:(I) =p2/2m
,
(2.8)
Here the quantity ~:~) is expressed in terms of another unknown quantity P0, the magnitude of a vector P0; and the normalization factor is denoted by x/~7,. An analysis of Eq. (2.6) for s>~2 shows the following [2]. For fermions Eq. (2.6) entails p, = 0. As should be expected, in the case of spinless fermions the condensate cannot be formed; and hereafter fermions will not be considered. As to bosons, ~:c,'~='sP2/2mwhile the normalization of q~(x,) follows from q~s(x.~) = X~cp{s-I):2Us(Xs) exp
P0
rk
,
where U,.(X,.)~ 1 if all the r j ' s entering into the set x~ diverge. Eq. (2.4) yields the equation for u~(x.,.) h2 " ~m Z VJ2u~(x') + / 1
ihpo + m ~
gu,.(X,.) - U,.(x.,)us(x,.) = 0.
(2.10)
/=l
In homogeneous media we are dealing with, the vector P0 disappears, in fact, from Eq. (2.10) [2]. For example, if s = 2, Eq. (2.10) is similar to the Schr6dinger equation for zero-energy eigenvalue 1~2 ~ ' 2 u a ( r ) -- U 2 ( r ) u 2 ( r ) = 0, m
r -- re
- rl ,
(2.1 1 )
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with the condition that u2(r) ~ 1 as [r[ ~ cx~. This is equivalent to a problem such as the scattering of a particle with the propagation vector k = 0 by a fixed force field. Eq. (2.11) together with the limiting condition can be reduced to the integral equation m f u2(r)+ ~ - -
U2(r')u2(r') , , ~ l ar = 1.
(2.12)
The closed hierarchy obtained above contains four arbitrary constants z, A, Pc and P0. In I such a hierarchy included two arbitrary constants z and A that were determined from thermodynamic considerations and the normalization condition. It is natural to resort to an analogous procedure to determine z, A, Pc and P0, which will be taken up in the next section. In order to clarify the physical bearing of P0, let us calculate the momentum P of the system by making use of a quantum mechanical formula and Eq. (I.2.1) (cf. Eqs. (I.3.5) and (I.3.6)):
P = /~U*(XN)
-ih ZV j
~P(XN)dXN=--ih
[VRt(r,r')]r,=rdr.
(2.13)
j=l
Upon inserting Rl(r,r ~) [2] we get P = p c V p 0 . Hence, the situation is analogous with that in which there are Arc--Pc V particles that move at a speed of po/m. Such a state of the system is thermodynamically equilibrium according to the form of the density matrices as given by Eq. (2.1), and for this reason the existence of a flow in the fuid is not accompanied by any dissipative processes.
3. The thermodynamics of a superfluid Bose system First of all we consider the above hierarchy at s = 1. Implying a spatially uniform medium when Pl = P - N/V, from the equation for Pl and (I.5.4) we obtain [2] v/-ff ,~3/2 P=Pc + 5-A~-•
(3.1)
We shall employ Pn = v/~Ao)z3/2/2 instead of A; then p = Pc + pn. If P0 ¢ 0 the system cannot be isotropic. We assume that the pair correlation function 9(r)=p2(r)/p 2 has a spatial dependence of the form 9 = 9 [ ( x 2 + yZ)l/2,Z] with the z-axis oriented along P0. We introduce the two functions
r
l
/
O(r) = r
0
with r =
Irl.
0
Note that in the case of a spherical symmetry
O(r) = O(r) = 9(r).
(3.3)
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The internal energy E is given by Eq. (I.3.6). Upon substituting R l ( r , r ' ) [2] one obtains, instead of Eq. (I.5.16), / ,
P~
E = 2--~p,.V + _2zPnV + 2r~Np / r 2 K ( r ) O ( r ) d r
.
(3.4)
tl
o
Inasmuch as the system is anisotropic, one must employ the stress tensor instead of the pressure. In the general case (gases, liquids, crystals), the stress tensor aij is [2]
(rij=-~
dr + ~ j
Rl(r,r')
;
c?xi
(pz(r',r' + r ) ) d r ,
(3.5)
r / ~g
where (.-.) denotes a space average according to
,/
(p2(r', r' + r)) = ~
pz(r', r' + r) d r ' .
(3.6)
For spatially homogeneous media when P2 = p 2 ( r 2 - r l ), one has (p2(r', r' + r)) = p2(r). Let p = - axx = - ayy be the pressure on a surface element normal to the x- or y-axis, and p + Ap = - azz the one on such an element normal to the z-axis. From Eq. (3.5), one has d
:rcp3 / r3 ~ r [g(t(r) - O(r)]dr , p =
-
(3.7)
-5-
o CX3
Ap = pcP--~ - ~p2
r3
[0(r) - 0(r)] dr.
(3.8)
0
Here, and in what follows, we use the notation ~ = pnz/p = ( l - f ~ . ) z where f~, = p,./p =
1 - Pn/P. According to the ideas of I we now turn to the second law of thermodynamics for quasi-static processes. Suppose that the system is enclosed in a cylinder whose axis is directed along P0, the length of the cylinder being L and its cross-section V/L [2]. The work done in changing the dimensions of the cylinder is 6A = p d V + ApV dL/L. Now the second law assumes the form
OdS=dE + 6A=-~dO
+
dV +
dL.
(3.9)
In addition to Eq. (I.5.19), the condition that the entropy S shall exist yields now two more equations [2]
0 OAp 00 _ Ap = O,
P ~Ap o--~p - A p = O .
(3.10)
The two equations of Eq. (3.10) have a unique simultaneous solution Ap=CoOp where Co is a constant independent of ~9 and p. In the isotropic high-temperature
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phase Ap = 0, therefore Co = 0, Consequently, always Ap = 0 even if P0 ~ 0, that is to say, notwithstanding the existence of a flow in the superfluid, Pascal's law is fulfilled. Now from Eq. (3.8), it follows that
pcp2= ~zmp2f r3~[O(r) - O(r)]dr.
(3.11)
0
By virtue of this the expression for E, Eq. (3.4), transforms to
E= ~?N + 2rcNp/r2K(r)O(r)dr+ 2NP o
r3
[O(r)-O(r)]dr.
(3.12)
o
Upon substituting Eqs. (3.7) and (3.12) into Eq. (I.5.19) and integrating by parts, one obtains the equation (cf. Eq. (I.5.20)) ~ 0F 2 0 • + 3p~pp - 2? Oc
= ~zP3 drr 2 K(r) 3 ( 0 - 0 ) - l Z p ~ p +
r
3
+~r
0
+r~r
c30
0~--~)+ 3p (~--~
fffip)] } .
(3.13)
We need also the Helmholtz free energy F. One may use (I.5.28) written in a general form [2]. Here, however, it is preferable to utilize the expression derived in Ref. [2]:
F(O,p)=2rcpNf drr2(2K+rdK) dr J f a o
d~O(r,O~2,p~3)
1 pOlo~o)3 2
oN + ~o
fJ
+ + ~oF(tgo, o p(Oo,y)-fi~ Po),
(3.14)
Po
where 00 and P0 are arbitrary constants. Let us now discuss Eq. (3.13). This equation is, in fact, an equation for two unknown functions ?(O,p) and z(O,p) instead of which it is more convenient to work with ?(O,p) and fc(zg,p). In addition to Eq. (3.13), it is necessary to obtain a second equation relating ? and fc. One can proceed from the fact that at given 0 and p the state of thermal equilibrium corresponds to a minimum of F. Minimization of F from Eq. (3.14) yields the equation [2]
Ofc
r 2 2K+r
O(r,?,f~,p)dr=O.
(3.15)
o
This provides the second equation for ? and ft. Moreover, Eq. (3.15) determines the phase transition line in the p-0 plane if one puts f~.=0, 0 = 9 and ? = r ( 0 , p ) with
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r(vg, p) relevant to the high-temperature phase studied in I. The equality ? = z on the line serves as boundary condition for Eq. (3.13). In reality, that line is a line where another solution of the above equations bifurcates off the high-temperature solution. The bifurcation may or may not correspond to a phase transition. This question as well as the relation with the Landau phenomenological theory of phase transitions are discussed in Ref. [2]. In this problem another situation is possible. Before the free energy reaches a minimum there may happen a breakdown of the condition of the mechanical stability of the system (@/c~V)o ~0. Just this situation occurs in an ideal Bose gas below the condensation point [2]. The case of an arbitrary system whose thermodynamics is governed not by the principle of minimum free energy but by the equation (@/?V),~ = 0 is also considered in Ref. [2].
4. Some consequences of the equation obtained In this section we shall adduce some consequences of the foregoing considerations, referring to Ref. [2] for details. The above equations permit one to get the following expression for the pair correlation function g ( r ) = p2(r)/p 2 (cf. Eq. (I.5.7)) .q(r) = P~u2(r)+
p
lfdqfdsn(s)v(r,q,s)[l+exp(~qr)].(4.1) 2ni(2v/-2nh)3p c
The functions v(r,q,s) and u2(r) are determined by Eq. (1.5.6) (one should write U2(r) instead of Uz(Ir])) and by Eq. (2.11) (or Eq. (2.12)), while U2(r) is specified by Eq. (I.3.1) with s = 2 where one must use some approximation or other for p3(rl, r:,r3) (see Section 5 of I). Hence, one has four equations for the four functions g, c., u: and U2. Once the equations are solved one can consider thermodynamic aspects and superfluid properties by leaning on the results of Section 3. Note that characteristic of the relevant state of the system are long-range spatial correlations decaying as I/jr I, which stems from Eqs. (2.12) and (4.1). The equations obtained can have spherically symmetric solutions analogously with the equations of 1. Then P0 = 0 according to Eqs. (3.3) and (3.11). Such a state of the system may be called the condensate phase without superfluidity. Therefore, tbrmation of a condensate does not necessarily lead to superfluidity. In particular, superfluidity cannot exist in an ideal gas and even in a slightly nonideal Bose system [2]. Superfluidity can be observed only if the interparticle interaction is sufficiently strong, in which case the equations obtained may have solutions without spherical symmetry. From this viewpoint let us discuss Eq. (2.11) that coincides with the Schr6dinger equation for zero-energy eigenvalue. Even if the potential is spherically symmetric, besides spherically symmetric solutions the Schr6dinger equation can, for special cases of the potential, have zero-energy solutions that are not spherically symmetric (see, e.g., Ref. [3, Section 15], where one considers bound states that appear with zero energy).
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Of course, an eigenvalue that is exactly zero can occur only in an exceptional case of the potential. However, the potential U2(r) is not a given quantity, and Uz(r) may adjust itself to the required form. The adjustment may be facilitated if the interaction potential K(r) has a real or virtual level near zero. Just such a level exists in the case of helium atoms [4,5]. Perhaps the superfluidity of helium is due to this peculiarity of the interatomic potential. On lowering the temperature there may appear various, both spherically symmetric and nonsymmetric, solutions of Eq. (2.11) which correspond to different values of the quantum number l, i.e., of the angular momentum in the usual interpretation of the Schr6dinger equation (note that P0 ~ 0 only if I # 0). Consequently, condensate phases of different types are possible, both superfluid and without superfluidity. One will observe the phase that provides an absolute minimum of the appropriate thermodynamic potential.
5. Hard spheres under the neglect of triplet correlations It is natural to try to solve the above equations in the case of a hard-sphere system wherein triplet correlations are neglected, treated in detail in I. If U 2 ( r ) = K ( r ) with K(r) defined by Eq. (I.6.1), Eq. (2.11) has a unique solution of the form a u2(r) -----1 -- - , r = Ir I , (5.1) r
subject to two conditions: u2(a) = 0 and ue(r) --* 1 as r --~ oc. The solution of Eq. (I.5.6) is the same as in I. Now Eq. (4.1) enables us to calculate y(r). Upon denoting the right-hand side of Eq. (I.6.3) by #r(r) we shall obtain [2] 9(r) = 1 - -2a ~ f c (1 - ~ r ) + ( 1
- f c ) [ y l ( r ) 1 -]
(5.2)
.
This function is spherically symmetric, which gives po = 0 on account of Eqs. (3.3) and (3.11 ). Hence, in the present case we have a condensate phase without superfluidity. Eq. (3.13) that determines ~(O,p) now becomes [2] 4rcah2p { 3m fc + (1 - f c ) H ( ? ) -~2 H (r)~--0 - [I - H(e) + fHl(f)]t9
,
(5.3)
where H ( ( ) is given by Eq. (1.6.6) with the upper sign, the prime over HI(f) denotes differentiation with respect to the argument, and f = a 2 m ~ - 2 ( 1 - f c ) -1. Now, it is necessary to find the point at which the above solution bifurcates off the high-temperature solution considered in I. Eq. (3.15) that determines this point assumes the form H(f') - 1 - f H ' ( f ) = 0.
(5.4)
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P
\
V1
Fig. 1. Isotherms of a hard-sphere Bose system under the neglect of triplet correlations. The specific volumes vt = l/pl and v2 = l/p2 are given by Eqs. (5.5) and (5.6), respectively.
However, an analysis shows that for the considered potential U2(r) always H ( ~ ) - 1 ( H ' ( ( ) > 0. Thus, Eq. (5.4) has no solution. Therefore, the bifurcation does not occur until the stability criterion ( @ / ~ V ) o < 0 breaks down. Afterwards, there emerges a state mentioned at the end of Section 3 and characterized by the equation (~p/OV)o = 0. The corresponding point is given by Eqs. (1.6.25) and (I.6.26). Those equations can be recast in the following parametric form:
--
Jzh 2aG-l(/~l )'
P!
= ~931"2 Go(fll
)'
(5.5)
fll being the parameter. As distinct to Eqs. (I.6.25) and (1.6.26) these equations are correct to the order a scx n 5/3. Below the point in question the isotherms have a horizontal part, as illustrated in Fig. 1 . As the specific volume v = l i p decreases from vl the condensate fraction J~ increases from zero. Ultimately it reaches the value f~ = 1 whereas values j~. > 1 make no physical sense. Hence, there is another special point v = v2 (or 0 = 02 ¢ 0) at which f , = 1. This point is given by
2aG-l(fl2)'
P2 =(2 + "~)6o~'"2G0(/~2),
(5.6)
where fi2 serves as parameter [2]. In a first approximation in a Eqs. (5.6) yield
r
P2
]
2/3
.
02 = [(2 + "~)~3o)J Note that to the same approximation Eq. (5.5) gives Oi = (pl/{3(o) 2/3.
(5.7)
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The behaviour of the isotherms at v < v2 when fc---1 is sketched in Fig. 1 and analysed in Ref. [2]. If n is small (recall that n = rca3p), the isotherm is presented by the solid line. It is typical of a first-order phase transition. If n2 > 6.04 × 10 -2, the isotherm has the form of the broken curve. In the last case we have two continuous phase transitions, at v = Vl and v = v2. It is not clear, however, whether this case can be observed in reality because triplet correlations that have been discarded may play an important part in the condensed region even if n is rather small [2]. A comparison of the present results with the known studies on the Bose condensation in a hard-sphere system [ 6 - 8 ] is made in Ref. [2]. Here we observe only that Lee and Yang [7] noted that in their work it was not possible to state exactly the order of the transition for a ¢ 0; this order was not determined uniquely in Refs. [6,8] too. In Ref. [9] Huang writes cautiously that the transition appears to be a second-order one while the argument of his own [6,8] that the transition may be of first order remains unrefuted. Let us point out some peculiarities of the condensate phase that follow from the present approach (see Ref. [2] for details). The properties of the condensate phase are considerably affected by details of correlations and of the interaction potential. This signifies that the properties should depend noticeably upon conditions of the experiment, impurities in the liquid, inhomogeneities, etc., which may account for discrepancies between the values of critical velocities in helium observed in different experiments [10]. Concluding this section we remark that in the case considered in the section Eq. (3.15) had no solution, which resulted in the isotherms of Fig. 1. At the same time the example of a weakly interacting system [2] shows that Eq. (3.15) may have a solution, so that the phase transition to a condensate phase may be second order as well, depending on the interaction potential.
6. Discussion and concluding remarks The present paper shows that within the scope of the approach proposed in I lies also such a phenomenon as superfluidity. The approach indicates that the phenomenon embodies symmetry breaking in a fluid which is due to formation of a spontaneous stationary flow, the state with the flow being thermodynamically equilibrium while the magnitude of the flow being determined by thermodynamic relations. This is analogous to the appearance of spontaneous magnetization below the Curie point in a ferromagnet, the value of the magnetization being given by thermodynamics. It should be emphasized that the quantity P0 which determines the flow appears in the theory quite naturally since there is no reason to put e(1)=0 in Eq. (2.7). Thus, no additional argument is needed to explain the existence of the frictionless flow so far as no dissipation of energy can occur in thermal equilibrium. Let us turn now to the question of how to elucidate superfluidity physically. We note first that incessant motion without any dissipation of energy is characteristic of quantum mechanics. As an example, we may mention the motion of electrons in an atom while
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the atom, let alone a molecule, can contain up to a hundred electrons. The motion may manifest itself externally by the existence of an orbital angular momentum L. Recall that not all electrons contribute to the total angular momentum although all the electrons constitute a unified dynamical system. A superfluid may be regarded as a gigantic atom whose radius is infinite, and in which for this reason the orbital motion is converted to a rectilinear one that is characterized by a linear momentum P instead of the angular momentum L. The gigantic atom differs from an ordinary one in that the role of electrons is played by neutral atoms which move in a self-consistent field, not all of them contributing to P (cf. the discussion of Eq. (2.13)). Here again we may draw a parallel between ferromagnetism and superfluidity. Ferromagnetism may be regarded as a macroscopic manifestation of the spin angular momentum while superfluidity is a peculiar macroscopic manifestation of the orbital angular momentum. In an ordinary atom both states with L ~ 0 and L = 0 are possible; analogously, the condensate phase may be either superfluid (P0 ¢ 0) or nonsuperfluid (P0 = 0). A similar view may be expressed as to explanation of superconductivity. In a superconductor charged particles, electrons, are entrained in the collective quantum motion under discussion, which gives rise to a supercurrent. From this point of view, high-temperature superconductivity is explained probably by a singular structure of relevant compounds, which favours the motion at high temperatures. In order to consider superconductivity in this way it is necessary first to extend the approach developed in I to systems containing nonzero spin particles, which will be the subject of a subsequent paper. In this context the following should be noted. Theories of superconductivity (superfluidity of electrons) based upon an idea that the phenomenon is due to spontaneous currents existing in a condition of thermodynamic equilibrium, which is analogous with the results of the present paper, were proposed as far back as the 1930s (for a review see Ref. [11], a critical analysis of such theories can be found in Ref. [12] as well). A characteristic paper along these lines is that of Landau [13]. Landau assumed that a superconductor contains local saturation currents flowing in different directions and yielding no resultant current in the absence of an external field. His study was not based upon quantum mechanics, being phenomenological, in essence, without convincing argumentation. All those theories were not sufficiently founded and were abandoned later. The present approach shows that there is a preferred direction specified by P0, which implies an anisotropy in the velocity distribution. In the recent experiments on Bose condensation in a gas of alkali atoms [14,15] one observed such an anisotropy. At the same time a direct comparison of our results with the experimental ones is impossible because in the present paper an idealized case is considered, in which the system occupies an infinite volume, whereas real volumes are always finite. Let us discuss manifestation of superfluidity in finite systems. In a confined volume flows must close upon themselves. It is quite possible that the volume will break up into cells in each of which the fluid will circulate along closed streamlines, forming a pattern similar to that observed, for example, in Rayleigh-B6nard convection (for a recent review see, e.g., Ref. [16]). Here again an analogy with a ferromagnet is in place: the
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cells in the superfluid are analogous to domains in the ferromagnet. At the same time there is a great difference. Formation of magnetic domains is energetically profitable because owing to them the magnetic energy of the specimen decreases. Curvature of the flow and formation of the cells in a superfluid will most likely be energetically unfavourable. Hence in some cases, especially just below the 2 point, the cells may not develop and one will observe a metastable condensate phase without superfluidity; in other cases, under appropriate circumstances, a cell pattern may arise. As to the motion of the superfluid near walls of the vessel we note that atoms of the walls may adjust themselves to the quantum motion since their temperature is low as well. When trying to visualize the flow pattern one may meet with some peculiarities for the superflow does not exert an extra pressure along the streamlines (see the discussion of Eq. (3.10)) as distinct from a flow in an ordinary liquid. The existence of the intrinsic flow explains readily the known fact that helium II crawls up the walls, out of an open vessel (see, e.g., Ref. [9]). This sheds further light on the fountain effect as well. The present ideas offer also a simple explanation for the existence of a critical velocity. If a superfluid in which the foregoing pattern has formed is subjected to an external action, be it a gradient of pressure or temperature, some of the closed flows will open and the superfluid will partially flow along the direction of the gradient. Here again we have an analogy with a ferromagnet in which an external magnetic field reorients the magnetic moments of the domains. To the saturation magnetization, when the specimen becomes one domain, will correspond a state of the superfluid in which the whole flow will be directed along the gradient. A further increase in the total flow is impossible without breaking the condition of thermodynamic equilibrium, which will result in viscosity. Hence, the critical velocity is determined by the formula V c = p o / m according to the discussion following Eq. (2.13), where P0 as a function of the temperature and density is given by Eq. (3.11). In narrow and long capillaries where the creation of closed flows is hindered superfluidity will manifest itself easier, which conforms to the experiment. The reasoning remains practically the same if the formation of the cells is energetically unfavourable; a temeperature or pressure gradient will again facilitate creation of a directional flow, and the critical velocity will again be given by Vc = po/m. In Landau's explanation of superfluidity [17] the critical velocity is of the same order of magnitude as the velocity of sound, that is, it is too large; in the present theory the critical velocity is not at all related with the velocity of sound as seen from Eq. (3.11). If superfluidity is regarded as spontaneous symmetry breaking, the critical velocity may be considered the order parameter vanishing at the transition point (see also Ref. [2]). A word should be said about the vortices. When the velocity of the flow exceeds the critical velocity, the equilibrium superfluid state breaks down, which gives rise to vortices. From this viewpoint the breakdown of superfluidity is the cause for creation of vortices. In Feynman's argument [18] this causal chain is reversed; the creation of vortices is the cause and the breakdown of superfluidity is the effect. It should be stressed that the cells discussed above have nothing to do with the vortices.
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Let us discuss now the phases that were called the condensate phases without superfluidity. We revert again to the analogy with an atom. If in the ground state of the atom the orbital angular momentum L is zero, an excited state can be such that L ¢ 0. In a superfluid the role of the exciting agent can be played by a temperature or pressure gradient. Consequently, the P0 = 0 state can be converted to a metastable P0 ~ 0 state. Once a quantum system goes to another state its subsequent behaviour does not depend upon the exciting agent (if there is no other transition). Therefore, in the emerging state of the superfluid the quantity P0 will be determined by internal properties of the superfluid, that is to say, by Eq. (3.11 ). On the whole, the properties of the metastable P0 ¢ 0 state should be analogous with those of a genuine superfluid. The question of lifetime of the metastable state lies beyond the scope of equilibrium statistical mechanics. In the case of a superconductor the role of the exciting agent in question may be played by not only an external voltage but also a contact potential.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
V.A. Golovko, Physica A 230 (1996) 658. V.A. Golovko, the full version of the present paper, unpublished. L.I. Schiff, Quantum mechanics, McGraw-Hill, New York, 1986. J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1964. Y.-H. Uang, W.C. Stwalley, J. Chem. Phys. 76 (1982) 5069. K. Huang, C.N. Yang, J.M. Luttinger, Phys. Rev. 105 (1957) 776. T.D. Lee, C.N. Yang, Phys. Rev. 112 (1958) 1419. K. Huang, in: J. de Boer, G.E. Uhlenbeck (Eds.), Studies in Statistical Mechanics, vol. 2, North-Holland Amsterdam, 1964, part A. K. Huang, Statistical Mechanics, 2nd edn, Wiley, New York, 1987. W.E. Keller, Helium-3 and Helium-4, Plenum, New York, 1969. L. Hoddeson, G. Baym, M. Eckert, Rev. Mod. Phys. 59 (1987) 287. V.L. Ginzburg, Usp. Fiz. Nauk 48 (1952) 25. L. Landau, Phys. Z. Sowjetunion 4 (1933) 43. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269 (1995) 198. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle. Phys. Rev. Lett. 75 (1995) 3969. A.V. Getling, Usp. Fiz. Nauk 161 (1991) 1 (English translation) Sov. Phys. Usp. 34 (1991) 101. L.D. Landau, J. Phys. (USSR) 5 (1941) 71. R.P. Feynman, in: C.J. Gorter (Ed.), Progress in Low Temperature Physics, vol. 1, lnterscience. New York: North-Holland, Amsterdam, 1955, ch. 2.
Physica A 246 (1997) 275-287
Superfluidity in the new quantum statistical approach V.A. Golovko Moscow Evening Metallurgical Institute, Lefortovsky Val 26, Moscow 111250, Russia
Received 23 April 1996; received in revised form 24 February 1997
Abstract This paper is a further development of the approach in quantum statistical mechanics proposed by the author. The hierarchy of equations for reduced density matrices obtained previously is extended to the case corresponding to the Bose condensation. The relevant state of the system with a condensate can be superfluid as well as nonsuperfluid. Special attention is given to the thermodynamics of superfluid systems. According to the results of the paper superfluidity is the state of a fluid whose symmetry is spontaneously broken because of a stationary flow. The state corresponds to thermodynamic equilibrium while the magnitude of the flow depends upon the temperature and is determined by thermodynamic considerations. The physical origin of superfluidity, peculiarities of the phenomenon in closed volumes and the critical velocity are discussed as well. PACS: 67.40; 64.60; 05.30 Keywords: Superfluidity; Bose condensation; Quantum statistical thermodynamics; Reduced density matrices
1. Introduction In a previous paper [1] (hereafter referred to as I) it was shown that properties o f an equilibrium quantum system can be treated on the basis o f a hierarchy o f equations for reduced density matrices. However, in the event o f a Bose system the examples considered in 1 show that the equations obtained have no solution at low temperatures, which points out that special investigation is needed in order to extend the approach o f I to that temperature region while bearing in mind that a correct physical theory o f Bose systems must necessarily lead to such a phenomenon as superfluidity. The present paper, which is a continuation o f the work begun in I, deals with the low-temperature behaviour o f a Bose system and especially with superfluidity. One may hope that this will also afford a deeper insight into the phenomenon. 0378-4371/97/$17.00 Copyright (~ 1997 Elsevier Science B.V. All rights reserved PHS0378-4371(97)00336-1
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In the present paper we consider the same systems of particles as in I and use the same notation. For the sake of convenience, when referring to an equation of I, we shall place I in front; so, for example, Eq. (1.2.7) will stand for Eq. (2.7) of I.
2. Basic equations The starting point of I is the hypothesis that at thermodynamic equilibrium reduced density matrices (we imply the time-independent term in Eq. (I.2.7)) are of the form (see Eq. (I.2.11))
Rs(xs, x~) ---- Z ns(e~s))~v(Xs)~h*(x',)"
(2.1)
In I it was assumed that the functions ns(~ s)) were smooth and even analytic. However, the smoothness of ns(e~s)) is not obligatory in the hypothesis expressed by Eq. (2.1). Let us suppose now that n~(e~s)) is mainly a smooth function of ~,s) except for a value v--v0 at which its magnitude is larger than that required by the smoothness. Then Eq. (2.1) can be rewritten as
Rs(xs, X~) =Rs(~')(Xs, Xs)+Rs ' (') (xs, x,) '
(2.2)
with (¢)
t
*
t
,% (x,,x,)=~os(x,)q~s (Xs),
(n)
t
Rs (xs' xs) = Z
n.~(e~s))lPv(Xs)Or*(x,),'
(2.3)
Y
where the superscript (c) means the condensate fraction and (n) the normal one (we resort to the terminology used in consideration of the Bose condensation of an ideal gas). In R} n) of Eq. (2.3) we have retained the term with v = v0 upon smoothing down n~(e~s)). Because of this the term RI c) appears in which the function g0s(Xs) differs from ~h~.0(Xs)in the normalization only: Ov0(Xs) is normalized to unity as in I while the normalization of q)s(Xs) includes the unknown quantity ns(e~i~,)). Since the functions ns(z) in Eq. (2.3) are smooth, Eqs. (I.2.16)-(I.2.24) remain valid for R~m(Xs,X~). As to the function (ps(Xs), according to Eq. (I.2.8) it obeys the equation ~2qgs(Xs)+ [e(s) - U,(xs)](ps(X,)= 0,
2m
(2.4)
j--1
in which e(s) is written for e~). When deducing Eqs. (I.2.14) and (I.3.1) no condition on n~(e~s)) was implied, therefore, they can be used as they stand. Analogously with Eq. (2.2), we write for diagonal elements ps(xs) = pSC)(xs) + p~')(x,), with Eq. (I.3.3) for pj')(x~).
p~C)(x~.) = kOs(xs)l =
(2.5)
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Thus, as in I we have again obtained a hierarchy of equations for diagonal elements of reduced density matrices, which contains p,(x,), Us(x,), vs(xs, ms,Z), and now q~(X,.) in addition. The equations are (2.5), (I.2.22), 0.3.1) and (2.4). In order to find n~.(z),e(.~) and the normalization of ~p.~(xs) we turn to the interrelation (I.4.1). By virtue of its linearity it can be satisfied by RI c) and RI ") separately. [t can be shown [2] that we again have Eqs. (I.4.15)-(I.4.17) for n~(z). Note that even in a condensed state n~.(z)is not of the form of a Bose distribution (cf. Appendix H of I). Now we turn to Eq. (I.4.1) as applied to R I"). Upon using the expression for R~,:/ of Eq. (2.3) and assuming that s <{ N we obtain ~ps_l(Xs
1)(/9 s. l ( X s, , _ l )
N1
/ ' ~p,.(x,_l,r~)q~,• (x,_l,r,.)dr, . t
(2.6)
v
Let us find q~l(rl) first. In the present paper we restrict ourselves to spatially homogeneous boundless systems, in which case we can put UI = 0 (see Eq. (1.5.1)). Then Eq. (2.4) at s = 1 becomes 2~1 V2 ~ol (r) + ~;(j)q01(r) = 0.
(2.7)
The solution of this equation, which gives p~l")= constant, is q~l(r)=x/p-Texp
( i~p0r )
~:(I) =p2/2m
,
(2.8)
Here the quantity ~:~) is expressed in terms of another unknown quantity P0, the magnitude of a vector P0; and the normalization factor is denoted by x/~7,. An analysis of Eq. (2.6) for s>~2 shows the following [2]. For fermions Eq. (2.6) entails p, = 0. As should be expected, in the case of spinless fermions the condensate cannot be formed; and hereafter fermions will not be considered. As to bosons, ~:c,'~='sP2/2mwhile the normalization of q~(x,) follows from q~s(x.~) = X~cp{s-I):2Us(Xs) exp
P0
rk
,
where U,.(X,.)~ 1 if all the r j ' s entering into the set x~ diverge. Eq. (2.4) yields the equation for u~(x.,.) h2 " ~m Z VJ2u~(x') + / 1
ihpo + m ~
gu,.(X,.) - U,.(x.,)us(x,.) = 0.
(2.10)
/=l
In homogeneous media we are dealing with, the vector P0 disappears, in fact, from Eq. (2.10) [2]. For example, if s = 2, Eq. (2.10) is similar to the Schr6dinger equation for zero-energy eigenvalue 1~2 ~ ' 2 u a ( r ) -- U 2 ( r ) u 2 ( r ) = 0, m
r -- re
- rl ,
(2.1 1 )
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with the condition that u2(r) ~ 1 as [r[ ~ cx~. This is equivalent to a problem such as the scattering of a particle with the propagation vector k = 0 by a fixed force field. Eq. (2.11) together with the limiting condition can be reduced to the integral equation m f u2(r)+ ~ - -
U2(r')u2(r') , , ~ l ar = 1.
(2.12)
The closed hierarchy obtained above contains four arbitrary constants z, A, Pc and P0. In I such a hierarchy included two arbitrary constants z and A that were determined from thermodynamic considerations and the normalization condition. It is natural to resort to an analogous procedure to determine z, A, Pc and P0, which will be taken up in the next section. In order to clarify the physical bearing of P0, let us calculate the momentum P of the system by making use of a quantum mechanical formula and Eq. (I.2.1) (cf. Eqs. (I.3.5) and (I.3.6)):
P = /~U*(XN)
-ih ZV j
~P(XN)dXN=--ih
[VRt(r,r')]r,=rdr.
(2.13)
j=l
Upon inserting Rl(r,r ~) [2] we get P = p c V p 0 . Hence, the situation is analogous with that in which there are Arc--Pc V particles that move at a speed of po/m. Such a state of the system is thermodynamically equilibrium according to the form of the density matrices as given by Eq. (2.1), and for this reason the existence of a flow in the fuid is not accompanied by any dissipative processes.
3. The thermodynamics of a superfluid Bose system First of all we consider the above hierarchy at s = 1. Implying a spatially uniform medium when Pl = P - N/V, from the equation for Pl and (I.5.4) we obtain [2] v/-ff ,~3/2 P=Pc + 5-A~-•
(3.1)
We shall employ Pn = v/~Ao)z3/2/2 instead of A; then p = Pc + pn. If P0 ¢ 0 the system cannot be isotropic. We assume that the pair correlation function 9(r)=p2(r)/p 2 has a spatial dependence of the form 9 = 9 [ ( x 2 + yZ)l/2,Z] with the z-axis oriented along P0. We introduce the two functions
r
l
/
O(r) = r
0
with r =
Irl.
0
Note that in the case of a spherical symmetry
O(r) = O(r) = 9(r).
(3.3)
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279
The internal energy E is given by Eq. (I.3.6). Upon substituting R l ( r , r ' ) [2] one obtains, instead of Eq. (I.5.16), / ,
P~
E = 2--~p,.V + _2zPnV + 2r~Np / r 2 K ( r ) O ( r ) d r
.
(3.4)
tl
o
Inasmuch as the system is anisotropic, one must employ the stress tensor instead of the pressure. In the general case (gases, liquids, crystals), the stress tensor aij is [2]
(rij=-~
dr + ~ j
Rl(r,r')
;
c?xi
(pz(r',r' + r ) ) d r ,
(3.5)
r / ~g
where (.-.) denotes a space average according to
,/
(p2(r', r' + r)) = ~
pz(r', r' + r) d r ' .
(3.6)
For spatially homogeneous media when P2 = p 2 ( r 2 - r l ), one has (p2(r', r' + r)) = p2(r). Let p = - axx = - ayy be the pressure on a surface element normal to the x- or y-axis, and p + Ap = - azz the one on such an element normal to the z-axis. From Eq. (3.5), one has d
:rcp3 / r3 ~ r [g(t(r) - O(r)]dr , p =
-
(3.7)
-5-
o CX3
Ap = pcP--~ - ~p2
r3
[0(r) - 0(r)] dr.
(3.8)
0
Here, and in what follows, we use the notation ~ = pnz/p = ( l - f ~ . ) z where f~, = p,./p =
1 - Pn/P. According to the ideas of I we now turn to the second law of thermodynamics for quasi-static processes. Suppose that the system is enclosed in a cylinder whose axis is directed along P0, the length of the cylinder being L and its cross-section V/L [2]. The work done in changing the dimensions of the cylinder is 6A = p d V + ApV dL/L. Now the second law assumes the form
OdS=dE + 6A=-~dO
+
dV +
dL.
(3.9)
In addition to Eq. (I.5.19), the condition that the entropy S shall exist yields now two more equations [2]
0 OAp 00 _ Ap = O,
P ~Ap o--~p - A p = O .
(3.10)
The two equations of Eq. (3.10) have a unique simultaneous solution Ap=CoOp where Co is a constant independent of ~9 and p. In the isotropic high-temperature
V.A. Golovko/Physica A 246 (1997) 2 7 5 ~ 8 7
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phase Ap = 0, therefore Co = 0, Consequently, always Ap = 0 even if P0 ~ 0, that is to say, notwithstanding the existence of a flow in the superfluid, Pascal's law is fulfilled. Now from Eq. (3.8), it follows that
pcp2= ~zmp2f r3~[O(r) - O(r)]dr.
(3.11)
0
By virtue of this the expression for E, Eq. (3.4), transforms to
E= ~?N + 2rcNp/r2K(r)O(r)dr+ 2NP o
r3
[O(r)-O(r)]dr.
(3.12)
o
Upon substituting Eqs. (3.7) and (3.12) into Eq. (I.5.19) and integrating by parts, one obtains the equation (cf. Eq. (I.5.20)) ~ 0F 2 0 • + 3p~pp - 2? Oc
= ~zP3 drr 2 K(r) 3 ( 0 - 0 ) - l Z p ~ p +
r
3
+~r
0
+r~r
c30
0~--~)+ 3p (~--~
fffip)] } .
(3.13)
We need also the Helmholtz free energy F. One may use (I.5.28) written in a general form [2]. Here, however, it is preferable to utilize the expression derived in Ref. [2]:
F(O,p)=2rcpNf drr2(2K+rdK) dr J f a o
d~O(r,O~2,p~3)
1 pOlo~o)3 2
oN + ~o
fJ
+ + ~oF(tgo, o p(Oo,y)-fi~ Po),
(3.14)
Po
where 00 and P0 are arbitrary constants. Let us now discuss Eq. (3.13). This equation is, in fact, an equation for two unknown functions ?(O,p) and z(O,p) instead of which it is more convenient to work with ?(O,p) and fc(zg,p). In addition to Eq. (3.13), it is necessary to obtain a second equation relating ? and fc. One can proceed from the fact that at given 0 and p the state of thermal equilibrium corresponds to a minimum of F. Minimization of F from Eq. (3.14) yields the equation [2]
Ofc
r 2 2K+r
O(r,?,f~,p)dr=O.
(3.15)
o
This provides the second equation for ? and ft. Moreover, Eq. (3.15) determines the phase transition line in the p-0 plane if one puts f~.=0, 0 = 9 and ? = r ( 0 , p ) with
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r(vg, p) relevant to the high-temperature phase studied in I. The equality ? = z on the line serves as boundary condition for Eq. (3.13). In reality, that line is a line where another solution of the above equations bifurcates off the high-temperature solution. The bifurcation may or may not correspond to a phase transition. This question as well as the relation with the Landau phenomenological theory of phase transitions are discussed in Ref. [2]. In this problem another situation is possible. Before the free energy reaches a minimum there may happen a breakdown of the condition of the mechanical stability of the system (@/c~V)o ~0. Just this situation occurs in an ideal Bose gas below the condensation point [2]. The case of an arbitrary system whose thermodynamics is governed not by the principle of minimum free energy but by the equation (@/?V),~ = 0 is also considered in Ref. [2].
4. Some consequences of the equation obtained In this section we shall adduce some consequences of the foregoing considerations, referring to Ref. [2] for details. The above equations permit one to get the following expression for the pair correlation function g ( r ) = p2(r)/p 2 (cf. Eq. (I.5.7)) .q(r) = P~u2(r)+
p
lfdqfdsn(s)v(r,q,s)[l+exp(~qr)].(4.1) 2ni(2v/-2nh)3p c
The functions v(r,q,s) and u2(r) are determined by Eq. (1.5.6) (one should write U2(r) instead of Uz(Ir])) and by Eq. (2.11) (or Eq. (2.12)), while U2(r) is specified by Eq. (I.3.1) with s = 2 where one must use some approximation or other for p3(rl, r:,r3) (see Section 5 of I). Hence, one has four equations for the four functions g, c., u: and U2. Once the equations are solved one can consider thermodynamic aspects and superfluid properties by leaning on the results of Section 3. Note that characteristic of the relevant state of the system are long-range spatial correlations decaying as I/jr I, which stems from Eqs. (2.12) and (4.1). The equations obtained can have spherically symmetric solutions analogously with the equations of 1. Then P0 = 0 according to Eqs. (3.3) and (3.11). Such a state of the system may be called the condensate phase without superfluidity. Therefore, tbrmation of a condensate does not necessarily lead to superfluidity. In particular, superfluidity cannot exist in an ideal gas and even in a slightly nonideal Bose system [2]. Superfluidity can be observed only if the interparticle interaction is sufficiently strong, in which case the equations obtained may have solutions without spherical symmetry. From this viewpoint let us discuss Eq. (2.11) that coincides with the Schr6dinger equation for zero-energy eigenvalue. Even if the potential is spherically symmetric, besides spherically symmetric solutions the Schr6dinger equation can, for special cases of the potential, have zero-energy solutions that are not spherically symmetric (see, e.g., Ref. [3, Section 15], where one considers bound states that appear with zero energy).
V.A. Golovko/Physica A 246 (1997) 275-287
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Of course, an eigenvalue that is exactly zero can occur only in an exceptional case of the potential. However, the potential U2(r) is not a given quantity, and Uz(r) may adjust itself to the required form. The adjustment may be facilitated if the interaction potential K(r) has a real or virtual level near zero. Just such a level exists in the case of helium atoms [4,5]. Perhaps the superfluidity of helium is due to this peculiarity of the interatomic potential. On lowering the temperature there may appear various, both spherically symmetric and nonsymmetric, solutions of Eq. (2.11) which correspond to different values of the quantum number l, i.e., of the angular momentum in the usual interpretation of the Schr6dinger equation (note that P0 ~ 0 only if I # 0). Consequently, condensate phases of different types are possible, both superfluid and without superfluidity. One will observe the phase that provides an absolute minimum of the appropriate thermodynamic potential.
5. Hard spheres under the neglect of triplet correlations It is natural to try to solve the above equations in the case of a hard-sphere system wherein triplet correlations are neglected, treated in detail in I. If U 2 ( r ) = K ( r ) with K(r) defined by Eq. (I.6.1), Eq. (2.11) has a unique solution of the form a u2(r) -----1 -- - , r = Ir I , (5.1) r
subject to two conditions: u2(a) = 0 and ue(r) --* 1 as r --~ oc. The solution of Eq. (I.5.6) is the same as in I. Now Eq. (4.1) enables us to calculate y(r). Upon denoting the right-hand side of Eq. (I.6.3) by #r(r) we shall obtain [2] 9(r) = 1 - -2a ~ f c (1 - ~ r ) + ( 1
- f c ) [ y l ( r ) 1 -]
(5.2)
.
This function is spherically symmetric, which gives po = 0 on account of Eqs. (3.3) and (3.11 ). Hence, in the present case we have a condensate phase without superfluidity. Eq. (3.13) that determines ~(O,p) now becomes [2] 4rcah2p { 3m fc + (1 - f c ) H ( ? ) -~2 H (r)~--0 - [I - H(e) + fHl(f)]t9
,
(5.3)
where H ( ( ) is given by Eq. (1.6.6) with the upper sign, the prime over HI(f) denotes differentiation with respect to the argument, and f = a 2 m ~ - 2 ( 1 - f c ) -1. Now, it is necessary to find the point at which the above solution bifurcates off the high-temperature solution considered in I. Eq. (3.15) that determines this point assumes the form H(f') - 1 - f H ' ( f ) = 0.
(5.4)
V.A. GolovkolPhysica A 246 (1997) 275-287
283
P
\
V1
Fig. 1. Isotherms of a hard-sphere Bose system under the neglect of triplet correlations. The specific volumes vt = l/pl and v2 = l/p2 are given by Eqs. (5.5) and (5.6), respectively.
However, an analysis shows that for the considered potential U2(r) always H ( ~ ) - 1 ( H ' ( ( ) > 0. Thus, Eq. (5.4) has no solution. Therefore, the bifurcation does not occur until the stability criterion ( @ / ~ V ) o < 0 breaks down. Afterwards, there emerges a state mentioned at the end of Section 3 and characterized by the equation (~p/OV)o = 0. The corresponding point is given by Eqs. (1.6.25) and (I.6.26). Those equations can be recast in the following parametric form:
--
Jzh 2aG-l(/~l )'
P!
= ~931"2 Go(fll
)'
(5.5)
fll being the parameter. As distinct to Eqs. (I.6.25) and (1.6.26) these equations are correct to the order a scx n 5/3. Below the point in question the isotherms have a horizontal part, as illustrated in Fig. 1 . As the specific volume v = l i p decreases from vl the condensate fraction J~ increases from zero. Ultimately it reaches the value f~ = 1 whereas values j~. > 1 make no physical sense. Hence, there is another special point v = v2 (or 0 = 02 ¢ 0) at which f , = 1. This point is given by
2aG-l(fl2)'
P2 =(2 + "~)6o~'"2G0(/~2),
(5.6)
where fi2 serves as parameter [2]. In a first approximation in a Eqs. (5.6) yield
r
P2
]
2/3
.
02 = [(2 + "~)~3o)J Note that to the same approximation Eq. (5.5) gives Oi = (pl/{3(o) 2/3.
(5.7)
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The behaviour of the isotherms at v < v2 when fc---1 is sketched in Fig. 1 and analysed in Ref. [2]. If n is small (recall that n = rca3p), the isotherm is presented by the solid line. It is typical of a first-order phase transition. If n2 > 6.04 × 10 -2, the isotherm has the form of the broken curve. In the last case we have two continuous phase transitions, at v = Vl and v = v2. It is not clear, however, whether this case can be observed in reality because triplet correlations that have been discarded may play an important part in the condensed region even if n is rather small [2]. A comparison of the present results with the known studies on the Bose condensation in a hard-sphere system [ 6 - 8 ] is made in Ref. [2]. Here we observe only that Lee and Yang [7] noted that in their work it was not possible to state exactly the order of the transition for a ¢ 0; this order was not determined uniquely in Refs. [6,8] too. In Ref. [9] Huang writes cautiously that the transition appears to be a second-order one while the argument of his own [6,8] that the transition may be of first order remains unrefuted. Let us point out some peculiarities of the condensate phase that follow from the present approach (see Ref. [2] for details). The properties of the condensate phase are considerably affected by details of correlations and of the interaction potential. This signifies that the properties should depend noticeably upon conditions of the experiment, impurities in the liquid, inhomogeneities, etc., which may account for discrepancies between the values of critical velocities in helium observed in different experiments [10]. Concluding this section we remark that in the case considered in the section Eq. (3.15) had no solution, which resulted in the isotherms of Fig. 1. At the same time the example of a weakly interacting system [2] shows that Eq. (3.15) may have a solution, so that the phase transition to a condensate phase may be second order as well, depending on the interaction potential.
6. Discussion and concluding remarks The present paper shows that within the scope of the approach proposed in I lies also such a phenomenon as superfluidity. The approach indicates that the phenomenon embodies symmetry breaking in a fluid which is due to formation of a spontaneous stationary flow, the state with the flow being thermodynamically equilibrium while the magnitude of the flow being determined by thermodynamic relations. This is analogous to the appearance of spontaneous magnetization below the Curie point in a ferromagnet, the value of the magnetization being given by thermodynamics. It should be emphasized that the quantity P0 which determines the flow appears in the theory quite naturally since there is no reason to put e(1)=0 in Eq. (2.7). Thus, no additional argument is needed to explain the existence of the frictionless flow so far as no dissipation of energy can occur in thermal equilibrium. Let us turn now to the question of how to elucidate superfluidity physically. We note first that incessant motion without any dissipation of energy is characteristic of quantum mechanics. As an example, we may mention the motion of electrons in an atom while
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the atom, let alone a molecule, can contain up to a hundred electrons. The motion may manifest itself externally by the existence of an orbital angular momentum L. Recall that not all electrons contribute to the total angular momentum although all the electrons constitute a unified dynamical system. A superfluid may be regarded as a gigantic atom whose radius is infinite, and in which for this reason the orbital motion is converted to a rectilinear one that is characterized by a linear momentum P instead of the angular momentum L. The gigantic atom differs from an ordinary one in that the role of electrons is played by neutral atoms which move in a self-consistent field, not all of them contributing to P (cf. the discussion of Eq. (2.13)). Here again we may draw a parallel between ferromagnetism and superfluidity. Ferromagnetism may be regarded as a macroscopic manifestation of the spin angular momentum while superfluidity is a peculiar macroscopic manifestation of the orbital angular momentum. In an ordinary atom both states with L ~ 0 and L = 0 are possible; analogously, the condensate phase may be either superfluid (P0 ¢ 0) or nonsuperfluid (P0 = 0). A similar view may be expressed as to explanation of superconductivity. In a superconductor charged particles, electrons, are entrained in the collective quantum motion under discussion, which gives rise to a supercurrent. From this point of view, high-temperature superconductivity is explained probably by a singular structure of relevant compounds, which favours the motion at high temperatures. In order to consider superconductivity in this way it is necessary first to extend the approach developed in I to systems containing nonzero spin particles, which will be the subject of a subsequent paper. In this context the following should be noted. Theories of superconductivity (superfluidity of electrons) based upon an idea that the phenomenon is due to spontaneous currents existing in a condition of thermodynamic equilibrium, which is analogous with the results of the present paper, were proposed as far back as the 1930s (for a review see Ref. [11], a critical analysis of such theories can be found in Ref. [12] as well). A characteristic paper along these lines is that of Landau [13]. Landau assumed that a superconductor contains local saturation currents flowing in different directions and yielding no resultant current in the absence of an external field. His study was not based upon quantum mechanics, being phenomenological, in essence, without convincing argumentation. All those theories were not sufficiently founded and were abandoned later. The present approach shows that there is a preferred direction specified by P0, which implies an anisotropy in the velocity distribution. In the recent experiments on Bose condensation in a gas of alkali atoms [14,15] one observed such an anisotropy. At the same time a direct comparison of our results with the experimental ones is impossible because in the present paper an idealized case is considered, in which the system occupies an infinite volume, whereas real volumes are always finite. Let us discuss manifestation of superfluidity in finite systems. In a confined volume flows must close upon themselves. It is quite possible that the volume will break up into cells in each of which the fluid will circulate along closed streamlines, forming a pattern similar to that observed, for example, in Rayleigh-B6nard convection (for a recent review see, e.g., Ref. [16]). Here again an analogy with a ferromagnet is in place: the
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cells in the superfluid are analogous to domains in the ferromagnet. At the same time there is a great difference. Formation of magnetic domains is energetically profitable because owing to them the magnetic energy of the specimen decreases. Curvature of the flow and formation of the cells in a superfluid will most likely be energetically unfavourable. Hence in some cases, especially just below the 2 point, the cells may not develop and one will observe a metastable condensate phase without superfluidity; in other cases, under appropriate circumstances, a cell pattern may arise. As to the motion of the superfluid near walls of the vessel we note that atoms of the walls may adjust themselves to the quantum motion since their temperature is low as well. When trying to visualize the flow pattern one may meet with some peculiarities for the superflow does not exert an extra pressure along the streamlines (see the discussion of Eq. (3.10)) as distinct from a flow in an ordinary liquid. The existence of the intrinsic flow explains readily the known fact that helium II crawls up the walls, out of an open vessel (see, e.g., Ref. [9]). This sheds further light on the fountain effect as well. The present ideas offer also a simple explanation for the existence of a critical velocity. If a superfluid in which the foregoing pattern has formed is subjected to an external action, be it a gradient of pressure or temperature, some of the closed flows will open and the superfluid will partially flow along the direction of the gradient. Here again we have an analogy with a ferromagnet in which an external magnetic field reorients the magnetic moments of the domains. To the saturation magnetization, when the specimen becomes one domain, will correspond a state of the superfluid in which the whole flow will be directed along the gradient. A further increase in the total flow is impossible without breaking the condition of thermodynamic equilibrium, which will result in viscosity. Hence, the critical velocity is determined by the formula V c = p o / m according to the discussion following Eq. (2.13), where P0 as a function of the temperature and density is given by Eq. (3.11). In narrow and long capillaries where the creation of closed flows is hindered superfluidity will manifest itself easier, which conforms to the experiment. The reasoning remains practically the same if the formation of the cells is energetically unfavourable; a temeperature or pressure gradient will again facilitate creation of a directional flow, and the critical velocity will again be given by Vc = po/m. In Landau's explanation of superfluidity [17] the critical velocity is of the same order of magnitude as the velocity of sound, that is, it is too large; in the present theory the critical velocity is not at all related with the velocity of sound as seen from Eq. (3.11). If superfluidity is regarded as spontaneous symmetry breaking, the critical velocity may be considered the order parameter vanishing at the transition point (see also Ref. [2]). A word should be said about the vortices. When the velocity of the flow exceeds the critical velocity, the equilibrium superfluid state breaks down, which gives rise to vortices. From this viewpoint the breakdown of superfluidity is the cause for creation of vortices. In Feynman's argument [18] this causal chain is reversed; the creation of vortices is the cause and the breakdown of superfluidity is the effect. It should be stressed that the cells discussed above have nothing to do with the vortices.
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Let us discuss now the phases that were called the condensate phases without superfluidity. We revert again to the analogy with an atom. If in the ground state of the atom the orbital angular momentum L is zero, an excited state can be such that L ¢ 0. In a superfluid the role of the exciting agent can be played by a temperature or pressure gradient. Consequently, the P0 = 0 state can be converted to a metastable P0 ~ 0 state. Once a quantum system goes to another state its subsequent behaviour does not depend upon the exciting agent (if there is no other transition). Therefore, in the emerging state of the superfluid the quantity P0 will be determined by internal properties of the superfluid, that is to say, by Eq. (3.11 ). On the whole, the properties of the metastable P0 ¢ 0 state should be analogous with those of a genuine superfluid. The question of lifetime of the metastable state lies beyond the scope of equilibrium statistical mechanics. In the case of a superconductor the role of the exciting agent in question may be played by not only an external voltage but also a contact potential.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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