On the Bose-Einstein condensation of a quantum liquid

Vol. 27 (1989)

REPORTS

ON THE BOSE-EINSTEIN

ON MATHEMATICAL

CONDENSATION

PHYSICS

OF A QUANTUM

No. 1

LIQUID*

J. MA~KOWIAK Institute

of Physics, N. Copernicus 87-100 Torun, Poland (Received

October

University,

IO, 1987)

,

A model of a noninteracting Boson liquid is discussed. The motion of the particles is locally ordered. It is shown that under a plausible conjecture on the thermodynamic limit of free energy of the system, it exhibits Bose-Einstein, condensation below some critical temperature T, and with the particle mass and density equal to those of liquid 4He, the local ordering in the liquid can be adjusted so that T, = 7”, the temperature of the I-transition of 4He. At T = OK the density of the condensed mode consisting of particles at rest is only a fraction of the total density. In this respect the model also behaves similarly as 4He.

The microscopic origins of the I-transition in liquid 4He are not fully understood. It is generally believed that this transition could be explained in terms of Bose-Einstein (BE) condensation in a weakly interacting Boson liquid (cf. e.g. Cl][3]), but several of its aspects are not clear, e.g. how does the I-type discontinuity of the specific heat of 4He arise and why does the density of the condensate in 4He approach only a fraction of the total density when the temperature is lowered to OK [4]. This note deals with the second property of 4He and with the temperature of the I-transition Tl = 2,19 K. A model of a noninteracting Boson liquid is presented and it is shown that the parameters characterizing the liquid (with particle mass m and density d equal to those of liquid 4He) can be adjusted so that BE condensation occurs at T, = T,. The model thus improves the approximation To io TAprovided by the ideal gas, since in an ideal gas with m and d equal to those of liquid 4He, To > TA [3]. At T = OK, similarly as in 4He, only a fraction of all the particles occupies the condensed mode corresponding to zero momentum. The model consists of n noninteracting identical bosons, each having mass m and spin zero, enclosed in a parallelepiped with volume IAl = L, L, L, under periodic boundary conditions. To distinguish the system from a free gas, let us take for the Hamiltonian the quantized version of the Hamiltonian describing a classical * Supported by the Polish Ministry of Sciences and Higher Education, Project CPB 01.03/3/2/87.

J. MACKOWIAK

12

liquid. A possible form of the latter was discussed in [S]. It consists of 12noninteracting particles in a mean field of the form -g(p, /3, u)cos(g(x), p), where p is the particle momentum, /I = (kT)-‘, v is the specific volume, and

y-‘g(x) is a random vector with constant distribution with respect to space coordinates x in /1 and cos(g(x), p) d enotes the cosine of the angle between g(x) and p. For the Hamiltonian of a quantum liquid consisting of n bosons in /i we take the quantized version of

where cl,(P) =

i

Y, = o

p32n

forPEP,

= CP,, P,+&

Pl ’ 0,

for p#p,,

s(x) being defined in the same way as y-‘g(x). Since the distribution of s(x) is constant, the probability P(ds(x)) that s(x) has a particular orientation at a fixed point XE~ is independent of this orientation and x and equals ds(47r)l. Hence,

where Sf is the 2-dimensional sphere with radius 1. Let us now find the free energy density of the system (l), denoted in the sequel by f@(n), D). W e s h a 11 assume that the volume IAl is large and IAl-+ co at constant particle density d = nl/il-‘. To calculate f@(n), /I), let us resort to the results of Thirring et al. [6]-[8] which provide a plausible conjecture about the form of the statistical sum of a large n-particle quantum system with a potential defined in the configuration space of particle positions. Their proof of asymptotic exactness of the Thomas-Fermi theory in the thermodynamic limit shows equality of this limit and the classical limit, in the sense that the trace Tr exp( -/?H) in the expression for the free energy density of a quantum Co_ulomb system, after passing to the thermodynamic limit, takes the form of a phase space integral of the classical density according to the exp(-PH(P,,..., P,, x1,..., x,)) (symmetrized or antisymmetrized statistics) with a factor depending on the dimension of the spin space. It is very likely that equality of these limits holds for other potentials, e.g. the one in (1). Below we apply this conjecture to calculate f(H(n), /I): lim f@(n), Ml-m = -lfmm

/I) = -,tmm

P-‘IN1ln

(pIAl)-‘lnTrS,exp(-/?PH(n))

J fi dxiJn j@I n-i=1

BOSE-EINSTEIN

CONDENSATION

OF A QUANTUM

13

LIQUID

where S, is the symmetrizer S,(l,...,

4 = (n!)-‘C~1S(1)...~nS(n). B

The integral J fI hiexP(-P R3”iZl

i

(l$/2m-

i

S~(Pj)cos(s(xj)~ Pj))s,(l,...,

n)

a=1

j=l

is obviously invariant under independent rotations of s(xJ, i = 1,. . . , n. Thus the random vectors s(xJ can be replaced by constant vectors ti with the same orientation at all X~EA. Consequently. limf(W&

P)

=-

,~~~~~-‘l~l-‘ln~...~exp(-~ Pl

=-

lim j?-‘l/11-‘ln I‘++~ eatI

where summation

Sabi)CoS(ti7

i i=l

Pn

(pf/2m-

J JJP(dt,)C...Cexp(-p cs:,” k Pl P”

i

(p?/2m--

i=l

Pi)))sn(l 7.. . ) n)T

in (3) runs over the admissible p = (2Kk,/L,,

2xk,fL,,

2nk,/L,),

(3) momenta kicZ1.

Thus the free energy density of the liquid (1) differs from that of the liquid with the Hamiltonian

Hc(n)= C (P?/2m-C Sa(Pi)CoS(ti? Pi>)= C hc(PiYti) i=l

a=1

(4)

i=l

(the probability that ti have particular orientations, each falling within dti on S:, being equal n I’(&)) only by additive terms which vanish in the thermodynamic limit.

J. MACKOWIAK

14

Direct calculation of limf(H,(ri), p) as I,41+ co from the expression (3) is extremely complicated. The difficulty can be circumvented by resorting to the equality of the free energy density for various ensembles in the thermodynamic limit [6]-[ lo]. In particular, one can expect that as in the case df noninteracting quantum gases with. non-stochastic potentials [lo], lim f(&(n), Ml-m where the fugacity

p) =

lim (~-llnz,+(~l~~)-lTrln(l-z,exp(-/?h,))), Ml-m

z, is given implicitly Inl-‘Trz,&

as the solution = d,

-zne)-l

of the equation

Q = exp(-b/z,).

The average particle density in the system is given by the expression (5), viz., Nn, P, z,) =

14-‘~G1exp(P4p, t)-1)-l,

(5) on the 1.h.s. of

(6)

P

where HP, t) = p2/2m-

i g,(Pbos(p, a=1

Q.

We shall investigate the variation of z, with fl in the thermodynamic constant particlk density as given by eq. (5): o(n, B, z,) = d

limit, under

(5)

and examine the necessary conditions for BE condensation in the liquid (4) in terms of /?. To this end we apply the technique of Landau and Wilde. In [l l] they demonstrated BE condensation of an ideal gas under arbitrary boundary conditions. After minor modifications their proof also applies to eq. (5). Below it is given in some details. Forjxed where q,(f, n) = infs(p, LEMMA 1.

n, /I, 4, {y,} eq. (5) has a unique solution z,E(O, exp(~~O(t, n))), t) and the infimum is taken over all admissible momenta.

P Proof: The sum in (6)’ donverges almost uniformly with respect to z, for O’< z, < exp(/3s0(t, n)) so o(n,‘/?, z,) is continuous in z, in this range for constant n, /_?,(y,>. Furthermore,

since each summand

on the r&s. of (6) has this property. Finally, p(n, p, 0) = 0 and n)), so the assertion holds, Q.E.D.

D(n, j?, z)+ co as z-+exp(~~~(t,

BOSE-EINSTEIN

CONDENSATION

In the thermodynamic

15

LIQUID

limit

limsO(t, n) = min{O, pf/2m-y and sO(t, n)JO. Thus n > N constitute a infinite set, (z,,} h as of this limit point.

OF A QUANTUM

,,...,

p,2/2m--yy,}

= min{O, 0 ,...,

O> = 0

to increasing values of the solutions z,, of eq. (5) corresponding subset {z”} of the interval (0, exp(j&,(t, N))). As a bounded at least one limit point. It suffices therefore to prove uniqueness To this end let us define

DE(n, fl, z):= 1nl-i

1

(z-lexp(/k(p,

t))- 1)-l

cfP,t)>E

for E > 0. Suppose z* is u limit point of {z,,}, so there thut z,,~--+z*. Then

LEMMA

such

2.

lim D,(Iz~, /I, z,J = (27~)~ nr+m for

0

<

z*

d

J

(z*-‘exp(/k(p,

exists

t))-lj-‘dp

n sequence

nk -+ cc

=:DE(/?, z*)

E(P*Q’C

1.

n,)), therefore 0 d z* ,< 1, and D,(nk, p, z,,) Clearly, 0 < .znk< exp(&(t, is well defined for all s > 0. The assertion results by writing DE(nk, p, z,J as a Riemann sum in terms of the admissible momenta and passing to the limit nk -+ a, Q.E.D. Proof:

D,(n, j3, z*)

provided

represents the density of particles with energy greater z* is proved to be unique. Thus, in case z* is unique, D,,(P, z*): = SUP D,(P,

than

.s,

z*)

c>o = DEmin(jj, z*) =

(21s))” J (z*-‘exp(/k(p,

t))- l)-ldp.

(7)

R3

is the density of excited particles in the infinite system and Do = d-D,, the density of particles occupying the ground state and, possibly, a number of lowest excited states. Since D,(p, z*) < d, therefore D,,@, z*) < d, implying Do@, z*) > 0. Let us examine the properties of D,X(pz z). For z < 1, the integrand on the r.h.s. of (7) can be expanded into a geometric series and integration of each term over the unit sphere yields 2 zk(~ky,)-1:ZI,,2(~k~~~pa~’ exp( -@p2/2m)p2dp+

0,,(/3, z) = (27c)-3’2 i

a=1

+2(2X)-2

k=l

f k=t

P=

Zk

J

R:/US, e

exp(-Wp2/2m)p2dp,

(8)

16

J. MACKOWIAK

where II&) = (2/rrx)l/‘sinhx. Clearly, for z < 1 the series (8) converges. To verify whether it converges for z = 1 let us find the asymptotic expression for the integral oo>p,>p,>o

K(P, PJ = “s’exp(-Pp2)p2& PO

for large p > 0. To this end let us write 1 WA PJ = -(P:-&exp(-@). 3 Then pp” =

-ln(3(p:-p~)-‘)‘/~(~ e-flP2p2dp)l/P PO

and lim& =pg as /3-+co. Since pi > pi for 0 < p < co, we may write pi = &+f(/?, pO) for 0 < p < co. Thenf@, pO) > 0 for 0 < I< co and limf(fl, pO) = 0 as /3+ co. To find the form off@, pe), let us note that the Laplace saddle-point method [12] yields 112PI P lim 2 -x j exp(-PP’)Q = 1. 0 P-00 0 Hence lim 4n:-‘/2/?3/2K(P, P’co . which implies that for large /? > 0 f(P, where C is a constant.

0) = 1

0) z $~-llnP+B-lC,

For p. > 0 we can thus write f(P? PO) = f(P? O)+cp,(B, PO),

where limcp,(fi,Po)=O

as /?+cc f(PT

or po+O, PO)

=

f(P?

(9)

or

O)cp,(P>

(10)

PO)7

where cp2(j3, 0) = 1 and hence limcp,(P, po) = c,(p,) > 0 as fi---+co. The correction term q~r(fi, po) in (9) would be significant only if it had the form cpl(PY PO) = B-S40(Po) ’ 0,

O
in which case for large /? > 0,

K(/?, po) z const~-312exp(-&~-~1-S~~o)).

(11)

For f of the form (10) for large p > 0,

WP, po) =

const B-

3w~0)/2~~~(

-pp;)_

(12)

BOSE-EINSTEIN CONDENSATION

OF A QUANTUM

17

LIQUID

In either case the series (8) converges for z = 1, which can be seen from the asymptotic form of the k th term of this series for large k. Thus B,X(j?, z) is well defined for 0 6 z < 1. From the representation (8) of D,, it also follows that ““(” W

‘) < 0

for 0 < z < 1 19

(13)

aDex@7 LJZ

‘)

for

(14)

>

0

0 < \

Z < \

1

and O,,(p,‘l)‘< co, limoex(fi, 1) = cc as p-0 and limD,,(/?, 1) = 0 as /3+co, Let us now define /3, = (kT,)-’ as the unique solution of the equation 0,&J,?

1) = d.

LEMMA 3. Zf T > T,, then all limit points z* of {z”} fiEfi z* E (0, 1). Conversely, if there is a limit point z* -c 1, then T > T,, so all limit points belong to the interval (0, 1).

Proof:

Denoting

by z* a limit point of z, and supposing o,,(P,

z*) G d = o,,(P,,

1) < o,,(B,

that T > T,, we get

1).

Thus by (14), z* < 1. Furthermore, d = I/i,,l-lC(z,‘exp(P&(p, P

t))-l)-l-+DeX(fi,

z*)

for z,,+z*.

(15)

Thus d = D,,(p, z*) and hence o,(/?, z*) = 0. Conversely, if the limit point z* satisfies z* < 1, then by (15), d = II,,@?, z*) which combined with (14) yields o,,(P,> 1) = d = D,,(P, z*) < o,,(B, implying

1)

T, < T, Q.E.D.

By (15), for T > T, any limit point satisfies the equation d = D,,(/?, z*), which together with (14) proves uniqueness of the limit point z* in this range of T’s. Moreover, O,,(/?, z*) = 0, so the liquid does not exhibit BE condensation in this range. LEMMA 4.

If

T G T,, then limz, = 1 as n + 00 and for T < T,, DO(/j, 1) > 0.

Proof: According to Lemma 3, the limit point z* of {zn} satisfies 0 < z < 1 if any only if T > T,. Thus for T d T,, z* = 1, and for T < T, Do@, 1) = d-Dex(P, by (13), Q.E.D.

1) = ~ex(P,, I)--D,AP,

1) > 0

(16)

18

J. MACKOWIAK

Thus at T, the liquid undergoes a second order phase transition, due to the continuity of z(d, T) and discontinuity of (az/aT) (d, T) at T, and for T < T, exhibits BE condensation by virtue of (16). The temperature T, of the transition is lower than the temperature of BE condensation in an ideal gas, since D,X(B, 1,

Yl?...,

Y,) >~ex(P,

if some y3 > 0

l,O,...,O)

and To, T, are defined as the unique solutions of the equations D,,((kT,)-‘, 1, 0,. . . 0) = d, Dex((kT,)-‘, 1, y1 ,... , y,) = d, resp ec t’lvely [ll]. For an ideal gas with particle mass and density of liquid 4He, T, = 3.2 K > TA [3], so as regards the transition temperature, the pre,sent model provides a better value. In fact, K can be lowered arbitrarily by shifting the intervals 9’ix to the right on the positive semi-axis and taking sufficiently large values of y, = pz/2m. As for the density of particles in the ground state at T = 0, one has limDo(fl,

1) = d

as /?--+co

since limD,,(/?, 1) = 0 as /I -+ (;o. Thus all particles of the liquid condense at T = 0 into the ground state which is (a+ I)-fold degenerate. The fraction of particles occupying at T = 0 each eigenstate corresponding to the lowest energy level is therefore equal (G+ 1)-l. At T = 0 the density of the condensed mode with particles at rest is e.qual d (G + 1)-l, which is only a fraction of the total density, similarly as in 4He [4]. Acknowledgement I am grateful

to Professor

W. Thirring

for his suggestions.

REFERENCES [l] R. P. Feynman, Phy.s. Rev. 94 (1954), 262. [Z] K. Huang, Statistical Mechanics, J. Wiley, New York, London, 1963. [3] N. N. Bogolyubov and N.N. Bogolyubov, Jr., Introduction to Quuntum Statistical Mechanics, Nauka, Moscow, 1984. [4] G. Baym, in: Muthemutical Methods in Solid State and Superfluid Theory, R. C. Clark, and G. H. Derrick (eds.), Oliver and Boyd, Edinburgh, 1969. [5] J. MaCkowiak, Physica 143 A (1987), 239. [6] W. Thirring, Quantum Mechanics of Large Systems, Springer, New York, 1982. 17) H. Narnhofer and W. Thirring, Ann. of’ Phys. (N. Y.) 134 (1981), ‘128. [S] B. Baumgartner, H. Narnhofer and W. Thirring, Ann. of Phys. (N. Y.) 150 (1983). 373. i9] J. Glim and A. Jaffe, Quuntum Physics, A Functional Integral Point @” View, Springer, New York, nHeidelberg, Berlin, 1981. [lo] J. MaCkowiak. Physica Scrip 38 (1988), 513. [11] L. J. Landau and I. E’. Wdde, Commun. Mnth. Phys. 70 (1974), 43. [127 A. Erdtlyi, Asymptotic Expansions, Dover Publications Inc., New York, 1956. rYc?e added in proof 1:;s equality (2) is trivial since H(n) is a multiplication

with s having a constant

distribution

independent

of xgA.

operator,

its eigenvalues

being equal