Thermodynamics of the Bose-system with a small number of particles
A theoretical description of the interacting Bose-system is proposed. It is based on the extrapolation of the results obtained for the systems with a small number of particles N = 2, 3, 4 , etc. to the bulk case of N = ∞ . It is shown that already the system with N = 12, 13 behaves almost as a b...
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Інститут фізики конденсованих систем НАН України
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| Cite this: | Thermodynamics of the Bose-system with a small number of particles / I.O. Vakarchuk, A.A. Rovenchak // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 431-447. — Бібліогр.: 21 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1204192025-06-03T16:30:14Z Thermodynamics of the Bose-system with a small number of particles Термодинаміка Бозе-системи з малим числом частинок Vakarchuk, I.O. Rovenchak, A.A. A theoretical description of the interacting Bose-system is proposed. It is based on the extrapolation of the results obtained for the systems with a small number of particles N = 2, 3, 4 , etc. to the bulk case of N = ∞ . It is shown that already the system with N = 12, 13 behaves almost as a bulk in a wide temperature range. Special attention is paid to the phase transition in these systems. The hard sphere potential is used in calculations. The sequence of heat capacity maxima is approximated as CmaxN / N ≃ 13.6 − aN⁻ε with ε = 0.0608 giving the value of bulk heat capacity as 13.6 while experimental value is close to 16. The temperature of λ -transition is estimated as 2.1–2.3 K versus experimental 2.17 K. Quite good qualitative and satisfactory quantitative agreement with the experimental data has been achieved. Запропоновано теоретичний опис взаємодіючої Бозе-системи, який ґрунтується на екстраполяції результатів, отриманих для систем з малим числом частинок N = 2, 3, 4 і т.д. на випадок N = ∞ . Показано, що вже система з N = 12, 13 поводиться майже як безмежна у широкому температурному діапазоні. Приділено особливу увагу фазовим переходам у цих системах. В обчисленнях використано потенціал твердих сфер. Послідовність максимумів теплоємности апроксимовано як CmaxN /N ≃ 13.6 − aN⁻ε з ε = 0.0608 , що дає для теплоємности безмежного зразка значення 13.6, експериментальне значення близьке до 16. Температуру λ -переходу оцінено як 2.1–2.3 K (експеримент дає 2.17 K). Досягнуто цілком добре якісне і задовільне кількісне узгодження з експериментальними даними. The authors appreciate Dr. Daniel G. Friend’s help with the helium-4 thermophysical data. 2001 Article Thermodynamics of the Bose-system with a small number of particles / I.O. Vakarchuk, A.A. Rovenchak // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 431-447. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 05.30.Jp, 74.25.Bt DOI:10.5488/CMP.4.3.431 https://nasplib.isofts.kiev.ua/handle/123456789/120419 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| language |
English |
| description |
A theoretical description of the interacting Bose-system is proposed. It is
based on the extrapolation of the results obtained for the systems with a
small number of particles N = 2, 3, 4 , etc. to the bulk case of N = ∞ .
It is shown that already the system with N = 12, 13 behaves almost
as a bulk in a wide temperature range. Special attention is paid to the
phase transition in these systems. The hard sphere potential is used in
calculations. The sequence of heat capacity maxima is approximated as
CmaxN / N ≃ 13.6 − aN⁻ε with ε = 0.0608 giving the value of bulk heat capacity
as 13.6 while experimental value is close to 16. The temperature of
λ -transition is estimated as 2.1–2.3 K versus experimental 2.17 K. Quite
good qualitative and satisfactory quantitative agreement with the experimental
data has been achieved. |
| format |
Article |
| author |
Vakarchuk, I.O. Rovenchak, A.A. |
| spellingShingle |
Vakarchuk, I.O. Rovenchak, A.A. Thermodynamics of the Bose-system with a small number of particles Condensed Matter Physics |
| author_facet |
Vakarchuk, I.O. Rovenchak, A.A. |
| author_sort |
Vakarchuk, I.O. |
| title |
Thermodynamics of the Bose-system with a small number of particles |
| title_short |
Thermodynamics of the Bose-system with a small number of particles |
| title_full |
Thermodynamics of the Bose-system with a small number of particles |
| title_fullStr |
Thermodynamics of the Bose-system with a small number of particles |
| title_full_unstemmed |
Thermodynamics of the Bose-system with a small number of particles |
| title_sort |
thermodynamics of the bose-system with a small number of particles |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120419 |
| citation_txt |
Thermodynamics of the Bose-system
with a small number of particles / I.O. Vakarchuk, A.A. Rovenchak // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 431-447. — Бібліогр.: 21 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT vakarchukio thermodynamicsofthebosesystemwithasmallnumberofparticles AT rovenchakaa thermodynamicsofthebosesystemwithasmallnumberofparticles AT vakarchukio termodinamíkabozesistemizmalimčislomčastinok AT rovenchakaa termodinamíkabozesistemizmalimčislomčastinok |
| first_indexed |
2025-12-01T18:06:41Z |
| last_indexed |
2025-12-01T18:06:41Z |
| _version_ |
1850330219050696704 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 3(27), pp. 431–447
Thermodynamics of the Bose-system
with a small number of particles
I.O.Vakarchuk, A.A.Rovenchak
Department for Theoretical Physics of Lviv National University,
12 Drahomanov Str., 79005, Lviv, Ukraine
Received May 28, 2001
A theoretical description of the interacting Bose-system is proposed. It is
based on the extrapolation of the results obtained for the systems with a
small number of particles N = 2, 3, 4 , etc. to the bulk case of N = ∞ .
It is shown that already the system with N = 12, 13 behaves almost
as a bulk in a wide temperature range. Special attention is paid to the
phase transition in these systems. The hard sphere potential is used in
calculations. The sequence of heat capacity maxima is approximated as
Cmax
N
/N ≃ 13.6− aN−ε with ε = 0.0608 giving the value of bulk heat ca-
pacity as 13.6 while experimental value is close to 16. The temperature of
λ -transition is estimated as 2.1–2.3 K versus experimental 2.17 K. Quite
good qualitative and satisfactory quantitative agreement with the experi-
mental data has been achieved.
Key words: Boson systems, finite systems, lambda-transition, specific
heat
PACS: 05.30.Jp, 74.25.Bt
1. Introduction
Liquid 4He is the only atomic system having its properties determined mainly
by quantum effects and the most essential among the latter is the statistical ef-
fect of identical particles. Due to it the macroscopic number of the 4He atoms can
have a zero momentum at some conditions, i. e., it can occupy a single quantum
state. The phenomenon of bosons accumulation at the lowest energy level is called
Bose-Einstein condensation (1924). According to London’s assumption (1938), it is
believed that superfluidity, λ-transition, and other unusual properties of liquid 4He
are a consequence of this very effect.
However, despite the lapse of time that has passed since the discovery of the
phenomena of λ-transition by Keesom and Clausius in 1932 and the superfluidity
by Kapitsa in 1938, nobody has managed to create a comprehensive microscopic
theory explaining the properties of strongly interacting Bose-systems such as 4He
c© I.O.Vakarchuk, A.A.Rovenchak 431
I.O.Vakarchuk, A.A.Rovenchak
within the whole temperature range of 0 < T < ∞.
Bogoliubov [1] was the first to show how one can build a theory of real helium
proceeding from the approximate second quantization techniques for the model of
weakly-imperfect Bose-gas. This theory works well for T < 1 K, especially while
T → 0.
The subsequent theoretical studies brought about the appearance of the quanti-
tative theory of the ground state of a strongly-imperfect many-boson system leading
to a good agreement with the experiment and to a theoretical description of low-
energy excitations which are responsible for the low-temperature behaviour of ther-
modynamic functions. The experimental investigations of the spatial structure by
means of X-ray and neutron diffraction and of the structure of the energy spectrum
and the condensate fraction by means of the inelastic neutron scattering revealed a
fairly good theoretical explanation.
Proceeding from of the experimental evidence, it was supposed for a long time
that 4He heat capacity had a logarithmic divergency at the transition temperature
Tλ. Brout showed in his early works [2] based on the Hartree–Fock method that it
was impossible to find an explanation of the observed heat capacity behaviour within
simple theoretical approaches. Feynman made an interesting notice regarding this
issue in his book [3]: “The explanation of this behaviour is left as an exercise for the
reader. If successful, publish!”
Feynman suggested that the λ-transition should be studied using the full N -
particle density matrix for the ideal Bose-gas with a phenomenologically introduced
effective mass for particles [4]. This approach leads only to a shift of the Bose-
condensation temperature in comparison with the ideal case leaving the form of the
heat capacity curve unchanged.
In the early 1970s the studies of the λ-transition were started in the renormaliza-
tion group (RG) approach for the theory with the degenerate two-component order
parameter. The rigorous grounds of this approach based on the functional integra-
tion techniques are given in [5,6]. They use the coherent states representation with
the complex order parameter the absolute value of which equals
√
N0 where N0 is
the number of atoms in Bose-condensate.
In this approach, the heat capacity critical exponent was calculated as α = 1/8. It
obviously does not correspond to the logarithmic divergence for which α equals zero.
However, taking into account that the calculated value of α is a small magnitude, the
experimental confirmation of this result required very precise measurements which
became available only recently [7]. Before it, there was even no certainty about the
sign of α. The measurements showed that the critical exponent α is negative, i. e.,
the heat capacity is finite at the transition temperature.
Recently this result was theoretically confirmed. Kleinert [8,9] obtained for α the
respective values of α = −0.01294 ± 0.00060 and α = −0.0120 ± 0.0009, while the
experiment gives α = −0.01285± 0.00038 [7]. One can find a bit different approach
to the critical exponents calculation in the paper by Campostrini et al. [10]. The
authors obtained the value of α = −0.0150 ± 0.0017. But, despite a very good
agreement of these results with the experiment, RG works only in the immediate
432
Bose-system with a small number of particles
vicinity of the transition point Tλ and gives no possibility of getting some information
of the thermodynamic functions behaviour in a wider near-critical range, e. g., for
|T − Tλ| ∼ 10−3 K.
In addition to these theoretical investigations, the numerical studies of 4He ther-
modynamic properties were conducted. The Diffusion Monte Carlo (DMC) tech-
niques are applied for the ground state calculations, and the Path Integral Monte
Carlo (PIMC) is used for finite temperatures. These techniques have been applied
since the 1960s. At present, PIMC is the most effective method. Its sole and es-
sential imperfection lies in the fact that it is impossible to use it for the ground
state (T = 0 K). But one can avoid this problem by considering the limit of the
results at T → 0 K. A detailed analysis of PIMC might be found in the work by
Ceperley [11]. It appears that the numerical approach provides a good agreement
with the experimental data in a wide temperature range on the whole, except for
the vicinity of the λ-transition point.
Thus, we have the situation when no satisfactory theoretical description of 4He
exists for a considerable temperature range of 1 K T < Tλ and T > Tλ, and the
only way out is the application of numerical techniques. It gives some space for
the investigations in this domain. Not claiming the solution of the problem, we still
propose here the model which, in our opinion, can near us to the understanding of
the processes in many-boson system via the consideration of systems with a small
number of particles.
For the latter, one can expect a much more precise description in terms of statis-
tical mechanics in comparison with what might be reached while directly considering
a many-particle system.
The aim of the present work consists in obtaining a thermodynamic description
of an interacting many-boson system based on the results for the systems with a
small number of particles N . We start our analysis from N = 2, 3, 4, etc. in order to
extrapolate these results for the case of N = ∞ after establishing some dependencies
on N and the temperature. We pay a special attention to the fact that even the
system consisting of a small number of particles exhibits the peculiarities on the
heat capacity curve such as the maximum which means the phase transition in the
limit of N → ∞. We will show at the same time that the system with N = 12, 13
tends to reveal qualitatively an “almost bulk” behaviour in a wide temperature
range.
The boson systems with a relatively small number of particles have been exper-
imentally obtained only in the last couple of years as particles in traps (see [12,13]
and references therein). It enables us to develop some methods for such systems and
a further comparison of the results with the experimental data.
We use the partition function formalism to calculate thermodynamic character-
istics of the free Bose-system. A method somewhat similar to this one was used in
works [12,13] for the study of the trapped finite Bose-systems: the recursion formu-
lae for the partition function proposed by P.Borrmann and G.Franke [14]. But our
expressions do not take the restriction for the system energy to be written as a sum
of one-particle energies. So, one can use them not only for the ideal gas but for the
433
I.O.Vakarchuk, A.A.Rovenchak
interacting system as well.
The paper is organized as follows. In section 2 some known initiating expressions
for calculating the thermodynamic function are given in short. In sections 3 and 4
we consider how the proposed method works in the case of an ideal Bose-gas the
results for which are well-known in the thermodynamic limit when the number of
particles N → ∞. The next step is to determine the asymptotic behaviour of our
results depending on the number of particles N . We will try to expand the found
regularities for the case of interacting bosons. The corresponding calculations for
the hard-sphere potential are given in section 5.
Generally, it is impossible to obtain a non-analyticity point of the heat capacity
function having a finite number of particles. Therefore, no “pure” phase transition
can be detected in this model. But we expect a certain qualitative agreement with
the experiment to be quite sufficient for the approximation used.
2. General statements
We consider the system of N particles with the Hamiltonian ĤN . The partition
function ZN = Sp e−βĤN where β is inverse temperature, β = 1/T .
In the coordinate representation we have
ZN =
∫
dr1 . . .
∫
drN RN (r1, . . . , rN | r1, . . . , rN), (1)
where RN is the density matrix. The particles are bosons of the mass m (4He atoms)
with their coordinates ri restricted in volume V being a cube with the periodic
boundary conditions. In this case RN is given by [15]
RN(r1, . . . , rN | r′1, . . . , r′N) =
= PN(r1, . . . , rN)
1
N !
1
λ3N
∑
Q
exp
(
− π
λ2
N∑
j=1
(rj − r
′
Qj
)2
)
, (2)
where Q is counting permutations of (1, . . . , N) and λ is the thermal de Broglie
wavelength
λ =
(
2πβ~2
m
)1/2
. (3)
The factor PN takes into account the interaction and will be described further.
The free energy FN is given by
FN = − 1
β
lnZN , (4)
total energy
EN =
∂
∂β
(βFN), (5)
434
Bose-system with a small number of particles
and heat capacity
CN = β2
[
Z ′′
N
ZN
−
(
Z ′
N
ZN
)2
]
, (6)
the primes mean derivation over β.
3. Partition function for the system of free particles
The ideal Bose-gas is a well-studied system for which the expressions for the
thermodynamic functions are known. Thus, it can be a good test example for the
applicability of our method and we are going to use this fact here.
On calculating the thermodynamic functions of the ideal Bose-system with a
small number of particles and comparing them with the well-known results for infi-
nite system one can detect some regularities showing the point when several particles
start to demonstrate the statistical behaviour.
One can find an expression for the partition function of N ideal bosons in the
explicit form. For this purpose we write the density matrix RN as follows:
RN(r1, . . . , rN | r1, . . . , rN) =
1
N !
1
λ3N
∆N , (7)
where
∆N =
∣∣∣∣∣∣∣∣∣
1 K12 K13 . . . K1N
K21 1 K23 . . . K2N
...
. . .
KN1 KN2 KN3 . . . 1
∣∣∣∣∣∣∣∣∣
+
, (8)
and
Kij ≡ K(rij) = e−πr2ij/λ
2
, rij = |rij|, rij = ri − rj. (9)
The subscript “+” means that every “−” in the expression for the determinant
should be substituted with “+”.
The partition function for free particles is
ZN =
1
N !
1
λ3N
∫
dr1 . . .
∫
drN ∆N . (10)
The “determinant” (8) might be written as [15]
∆N = ∆
(1)
N−1 +
∑
1<i6N
K1iKi1∆
(1,j)
N−2
+
∑
1<i<j6N
(K1iKijKj1 +K1jKjiKi1)∆
(1,i,j)
N−3 + . . . , (11)
435
I.O.Vakarchuk, A.A.Rovenchak
The superscripts in ∆N−l mean the lack of the correspondent rows and columns
in the expression for ∆N−l.
Integrating (10) we obtain
ZN =
1
N
{
V
λ3
ZN−1 +
1
(λ3)2
ZN−2
∫
dr1
∫
dr2 K12K21
+
1
(λ3)3
ZN−3
∫
dr1
∫
dr2
∫
dr3 K12K23K31 + . . .
}
, (12)
After the Fourier transformation
K(r) =
1
V
∑
q
e−iqrKq, (13)
Kq =
∫
dr eiqrK(r) = λ3e−λ2q2/4π (14)
the partition function is
ZN =
1
ρλ3
N∑
l=1
ZN−l
l3/2
, Z0 ≡ 1. (15)
One can use this expression to calculate the partition function of the system. We
will make such calculations in the next section.
4. Main results for the free particles
It is possible to calculate heat capacities CN for N = 1, 2, 3, . . . particles
sequentially using expressions (6) and (15). In addition, we can show explicitly the
low- and high-temperature asymptotics for CN , namely:
CN
N
∣∣∣∣
T→∞
=
3
2
, (16)
CN
N
∣∣∣∣
T→0
=
3
2
1
N
+ ANT
3/2, AN = const with respect to T. (17)
In the case of N = ∞ these temperature dependencies coincide with the known
results for the ideal Bose-gas [16]. For low temperatures one has
CN
N
∣∣∣∣
N=∞
=
15
4
ζ(5/2)
ζ(3/2)
(
T
Tc
)3/2
; (18)
the high-temperature limit being 3/2. The critical temperature T c (the non-analy-
ticity point on the heat capacity curve) in this case is defined from the equation
ρλ3
c = ζ(3/2), (19)
436
Bose-system with a small number of particles
Table 1. Main results for the ideal systems
N ρ
[
λ
(N)
c
]3
Cmax
N /N N ρ
[
λ
(N)
c
]3
Cmax
N /N
1 — 1.500 90 2.185 1.747
2 1.131 1.532 100 2.200 1.753
3 1.297 1.553 200 2.286 1.783
5 1.490 1.582 300 2.328 1.799
7 1.606 1.602 400 2.354 1.809
10 1.717 1.624 500 2.373 1.816
15 1.830 1.649 600 2.387 1.822
20 1.902 1.666 700 2.399 1.827
30 1.993 1.690 800 2.408 1.831
40 2.050 1.706 900 2.416 1.834
50 2.091 1.718 1000 2.423 1.837
60 2.123 1.727 1100 2.429 1.839
70 2.147 1.735 1200 2.434 1.842
80 2.168 1.742 ∞ 2.612 1.926
where λc is the thermal de Broglie wavelength for Tc, and ζ is the Rieman ζ-function,
ζ(3/2) = 2.612 . . . . Taking ρ = 0.02185 A
◦
−3 for liquid 4He we obtain Tc = 3.138 K.
In this work we consider the systems with different N from 2 to 1200. As was
mentioned above it is impossible to obtain the non-analyticity in the heat capacity,
so we take the point of CN maximum as the critical temperature T
(N)
c .
It appears that those values could be described by the linear dependence
ρ
[
λ(N)
c
]3 ≃ 2.61− c
N1/3
, c ≃ 1.9 (20)
with λ
(N)
c being thermal de Broglie wavelength for T
(N)
c . One can easily notice that
2.61 is very close to ζ(3/2) = 2.612 . . . .
We also have a similar correlation for the CN maxima Cmax
N /N . So, one should
expect that the limit N → ∞ gives the correct curve (see figures 1 and 2).
The results are also given in table 1.
As one can see from figure 1, good qualitative and satisfactory quantitative
agreement is reached at N = 100. Although, typical elements of the heat capacity
curve appear at smaller N , e. g., a clearly defined maximum is found at N = 10–15.
As we shall show further, these very values might be considered as the limit between
“individual” and statistic behaviour of the system with N particles.
437
I.O.Vakarchuk, A.A.Rovenchak
Figure 1. Heat capacities in the ideal case.
Figure 2. Temperature of the CN/N maxima in the ideal case. Solid line shows
dependence (20), circles correspond to the data from table 1.
438
Bose-system with a small number of particles
5. Interaction in the hard-spheres potential approximation
In a many-body system, it is a very complicated problem to take into account the
interaction and it has not as yet been completely solved. Specifically, the systems
with a small number of particles make it possible to consider this problem from the
first principles, i. e., to include the interaction into the calculations explicitly. From
the mathematical point of view, the computations become heavily complicated while
the number of particles increases, and the direct interaction account soon turns out
to be impossible.
We will proceed from the expression for the density matrix RN in the free par-
ticles case. In the systems like 4He small interatomic distances are essential when
atoms behave almost as hard spheres. Thus, the wavefunction should vanish rapidly
as soon as the distance between any two atoms approaches the hard sphere diameter.
Due to this we shall take the interaction into account by the substitution: of ∆N
in (10) with ∆NPN where PN is a function of the particles coordinates:
ZN =
1
N !
1
λ3N
∫
dr1 . . .
∫
drN ∆NPN . (21)
For the sake of simplicity we suppose the interaction to have a pair nature since
the obtained results are expected to be rather qualitative than quantitative. In the
case of hard spheres potential we have
PN(r1, . . . , rN) =
N−1∏
i=1
N∏
j=2
P (i, j),
P (i, j) = P (|ri − rj|) = P (rij),
P (i, j) = 1 + f(i, j) =
{
0, rij 6 a
1, rij > a
, (22)
a is the hard sphere diameter.
One can write the partition function as
ZN =
1
ρλ3
N∑
l=1
b′l(N)ZN−l, Z0 ≡ 1, (23)
λ =
(
2πβ~2
m∗
)1/2
, (24)
where m∗ is the effective mass of 4He atom, m∗ ≃ 1.7m [17].
We integrate (21) in two steps. The first one is the circular approximation (CA)
and the second one is the free-volume approximation (FVA).
1) CA is to the effect that only the interparticle interactions in the circle 1—2—
3—. . .—N—1 are taken into account. It means we neglect the interaction between
the first and the third, fourth, etc. particles, between the second and fourth, fifth, etc.
439
I.O.Vakarchuk, A.A.Rovenchak
particles, . . . . This makes it possible to integrate (21) using Fourier transformation
as we did in (10). Using also FVA we obtain for the coefficients b′l(N):
b′l(N) =
[
1− v0
V
(N − l)
]2 [
1− v0
V
(N − 3)
] l−1∏
k=3
[
1− v0
V
(N − k)
]
Il−1, l > 3
[
1− v0
V
(N − l)
]l
Il−1, l 6 3
. (25)
Here we have used the following designations:
v0 =
4
3
πa3, (26)
In =
8(4π)n−1
∞∫
0
dQ Q2
(
1
4π
e−Q2/4π − 1
Q
∫ a/λ
0
x sinQx e−πx2
dx
)n+1
, n > 0
1, n = 0
. (27)
2) In FVA we indirectly take into account the interactions neglected in CA. For
this purpose each integral over ri, i = 1, . . . , l is multiplied by the factor (1−nv0/V )
where n equals the number of multipliers P (i, j) which are not taken into account
in the Fourier transformation. To clarify the above statement we consider the most
simple non-trivial example with N = 4. According to (25) we have:
• b′1(4) =
[
1− 3v0
V
]
I0. Integral over r1 “hooks” the following three multipliers:
P (1, 2), P (1, 3), P (1, 4).
• b′2(4) =
[
1− 2v0
V
]2
I1. Integral over r1 “hooks” P (1, 2), P (1, 3), P (1, 4) but
P (1, 2) is included into I1. Integral over r2 “hooks” more P (2, 3) and P (2, 4).
• b′3(4) =
[
1− v0
V
]3
I2. Each integral over ri “hooks” additional P (i, 2).
• b′4(4) =
[
1− v0
V
]2
I3. P (1, 3) and P (2, 4) are not included into I3.
Thus, FVA improves the circular approximation. But it is not possible to use it
for large l and N because in this case a big number of P (i, j) multipliers will not be
integrated in the proper way but via formal factors.
In the calculations we assume the hard sphere diameter a = 2.1 A
◦
. The heat
capacity curves exhibit a fast approaching of the maximum point to a value from
the region of 2.2–2.3 K while N is increasing. The exact λ-transition temperature is
2.17 K.
440
Bose-system with a small number of particles
Figure 3. Heat capacities in the free volume and circular approximation. The
circles show the NIST CV data [18].
As has already been mentioned, in the ideal case the linear (over N −ε) depen-
dence fits the sequence of CN/N maxima quite well
Cmax
N
N
≃ Cmax
∞
− aN−ε, ε = 1/3. (28)
Assuming the same form for the hard spheres system one can find Cmax
∞
= 13.6 and
ε = 0.0608 making the maxima sequence as follows:
Cmax
N
N
≃ 13.6− 12.9N−ε, ε = 0.0608. (29)
Although small N values of 12, 13 do not allow one to identify the behaviour
at N → ∞ we obtain the values of heat capacity maximum from the range of
experimental data [7,18,19].
One can write the exact expression for the partition function similar to that in
the case of CA and FVA:
ZN =
1
ρλ3
N∑
l=1
b∗l (N)ZN−l, Z0 ≡ 1. (30)
From expansion (11) and formula (21) we obtain (without approximations):
b∗l (N) =
1
V
1
(N − l)!
1
(λ3)N−1
∫
dr1 . . .
∫
drl K12 . . .Kl−1,lKl1
×
l−1∏
i=1
l∏
j=i+1
P (i, j) BlN (1, . . . , l), (31)
441
I.O.Vakarchuk, A.A.Rovenchak
BlN(1, . . . , l) =
∫
drl+1 . . .
∫
drN
∆
(1,...,l)
N−l P
(1,...,l)
N−l
l∏
i=1
N∏
j=l+1
P (i, j)
ZN−l
, (32)
P
(1,...,l)
N−l =
N−1∏
i=l+1
N∏
j=i+1
P (i, j). (33)
We also introduce the following designations to simplify further records:
R(1, 2) =
∫
dr3 P (1, 3)P (2, 3), (34)
K̃ij = KijP (i, j). (35)
One can write the coefficients b∗l (N) independently of the interatomic potential,
i. e. for any PN :
b∗1(1) = 1, (36)
b∗1(2) =
1
V 2
∫
dr1
∫
dr2 P (1, 2), (37)
b∗1(3) =
∫
dr1
∫
dr2 P (1, 2)R(1, 2) (1 +K12K21)
∫
dr1
∫
dr2 P (1, 2) (1 +K12K21)
, (38)
b∗2(2) =
1
V
1
λ3
∫
dr1
∫
dr2 K12K21P (1, 2), (39)
b∗2(3) =
1
V 2
1
λ3
∫
dr1
∫
dr2 K12K21P (1, 2)R(1, 2), (40)
b∗3(3) =
1
V
1
(λ3)2
∫
dr1
∫
dr2
∫
dr3 K̃12K̃23K̃31. (41)
These expressions might be written via one-dimensional integrals:
b∗1(2) =
4π
V
∫
∞
0
P (r12)r
2
12 dr12, (42)
b∗2(2) =
4π
λ3
∫
∞
0
K2
12P (r12)r
2
12 dr12, (43)
R(r12) =
8
r12
∫
∞
0
dq
q
sin qr12
{∫
∞
0
rP (r) sin qr dr
}2
, (44)
b∗3(3) =
32π
(λ3)2
∫
∞
0
dq
q
{∫
∞
0
K12P (r12)r12 sin qr12 dr12
}3
, (45)
b∗2(3) =
4π
V λ3
∫
∞
0
K2
12P (r12)R(r12)r
2
12 dr12. (46)
In the hard spheres approximation, the integration is made in a simple way. We
obtain
b∗1(1) = 1, (47)
442
Bose-system with a small number of particles
b∗1(2) = 1− v0
V
, (48)
b∗2(2) =
1
23/2
{
1− erf
(a
λ
√
2π
)}
+
a
λ
e−2πa2/λ2
, (49)
b∗1(3) =
V − 3v0 +
81
32
v2
0
V
+ λ3b∗2(3)
V b∗1(2) + λ3b∗2(2)
, (50)
b∗2(3) =
V − v0
V
{a
λ
(
e−2πa2/λ2 − 2e−8πa2/λ2
)
+
1
23/2
(
erf
(
2
a
λ
√
2π
)
− erf
(a
λ
√
2π
))}
+
πa2λ
V
{
a2
λ2
(
e−2πa2/λ2 − 4e−8πa2/λ2
)
+
1
2π
(
e−2πa2/λ2 − e−8πa2/λ2
)}
+
πλ3
12V
{
e−2πa2/λ2
(
a4
λ4
+
1
π
a2
λ2
+
1
2π2
)
−e−8πa2/λ2
(
16
a4
λ4
+
4
π
a2
λ2
+
1
2π2
)}
+
V − 2v0
V
{
1
23/2
(
1− erf
(
2
a
λ
√
2π
))
+ 2
a
λ
e−8πa2/λ2
}
, (51)
b∗3(3) = 32π
∫
∞
0
dQ
Q
{
Q
4π
e−Q2/4π −
∫ a/λ
0
x sinQx e−πx2
dx
}3
. (52)
Our R coincides with the corresponding function from [20]:
R(r12) =
∫
dr3 (1 + f13)(1 + f23) = V − 2v0 + g1(r12), (53)
The function g1(r12) is given by
g1(r) =
∫
dr3 f13f23 =
{
2
3
πa3
(
2− 3
2
r
a
+ 1
8
(
r
a
)3)
, r < 2a
0, r > 2a
. (54)
Meanwhile we cannot write exact expressions for greater N . The comparison of
the exact results and those in CA and FVA is presented in table 2.
Our expressions for the partition function have the correct form Z
(cl)
N in the
classical limit (~ → 0) [21].
Z
(cl)
N =
1
N !
1
λ3N/2
∫
dr1 . . .
∫
drN
∏
16i<j6N
P (i, j). (55)
It means that the proposed method is applicable to a wide temperature range.
6. Conclusion
It is shown that the proposed method leads to a good agreement between the
calculated results and the experimental data. We obtained not only qualitative but
even satisfactory quantitative fit as well.
443
I.O.Vakarchuk, A.A.Rovenchak
Table 2. Results for the interacting systems
Approximation Exact values
N
T
(N)
c Cmax
N /N T
(N)
c Cmax
N /N
1 — 1.50 — 1.50
2 2.86 1.67 2.86 1.67
3 2.54 1.78 2.19 1.83
4 2.28 1.90 — —
5 2.25 2.01 — —
6 2.24 2.11 — —
7 2.23 2.20 — —
8 2.23 2.29 — —
9 2.23 2.37 — —
10 2.23 2.44 — —
11 2.23 2.50 — —
12 2.23 2.56 — —
13∗ 2.23 2.61 — —
18 2.23 2.82 — —
∞ 2.2–2.3 13.6∗∗ — —
∗ We suppose that the system of 12–13 particles behaves almost as a bulk since
the atoms in 4He are packed compactly enough to say that one atom is sur-
rounded by 12 atoms. Therefore, 13 atoms constitute the first closed sphere
(see figure 5).
∗∗ The experimental value is ≃ 16. The heat capacity in the λ-point was believed
to be infinite for a long time but recent experiments show the finite peak [7].
444
Bose-system with a small number of particles
Figure 4. Cmax
N /N values for the hard spheres system. Solid line shows depen-
dence (29), circles correspond to the data from table 2.
(a) (b)
Figure 5. Helium-4 atoms as hard spheres. (a) The first closed sphere of 12 atoms
surrounding 1 central atom. One atom is not seen since it is situated exactly
behind the central atom. (b) Solid line shows the “surface of atom”, dashed
line shows the “surface of alien atoms”, dashed-dotted line shows “surface of
inaccessible space”. Thus, assuming the volume of “inaccessible space” 4πR3/3
to be the volume occupied by 1 atom one can obtain the hard sphere diameter
a ≃ 2.2 A
◦
.
445
I.O.Vakarchuk, A.A.Rovenchak
The expansions over 1/N obtained in this work demonstrate the existence of
some parameter ε having the value of 1/3 for the ideal system. We estimated ε =
0.0608 from the numerical analysis of the results. The theoretical calculation of this
parameter will be our next problem along this line of research.
Acknowledgements
The authors appreciate Dr. Daniel G. Friend’s help with the helium-4 thermo-
physical data.
References
1. Bogoliubov N.N. // J. Phys. (USSR), 1947, vol. 9, p. 23.
2. Brout R. // Phys. Rev., 1963, vol. 131, p. 899.
3. Feynman R.P. Statistical Mechanics. Mass., W. A. Benjamin, 1972.
4. Feynman R.P. // Phys. Rev., 1953, vol. 91, p. 1291, p. 1301.
5. Langer I.S. // Phys. Rev., 1968, vol. 167, p. 183.
6. Vakarchuk I.A. // Teor. Mat. Fiz., 1978, vol. 35, p. 76; Teor. Mat. Fiz., 1978, vol. 36,
p. 122 (in Russian); Theor. Math. Phys., 1978, vol. 35, p. 327.
7. Lipa J.A., Swanson D.R., Nissen J.A. // Phys. Rev. Lett., 1996, vol. 76, p. 944.
8. Kleinert H. // Phys. Rev. D, 1999, vol. 60, p. 085001.
9. Kleinert H. // preprint cond-mat/9906107.
10. Campostrini M., Pelissetto A., Rossi P., Vicari E. // Phys. Rev. B, 2000, vol. 61,
p. 5905.
11. Ceperley D.M. // Rev. Mod. Phys., 1995, vol. 67, p. 279.
12. Deng Wenji, Hui P.M. // Solid State Communications, 1997, vol. 104, p. 797.
13. Schmidt H.-J., Schnack J. // Physica A, 1998, vol. 260, p. 479.
14. Borrmann P., Franke G. // J. Chem. Phys., 1993, vol. 98, p. 2484.
15. Vakarchuk I.O. // J. Phys. Stud. (Lviv), 1996, vol. 1, p. 25 (in Ukrainian); J. Phys.
Stud. (Lviv), 1997, vol. 1, p. 156 (in Ukrainian).
16. Huang K. Statistical Mechanics. New-York, Wiley, 1987.
17. Vakarchuk I.O. // Visn. Lviv. un-tu, Ser. fiz., 1993, vol. 26, p. 29 (in Ukrainian).
18. Arp V.D., McCarty R.D., Friend D.G. // Natl. Inst. Stand. Technol., Tech. Note 1334
(revised), 1998.
19. Coleman M., Lipa J.A. // Phys. Rev. Lett., 1995, vol. 74, p. 286.
20. Nijboer B.R.A., Van Hove L. // Phys. Rev., 1952, vol. 85, p. 777.
21. Isihara A. Statistical Physics. New York, Academic Press, 1971.
446
Bose-system with a small number of particles
Термодинаміка Бозе-системи з малим числом
частинок
І.О.Вакарчук, А.А.Ровенчак
Кафедра теоретичної фізики,
Львівський національний університет імені Івана Франка,
79005 Львів, вул. Драгоманова, 12
Отримано 28 травня 2001 р.
Запропоновано теоретичний опис взаємодіючої Бозе-системи, який
ґрунтується на екстраполяції результатів, отриманих для систем з
малим числом частинок N = 2, 3, 4 і т.д. на випадок N = ∞ . Пока-
зано, що вже система з N = 12, 13 поводиться майже як безмежна у
широкому температурному діапазоні. Приділено особливу увагу фа-
зовим переходам у цих системах. В обчисленнях використано потен-
ціал твердих сфер. Послідовність максимумів теплоємности апрок-
симовано як Cmax
N
/N ≃ 13.6 − aN−ε з ε = 0.0608 , що дає для тепло-
ємности безмежного зразка значення 13.6, експериментальне зна-
чення близьке до 16. Температуру λ -переходу оцінено як 2.1–2.3 K
(експеримент дає 2.17 K). Досягнуто цілком добре якісне і задовіль-
не кількісне узгодження з експериментальними даними.
Ключові слова: бозонні системи, скінчені системи,
лямбда-перехід, питома теплоємність
PACS: 05.30.Jp, 74.25.Bt
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