The relation between fractional statistics and finite bosonic systems in one-dimensional case
The equivalence is established between the one-dimensional (1D) Bose-system with a finite number of particles and the system obeying the fractional (intermediate) Gentile statistics, in which the maximum occupation of single-particle energy levels is limited. The system of 1D harmonic oscillators is...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1171412017-05-21T03:02:37Z The relation between fractional statistics and finite bosonic systems in one-dimensional case Rovenchak, A. Низкоразмерные и неупорядоченные системы The equivalence is established between the one-dimensional (1D) Bose-system with a finite number of particles and the system obeying the fractional (intermediate) Gentile statistics, in which the maximum occupation of single-particle energy levels is limited. The system of 1D harmonic oscillators is considered providing the model of harmonically trapped Bose-gas. The results are generalized for the system with power energy spectrum. 2009 Article The relation between fractional statistics and finite bosonic systems in one-dimensional case / A. Rovenchak // Физика низких температур. — 2009. — Т. 35, № 5. — С. 510-513. — Бібліогр.: 32 назв. — англ. 0132-6414 PACS: 05.30.Ch, 05.30.Jp, 05.30.Pr http://dspace.nbuv.gov.ua/handle/123456789/117141 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Rovenchak, A. The relation between fractional statistics and finite bosonic systems in one-dimensional case Физика низких температур |
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The equivalence is established between the one-dimensional (1D) Bose-system with a finite number of particles and the system obeying the fractional (intermediate) Gentile statistics, in which the maximum occupation of single-particle energy levels is limited. The system of 1D harmonic oscillators is considered providing the model of harmonically trapped Bose-gas. The results are generalized for the system with power energy spectrum. |
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Article |
| author |
Rovenchak, A. |
| author_facet |
Rovenchak, A. |
| author_sort |
Rovenchak, A. |
| title |
The relation between fractional statistics and finite bosonic systems in one-dimensional case |
| title_short |
The relation between fractional statistics and finite bosonic systems in one-dimensional case |
| title_full |
The relation between fractional statistics and finite bosonic systems in one-dimensional case |
| title_fullStr |
The relation between fractional statistics and finite bosonic systems in one-dimensional case |
| title_full_unstemmed |
The relation between fractional statistics and finite bosonic systems in one-dimensional case |
| title_sort |
relation between fractional statistics and finite bosonic systems in one-dimensional case |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2009 |
| topic_facet |
Низкоразмерные и неупорядоченные системы |
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http://dspace.nbuv.gov.ua/handle/123456789/117141 |
| citation_txt |
The relation between fractional statistics and finite bosonic systems in one-dimensional case / A. Rovenchak // Физика низких температур. — 2009. — Т. 35, № 5. — С. 510-513. — Бібліогр.: 32 назв. — англ. |
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Физика низких температур |
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2025-07-08T11:42:58Z |
| last_indexed |
2025-07-08T11:42:58Z |
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| fulltext |
Fizika Nizkikh Temperatur, 2009, v. 35, No. 5, p. 510–513
The relation between fractional statistics and finite
bosonic systems in one-dimensional case
Andrij Rovenchak
Department for Theoretical Physics, Ivan Franko National University of Lviv, 12 Drahomanov Str., Lviv UA-79005, Ukraine
E-mail: andrij.rovenchak@gmail.com
Received November 13, 2008, revised December 8, 2008
The equivalence is established between the one-dimensional (1D) Bose-system with a finite number of
particles and the system obeying the fractional (intermediate) Gentile statistics, in which the maximum oc-
cupation of single-particle energy levels is limited. The system of 1D harmonic oscillators is considered pro-
viding the model of harmonically trapped Bose-gas. The results are generalized for the system with power
energy spectrum.
PACS: 05.30.Ch Quantum ensemble theory;
05.30.Jp Boson systems;
05.30.Pr Fractional statistics systems (anyons, etc.).
Keywords: finite bosonic systems, traped Bose-gas, fractional statistics.
1. Introduction
The observation of Bose–Einstein condensation
(BEC) in ultracold trapped alkali gases [1,2] gave new
stimulus to the study of this quantum phenomenon. In
particular, the effects of a finite number of particles on
BEC of an ideal gas was discussed in theoretical works
[3–8], where the corrections to the bulk properties were
found.
The aim of the present study is to propose the descrip-
tion of a bosonic system with a finite number of particles
by means of finding such a model system, for which
the treatment might be mathematically simpler. To some
extent, such an approach has common features with find-
ing boson–fermion equivalence in ideal gases [9] or
Tonks–Girardeau gas [10] achieved experimentally in
2004 [11,12]. The anyon–fermion mapping is also known
in the application to ultracold gases [13].
Haldane’s exclusion statistics [14] was considered by
Berg�re [15]. The connection of the exclusion (anyon)
statistics parameter and the interaction in one-dimensi-
onal systems was studied in Refs. 16–18. Recently, a com-
binatorial interpretation of exclusion statistics was given
by Comtet et al. [19].
Another approach is seen in a different type of the frac-
tional statistics, which is formally understood as an inter-
mediate one between Fermi and Bose statistics. Namely,
the maximum occupation of a particular energy level is
limited to M, with M �1 corresponding to the fermionic
distribution and M �� being the bosonic one, respec-
tively. This statistics is known as the Gentile statistics
[20–22]. If the relation between the number of particles N
in the real system and the parameter M in the model one
can be found, the stated problem is solved.
For simplicity, a one-dimensional system is consid-
ered. The paper is organized as follows. Microcanonical
approach for harmonic oscillators with single-particle en-
ergy levels given by � �m m� � is considered in Sec. 2. The
oscillators, unlike classical particles, are indistinguish-
able reproducing thus a quantum case. Physically, this
corresponds to bosons trapped in a highly asymmetric
harmonic trap. In Sec. 3, the same system is treated within
canonical and grand-canonical approaches. Section 4
contains the generalization of obtained results for the sys-
tem with power energy spectrum �m
sm� . Short discus-
sion in Sec. 5 concludes the paper.
2. Microcanonical approach
The number of microstates à E( ) in the system of 1D
oscillators is the number of ways to distribute the energy
E n� �� over the (indistinguishable) particles. Such a
problem reduces to the problem in number theory known
as the partition of an integer [23–25]. An asymptotic ex-
© Andrij Rovenchak, 2009
pression for (unrestricted) partition is given by the well-
known Hardy–Ramanujan formula [26]:
p n
n
n( ) �
1
4 3
2 3e� / . (1)
Thus one obtains:
à E
E
E( )
/
�
1
4 3
2 3
�
�
�
� �e / / . (2)
Using the entropy S � ln from the definition of the
temperature1/ /T dS dE� the following equation of state
is obtained:
E
T
� �
�
�
�
�
�
�
2 2
6
�
�
. (3)
As the energy E is an extensive quantity, E N� , where N
is the number of particles, the thermodynamic limit
�N � const follows immediately from the above equation.
The same result also might be obtained from different
considerations [27].
If one considers a finite system of bosons or a system
of particles obeying fractional statistics the number of
ways to distribute the energy E n� �� over N particles is
the problem of restricted partitions of an integer number n
[25]. For convenience, hereafter �� is the unit of both
energy and temperature.
The expression for the finite system is given by the
number of partitions of n into at most N summands and
asymptotically equals [28]:
à n
n
nn
N
n
fin e e( ) exp�
�
�
�
��
�
�
�
��
1
4 3
62 3 6�
�
�
/ . (4)
The result reducing to the fractional statistics was con-
sidered by Srivatsan et al. [29], it corresponds to the num-
ber of partitions of n where every summand appears at
most M times:
à n
n
n
M M
frac exp( ) /�
�
�
�
�
�
�
�
��
�
�
�
��
�1
4 3
2 3 1
1
1
1
1 2
�
/
�
�
�
1 2/
.
(5)
As one can see, Ã E( ) from Eq. (2) constitutes the lead-
ing factor both in Eqs. (4) and (5). The respective entro-
pies are
Sfin = ln Ãfin = ln à + �Sfin and Sfrac = ln Ãfrac = ln à + �Sfrac .
(6)
Comparing the corrections �S fin and �S frac , one finds the
equivalence condition linking the maximum occupation
parameter M and the number of particles N :
M
N
n
�
�
�
�
�exp
�
6
.
3. Canonical and grand-canonical approach
It is straightforward to show that in the case of the de-
fined fractional statistics the occupation number of the
energy level � equals [20–22]
f T
M
M T M T
( , , )
( ) ( )( )
� �
� � � �
�
�
�
1
1
1
11e e/ /
, (7)
where� is the chemical potential and T is the temperature.
The chemical potential is related to the number of par-
ticles � as follows:
� �
�
�
� f TM
j
j
0
( , , )� � (8)
and energy E equals
E f Tj
j
M j�
�
�
�� � �
0
( , , ) . (9)
However, the case of a finite system is much easier to
implement in the canonical approach. It is possible to
show that the partition function of N indistinguishable 1D
oscillators is given by (cf. [24])
ZN
j T
j
N
�
�
�
�
�
�
� 1
1
1
e / , (10)
from which the energy EN can be calculated. In the limit
of large N the leading term is given by
E E
N
N
N T
T
�
Bose
e
e
/
/1 1
, (11)
where EBose is the energy of an infinite bosonic system.
For the fractional-statistics system the grand-canoni-
cal approach is used. The fugacity z T� e� / is represented
as z z z� �Bose � with z Bose satisfying
� �
� 1
11z i T
i Bose e
� /
. (12)
It is found that �z M�1/ in the limit of large M, from
which the correction to the energy given by Eq. (9) fol-
lows:
E E
M
M
�Bose
1
. (13)
Comparing Eqs. (11) and (13) one obtains the following
relation between the parameters M and N :
M
N
N T�
1
e / . (14)
In the exponent, the temperature T is related to the energy
level n of (6) via Eq. (3) (with E n� ). Result (14) thus re-
produces the microcanonical one (6) up to the negligible
factor of 1/ N — it must be taken into account that
The relation between fractional statistics and finite bosonic systems in one-dimensional case
Fizika Nizkikh Temperatur, 2009, v. 35, No. 5 511
only leading terms were preserved in the logarithms of (4)
and (5).
4. Power energy spectrum
In this section, a general power energy spectrum
� � ams (s � 0) is considered. By choosing appropriate en-
ergy units, one can set the constant a �1. In fact, only s �1
and s � 2 cases are realized in real physical systems [29],
but other values can effectively occur in some exotic
model systems or in the density of states of a system con-
fined by an external potential within a WKB approach.
To obtain à nfin ( ) for arbitrary s it is worth to recall
briefly the derivation of the expression for restricted par-
titions from [25].
Partition function Z( )� and the number of microstates
à E( ) are related via the Laplace transform:
Z à E dEE( ) ( )� ��
�
�
0
e , Ã E
i
Z d
i
i
E( ) ( )�
�
� �
�
1
2�
� ��e .
(15)
The entropy S( )� equals
S E Z( ) ln ( )� � �� � . (16)
For energy spectrum �m
sm� the partition function is
Z
m
ms
( ) ( )� ��
�
�
�
1
11 e . (17)
Using the saddle-point method, one can evaluate à E( )
(15) as follows:
à E
S
S
( )
exp [ ( )]
( )
�
��
�
� �
0
02
. (18)
The entropy S( )� , after applying the Euler–Maclaurin
summation formula, can be expressed in such a form
S E E
C s
m
m
s
s
( ) ln
( )
ln ,
/
� � �
�
���
�
�
�
� � � � �
�
�
�
1
1
1
1
2
e �
(19)
where
C s
s s
( ) � ��
�
�
� ��
�
�
�� 1
1
1
1
, (20)
�( )z and ( )z being Euler’s gamma-function and Rie-
mann’s zeta-function, respectively.
The stationary point� 0 is
� !0
1
1� �
�
�
� �
�
�C s
sE
E
s s
s
s s( )
/ ( )
/ ( ). (21)
Thus, the number of microstates is
à E
s
s
E s Es
s
s
s
s
s( )
( )
exp ( )
( )/
( )�
�
�
"
�
�
� �!
�
!
2 1
1
1 2
3 1
2 1
1
1
#
$
$
%
&
'
'
.
(22)
Substituting E with n one can obtain the well-known
Hardy–Ramanujan formula [26] for the number of parti-
tions of an integer n into the sum of sth powers.
When the number of particles N in the system is finite,
the correction to the above formula must be found. In this
case, the partition function equals
ln ( ) lnZN
m
N
ms
� ��
�
�
�
�
�
�
1
1 e (23)
and for the entropy one has
S E
m
N
ms
fin e( ) ln� � ��
�
�
�
�
�
�
1
1 . (24)
After simple transformations it is easy to obtain the fol-
lowing:
S S
s s
N
s
s
fin ( ) ( ) ,
/
� �
�
��
�
�
�
�
1 1
1
� , (25)
where �( , )a x is incomplete �-function. Thus,
à E à E
s s
N
s
s
fin ( ) ( ) exp ,
/
�
�
�
�
�
"
#
$
$
%
&
'
'
1 1
0
1 0
�
�� . (26)
Applying the asymptotic expansion for �( , )a x [30,
Eq. (6.5.32)], we finally arrive at the following:
à E à E
s
N s N s
fin e( ) ( ) exp�
"
#
$
%
&
'
1
0
1 0
�
�
. (27)
Substituting E with n one obtains the result for re-
stricted partitions
à n à n
s
n N
s
s s s N ns
s s s
fin e( ) ( ) exp / ( )
/( )
�
"
#
$
�
�1 1 1
1
!
! %
&
' ,
(28)
cf. also Eq. (17) from [31]. For this function, the notation
p nN
s ( ) is traditionally used, note, however, that in the
problem of integer partitions s must be integer. For s �1
the obtained expression reduces to that of Erd�s and
Lehner [28], see Eq. (4).
The fractional-statistics result can be directly taken
from [29]
à n
s
s M
s
s s
s s
frac ( )
( ) ( )( )/ /
/ (
�
�
�
�
�
�
�
��
�
!
�2 1
1
1
11 2 1
1)
(
(
�
�
�
�
�
� �
�
�
�
n
M
s n
s
s
s s
s s3 1
2 1
1
1
1
1
1
1( )
/
/ ( )
exp
( )
( )!
1
1s�
"
#
$
$
%
&
'
'
. (29)
To obtain the relation between the parameters M and
N , one can again consider the entropies S Ãfrac frac� ln
and S Ãfin fin� ln :
512 Fizika Nizkikh Temperatur, 2009, v. 35, No. 5
Andrij Rovenchak
S S
s
N
S S
s
s
M
s N
s s
s
frac
fin
e
�
�
�
1
1
0
1
1
0
�
!
�
,
,/
(30)
where S � ln .
Dropping the constants, the following result is ob-
tained:
M n N n Ns
s
s s
s
s
s s1
1
1 1 1/ exp�
�
�
�
��
�
�
�
��
�
�! . (31)
It is interesting to find in this general case the connec-
tion between energy E and temperature T from the defini-
tion 1/ /T dS dE� :
1
1 1
T
E E Ts
s
s
s
s
s� ) �
� �! ! . (32)
Thus, the leading contribution in the relation of M and N
(31) is
M
sN
T
s
�
�
�
�
�
�
exp , (33)
which is compatible with (14).
5. Discussion
To summarize, the equivalence is established between
the finite bosonic system and the system obeying frac-
tional (intermediate) Gentile statistics in the case of
one-dimensional harmonic trap. This approach is ex-
tended to a general power energy spectrum. While the ex-
pressions for two-dimensional (2D) partitions are also
known [23], the application to asymmetric (elliptical)
traps as well as the generalization for arbitrary 2D
systems needs additional study.
Interacting systems are of special interest now. Weak
interactions are known not to change the properties of a
Bose-system drastically. Thus, one can use, e.g., a slight-
ly modified excitation spectrum [32] and, upon calculat-
ing the properties of a model fractional-statistics system,
obtain the results for a finite one from the established
equivalence.
I am grateful to my colleagues Prof. Volodymyr Tka-
chuk, Dr. Taras Fityo, and Yuri Krynytskyi for useful dis-
cussions and comments regarding the presented material.
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The relation between fractional statistics and finite bosonic systems in one-dimensional case
Fizika Nizkikh Temperatur, 2009, v. 35, No. 5 513
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