Ideal Bose-gas in nonadditive statistics
The paper analyzes an approach to the generalization of the conventional Bose–Einstein statistics based on the nonadditive entropy of Tsallis. A detailed derivation of thermodynamic functions is presented. The calculations are made for the specific heat of two model systems, namely, the ideal three...
Saved in:
| Published in: | Физика низких температур |
|---|---|
| Date: | 2018 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2018
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/176264 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Ideal Bose-gas in nonadditive statistics / A. Rovenchak// Физика низких температур. — 2018. — Т. 44, № 10. — С. 1308-1315. — Бібліогр.: 35 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-176264 |
|---|---|
| record_format |
dspace |
| spelling |
Rovenchak, A. 2021-02-04T07:56:24Z 2021-02-04T07:56:24Z 2018 Ideal Bose-gas in nonadditive statistics / A. Rovenchak// Физика низких температур. — 2018. — Т. 44, № 10. — С. 1308-1315. — Бібліогр.: 35 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/176264 The paper analyzes an approach to the generalization of the conventional Bose–Einstein statistics based on the nonadditive entropy of Tsallis. A detailed derivation of thermodynamic functions is presented. The calculations are made for the specific heat of two model systems, namely, the ideal three-dimensional gas obeying the nonadditive modification of the Bose–Einstein statistics and the system with linear excitation spectrum attempted as a qualitative approximation of liquid ⁴He thermodynamics. Проаналізовано підхід до узагальнення традиційної статистики Бозе–Ейнштейна на основі неадитивної ентропії Цалліса. Подано докладне виведення термодинамічних функцій. Зроблено розрахунки для питомої теплоємності двох модельних систем, а саме: ідеального тривимірного газу, що підкоряється неаддитивній модифікації статистики Бозе–Ейнштейна, та системи з лінійним спектром елементарних збуджень, прийнятої за якісне наближення термодинаміки рідкого ⁴He. Проанализирован подход к обобщению традициионной статистики Бозе–Эйнштейна на основе неаддитивной энтропии Цаллиса. Представлен подробный вывод термодинамических функций. Сделаны расчеты для удельной теплоемкости двух модельных систем, а именно: идеального трехмерного газа, подчиняющегося неаддитивной модификации статистики Бозе–Эйнштейна, и системы с линейным спектром элементарных возбуждений, принятой как качественное приближение термодинамики жидкого ⁴He. This work was partly supported by Project FF-30F (No. 0116U001539) from the Ministry of Education and Science of Ukraine. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Актуальні проблеми квантових рідин та кристалів Ideal Bose-gas in nonadditive statistics Ідеальний бозе-газ з неадитивною статистикою Идеальный бозе-газ с неаддитивной статистикой Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Ideal Bose-gas in nonadditive statistics |
| spellingShingle |
Ideal Bose-gas in nonadditive statistics Rovenchak, A. Актуальні проблеми квантових рідин та кристалів |
| title_short |
Ideal Bose-gas in nonadditive statistics |
| title_full |
Ideal Bose-gas in nonadditive statistics |
| title_fullStr |
Ideal Bose-gas in nonadditive statistics |
| title_full_unstemmed |
Ideal Bose-gas in nonadditive statistics |
| title_sort |
ideal bose-gas in nonadditive statistics |
| author |
Rovenchak, A. |
| author_facet |
Rovenchak, A. |
| topic |
Актуальні проблеми квантових рідин та кристалів |
| topic_facet |
Актуальні проблеми квантових рідин та кристалів |
| publishDate |
2018 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| title_alt |
Ідеальний бозе-газ з неадитивною статистикою Идеальный бозе-газ с неаддитивной статистикой |
| description |
The paper analyzes an approach to the generalization of the conventional Bose–Einstein statistics based
on the nonadditive entropy of Tsallis. A detailed derivation of thermodynamic functions is presented. The calculations are made for the specific heat of two model systems, namely, the ideal three-dimensional gas obeying
the nonadditive modification of the Bose–Einstein statistics and the system with linear excitation spectrum attempted as a qualitative approximation of liquid ⁴He thermodynamics.
Проаналізовано підхід до узагальнення традиційної статистики Бозе–Ейнштейна на основі неадитивної ентропії Цалліса. Подано докладне виведення термодинамічних функцій.
Зроблено розрахунки для питомої теплоємності двох модельних систем, а саме: ідеального тривимірного газу, що підкоряється неаддитивній модифікації статистики Бозе–Ейнштейна,
та системи з лінійним спектром елементарних збуджень,
прийнятої за якісне наближення термодинаміки рідкого ⁴He.
Проанализирован подход к обобщению традициионной
статистики Бозе–Эйнштейна на основе неаддитивной энтропии Цаллиса. Представлен подробный вывод термодинамических функций. Сделаны расчеты для удельной теплоемкости
двух модельных систем, а именно: идеального трехмерного
газа, подчиняющегося неаддитивной модификации статистики Бозе–Эйнштейна, и системы с линейным спектром элементарных возбуждений, принятой как качественное приближение термодинамики жидкого ⁴He.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/176264 |
| citation_txt |
Ideal Bose-gas in nonadditive statistics / A. Rovenchak// Физика низких температур. — 2018. — Т. 44, № 10. — С. 1308-1315. — Бібліогр.: 35 назв. — англ. |
| work_keys_str_mv |
AT rovenchaka idealbosegasinnonadditivestatistics AT rovenchaka ídealʹniibozegazzneaditivnoûstatistikoû AT rovenchaka idealʹnyibozegazsneadditivnoistatistikoi |
| first_indexed |
2025-11-25T22:34:38Z |
| last_indexed |
2025-11-25T22:34:38Z |
| _version_ |
1850570937293864960 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 10, pp. 1308–1315
Ideal Bose-gas in nonadditive statistics
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 January 25, 2018, published online August 28, 2018
The paper analyzes an approach to the generalization of the conventional Bose–Einstein statistics based
on the nonadditive entropy of Tsallis. A detailed derivation of thermodynamic functions is presented. The calcu-
lations are made for the specific heat of two model systems, namely, the ideal three-dimensional gas obeying
the nonadditive modification of the Bose–Einstein statistics and the system with linear excitation spectrum at-
tempted as a qualitative approximation of liquid 4He thermodynamics.
Keywords: Bose–Einstein statistics, Tsallis entropy, nonadditive statistics, ideal Bose-gas, 4He.
1. Introduction
A generalization of the classical Boltzmann–Gibbs en-
tropy was suggested by Tsallis [1]. The proposed nonaddi-
tive entropy might be relevant in descriptions of systems
with long-range interactions, for non-Markovian processes
or systems with “memory” and some others [2]. It has also
been used in the description of complex systems beyond
physics [3], for instance, to model DNA sequences [4],
financial data [5] or distributions in linguistics [6].
In the present paper we will focus on the application of
a nonadditive entropy for generalizations of quantum dis-
tributions [7–10]. For integrity, a presentation of the calcu-
lation scheme involving the density of states for a power-law
excitation spectrum is followed by an introductory infor-
mation about the Tsallis statistics and the description of one
possible generalization of the Bose-distribution. A detailed
mathematical derivation of thermodynamic quantities is
complemented by calculations for two systems a three-di-
mensional ideal Bose-gas and a rough model of 4He.
We will consider the following simple scheme for the
calculation of thermodynamic functions. The total number
of particles is given as the sum of mean occupation num-
bers ( , , )jn T zε over all the states with energies jε ,
= ( , , ),j
j
N n T zε∑ (1)
where T is temperature and z is fugacity related to the
chemical potential µ via /= e Tz µ .
The dependence ( , )z T N being the solution of Eq. (1)
can be used to calculate the total energy of the system
= ( , , )j j
j
E n T zε ε∑ (2)
as a function of temperature T and the number of particles N.
Further on, we can obtain the equation of state, calculate
heat capacities and other thermodynamic quantities.
2. Density of states
For generality, let us consider a system of spinless bos-
ons with the following elementary excitation spectrum
= ,b
p apε (3)
where | |p ≡ p is the momentum absolute value and a, b
are the spectrum parameters. In a D -dimensional space,
2 2
1= Dp p p+ + .
Suppose the particles reside in volume D . If it is mac-
roscopically large, the summation over states can be substi-
tuted by the integration over the phase space:
= ( , , ) = ( , , ),j p
j
N n T z d n T zε τ ε∑ ∫ (4)
where the phase space element
1 1
( )( )= , with ( ) = , ( ) = .
(2 )
D DD
dp dqd dp dp dp dq dq dqτ
π
For particles with spin σ , additional multiplicity factor of
(2 1)σ+ would occur.
This relation can be rewritten by introducing the density
of states function ( )g ε :
0
= ( ) ( , , ).N d g n T z
∞
ε ε ε∫ (5)
© Andrij Rovenchak, 2018
Ideal Bose-gas in nonadditive statistics
There is no dependence on coordinates in the spectrum
pε , so the integration over coordinates ( )dq is trivial:
1= ( ) ( ) = ( ) .
(2 ) (2 )
D
D Dd dp dq dpτ
π π∫ ∫ ∫ ∫
Moreover, as the spectrum pε depends only on the abso-
lute value of the momentum p, the integration over mo-
menta in (4) can be done in (hyper-)spherical coordinates,
1
1 = D
D Ddp dp p dpd− Ω :
1
0
= ( , , ),D
D D pN dp p n T z
∞
−Ω ε∫ (6)
where /2= 2 / ( / 2)D
D DΩ π Γ is the solid hyper-angle in
the D -dimensional space.
Making the change of variables = bapε we finally ob-
tain the density of states in the following form [11, p. 150]:
/2
1 12( ) = = ,
( / 2) ( / 2 1)(2 ) (2 )
D s s
s sD
D D D
a sag
D Db
− −
− −π
ε ε ε
Γ Γ +π π
(7)
where = /s D b. In particular, for a gas of free particles
with 2= / 2p p mε one gets [12]
/2
/2 1
2( ) = .
( / 2) 2
D
DD mg
D
−
ε ε Γ π
(8)
Note that for a system of particles trapped to a harmon-
ic potential 2 2
1 1( , , ) = ( ) / 2D DV q q m ω + +ω the density
of states can be shown [12] to have the form
1
1
1 1( ) = , where = .
( ) ( )
D D
DDg
D
−ε ε ω ω ω
Γ ω
(9)
For convenience, we will introduce a shorthand nota-
tion for the constant in the density of states,
11 = , so that ( ) = .
( / 2 1) (2 )
s
s
DD
sa A g A
D
−
−ε ε
Γ + π
(10)
Equation (5) thus becomes as follows:
1
0
= ( , , ).s
D
N A d n T z
∞
−εε ε∫
(11)
Note that the consideration of thermodynamic function
implies the so called thermodynamic limit, / = constDN
as N →∞ and D →∞ . For a system of harmonic oscilla-
tors this condition is written as = constDNω as N →∞
and 0ω→ .
3. Nonadditive statistics of Tsallis
In this section, we will briefly introduce an approach to
generalize the conventional Boltzmann–Gibbs statistics.
For W microstates with probabilities jp the generalized
entropy as proposed by Constantino Tsallis [1] is given by
=1 =1
1= 1 , whereas = 1,
1
W W
q
q jj
j j
S p p q
q
− ∈
−
∑ ∑ . (12)
In the limit of 1q → one easily recovers the Boltzmann–
Gibbs entropy:
( 1) ln1 = e 1 ( 1) ln
q pq j
jjp q p
−− + −
so that
=1 =1
1= 1 = = ln
1
W W
q
q j jj
j j
S p p p
q
− −
−
∑ ∑
as expected.
While the conventional entropy is additive,
( ) = ( ) ( )S A B S A S B+ + in the case of a system split into
two subsystems A and B , the entropy from Eq. (12) can be
shown to satisfy [2]
( ) = ( ) ( ) (1 ) ( ) ( ),q q q q qS A B S A S B q S A S B+ + + − (13)
meaning it is not an additive quantity. So, the parameter q
might be considered as a measure of nonadditivity.
Similarly to the conventional entropy, qS is maximal if
all the probabilities are equal ( = 1/jp W for all j ):
1 1= .
1
q
q
WS
q
− −
−
(14)
In the limit of 1q → , the well-known relation = lnS W
follows.
Using the so-called q-logarithm,
1
1
1ln , whereas ln = ln ,
1
q
q
xx x x
q
− −
≡
−
(15)
one can write the Tsallis entropy in the following Boltz-
mann-like form:
= ln .q qS W (16)
An inverse to the q-logarithm is given by the following
function:
1/(1 )
1/(1 )
exp( ), for = 1,
e = [1 (1 ) ] , for 1 and 1 (1 ) > 0,
0 , for 1 and 1 (1 ) 0,
x q
q
q
x q
q x q q x
q q x
−
−
+ − ≠ + −
≠ + − ≤
(17)
which is known as the Tsallis q-exponential [13]. Its
graphs are shown in Fig. 1.
There is a number of approaches based on the nonaddi-
tive entropy of Tsallis [14–17]. It appears in particular that
the link between the entropy, energy, and temperature can-
not be trivially replicated from the Boltzmann–Gibbs sta-
tistics. Strategies to generalize the Bose–Einstein and Fer-
mi–Dirac distributions also vary [7,9,18,19].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 10 1309
Andrij Rovenchak
4. Generalization of the Bose-statistics
We will demonstrate a rather simple approach to general-
ize the quantum Bose-distribution. The Gibbs factor
( )/ 1 /e = eT Tzε−µ − ε will be substituted to 1 /e T
qz− ε . Note that
the Tsallis q-exponentials do not obey the factorization rule,
e e e ,x y x y
q q q
+ ≠ (18)
but satisfy the following relations instead [20]:
1 1 2
1 (1 )/(e e ) = (e ) (1 ) , e = e .
p pxx y q x y q x
q q q q q pq xy− + −
− −
+ − (19)
We thus cannot relate the chemical potential and fugacity
by a simple link, /e T
qz µ≠ . On the other hand, such an ap-
proach facilitates comparisons between various types of
fractional statistics via virial and cluster expansions [11],
as shown for several nonadditive modifications in [21–23].
The generalized nonadditive Bose-distribution applied
in this work is as follows
1 /
1( , , ) = .
e 1T
q
n z T
z− ε
ε
−
(20)
There is only a seemingly subtle difference between this
definition and, for instance, the one proposed in [18,24],
where the fugacity was introduced as /= e T
q qz µ and the oc-
cupation numbers were
( )/
1( , , ) = .
e 1T
q
n T
ε−µ
ε µ
−
(21)
However, this difference significantly affects the high-tem-
perature behavior, as we will see further.
5. Critical temperature
As 0e = 1q , we observe a situation similar to that in ordinary
Bose-distribution. Namely, the ground-state contribution is
1
1(0, , ) = ,
1
n z T
z− −
(22)
which tends to infinity at 1z → . Physically it means that
the occupation of the ground state becomes macroscopical-
ly large, as in the ordinary Bose–Einstein condensation. On
the other hand, this contribution is neglected if we consider
the density of states (7) at > 1s since ( )g ε vanishes as
0ε → . As a consequence, we should write the ground state
occupation 0 (0, , )N n z T≡ explicitly,
1
0 1 /
0
= ,
e 1
s
D T
q
dN N A
z
∞ −
− ε
ε ε
+
−∫ (23)
where 0 = 0N (i.e., it is a microscopic number, 0N N )
and < 1z for temperatures above some critical value cT
and 0 > 0N and = 1z below cT . The critical temperature is
defined by the condition
1
/
0
= .
e 1
s
D Tcq
dN A
∞ −
ε
ε ε
−
∫ (24)
Eliminating the temperature dependence in the integrand
we obtain
1
0
= .
e 1
s
s
c x
D q
N x dxAT
∞ −
−∫
(25)
Evaluation of the integral in the definition of the critical
temperature can be made similarly to the ordinary Bose-
statistics. So,
11
1 1
1
=10 0 0
(e )
= = (e ) .
e 1 1 (e )
xs qs s x k
qx x
kq q
x dx dxx dxx
−∞ ∞ ∞ ∞−
− − −
−− −
∑∫ ∫ ∫
For = 1q this yields
1
0
= ( ) ( ),
e 1
s
x
x dx s s
∞ −
ζ Γ
−∫
where Riemann’s zeta-function reads (for > 1s ):
=1
1( ) = .s
k
s
k
∞
ζ ∑ (26)
Other cases, namely < 1q and > 1q , should be treated a bit
differently.
For < 1q , the argument 1 (1 )q x+ − remains positive for
all 0x ≥ , so
[ ]
1
1 1
=10 0
= 1 (1 )
e 1
s k
s qx
kq
x dx dxx q x
∞ ∞∞−
−−
−+ − =
−
∑∫ ∫
=1
1= B , ,
1(1 )s
k
ks s
qq
∞
− −−
∑ (27)
where the beta-function B( , ) = ( ) ( ) / ( )u u uΓ Γ Γ +v v v .
Fig. 1. (Color online) Tsallis q -exponential ex
q for different va-
lues of the parameter q .
1310 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 10
Ideal Bose-gas in nonadditive statistics
For > 1q , one has to set the upper limit of integration
0 = 1/ ( 1)x q − , where 1 (1 )q x+ − becomes negative. So,
[ ]
01
1 1
=10 0
= 1 (1 )
e 1
xs k
s qx
kq
x dx dxx q x
∞ ∞−
−−
−+ − =
−
∑∫ ∫
=1
1= B , 1 ,
1( 1)s
k
ks
qq
∞
+ −−
∑ (28)
To be consistent with the = 1q case we can formally in-
troduce a q-generalized zeta-function in the following man-
ner, cf. [18,24]:
=1
=1
( ) for = 1,
1( ) = B , for < 1,
1(1 ) ( )
1 B , 1 for > 1,
1( 1) ( )
q s
k
s
k
s q
ks s s q
qq s
ks q
qq s
∞
∞
ζ
ζ − −− Γ
+ −− Γ
∑
∑
(29)
so that
1
0
= ( ) ( )
e 1
s
qx
q
x dx s s
∞ −
ζ Γ
−∫ (30)
with the definition of the critical temperature in Eq. (25)
becoming consistent with the ordinary Bose-condensation
temperature:
= ( ) ( ).s
c q
D
N AT s sζ Γ
(31)
As a bonus from such a form, the factor of ( )sΓ explicitly
present in the above formula cancels the same factor in the
denominators of the density of states given by Eqs. (8), (9).
Note that from the asymptotic behavior of the beta-func-
tion at large v while keeping u fixed,
B( , ) ( ) ,uu u −Γv v
one easily recovers Eq. (26) from ( )q sζ in the limits of
1q → . Unfortunately, a general closed-form expression for
( )q sζ without infinite summations has not been found.
6. Low temperature
For temperatures below the critical one cT , where = 1z ,
the calculations are quite simple. From Eq. (23) using (30)
we obtain
1
0 0/
0
= = ( ) ( ),
e 1
s
s
D D qT
q
dN N A N A T s s
∞ −
ε
ε ε
+ + ζ Γ
−∫ (32)
or applying the critical temperature definition (31),
0 = 1 ,
s
c
N T
N T
−
(33)
which coincides with the expression for the condensate
fraction (relative number of particles with zero momenta)
in ordinary ideal Bose-gas.
The calculation of energy is also simple
1
/
0
= = ( 1) ( 1),
e 1
s
s
D D qT
q
dE A A T s s
∞
+
ε
ε ε
ζ + Γ +
−∫ (34)
which can be written as
( 1)
= ,
( )
s
q
q c
s TE s NT
s T
ζ +
ζ
(35)
again being consistent with the energy of an ideal Bose-gas
at = 1q .
The isochoric heat capacity is thus proportional to sT :
( 1)
= = ( 1) .
( )
s
q
V
q cV
sE TC s s N
T s T
ζ + ∂ + ∂ ζ
(36)
The calculations above cT would require a more elabo-
rated approach involving in particular a q-generalization of
the polylogarithm function ,Li ( )q s z . For instance, at < 1q ,
the number of particles is defined by the integral
[ ]
1
1 11
=10 0
= 1 (1 )
e 1
s k
k s qx
kq
x dx z dxx q x
z
∞ ∞∞−
−−
−−
+ − =
−
∑∫ ∫
,
=1
1= B , ( )Li ( )
1(1 )
k
q ss
k
kz s s s z
qq
∞
− ≡ Γ −−
∑ , (37)
and similarly for > 1q . This new function reduces to the
ordinary polylogarithm at 1q → ,
1,
=1
Li ( ) Li ( ) = .
k
s s s
k
zz z
k
∞
≡ ∑ (38)
However, no simple recursive relations can be written for
derivatives ,Li ( )q s z , unlike the undeformed case ( = 1q ) or
the approach of [18,24]. Consequently, only a cumbersome
expression would be obtained for the heat capacity calling
rather for simple numerical evaluation instead.
In particular, one can show that the behavior of the iso-
choric heat capacity at the critical temperature is qualita-
tively the same as in the ordinary ideal Bose-system [25].
Namely, VC is continuous for 1 < < 3 / 2s , there is a cusp
on the VC curve for = 3 / 2s , and there is a discontinuity
(a finite jump) of VC for > 3 / 2s .
7. High temperatures and classical limit
At high temperatures, fugacity tends to zero, so approxi-
mately
( ) 11 /
0
= e .s T
D qN z A d
∞ −− εεε∫ (39)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 10 1311
Andrij Rovenchak
In the same approximation, energy is given by
( ) 1/
0
= e .s T
D qE z A d
∞ −εεε∫ (40)
The change of variables = /x Tε yields
( ) 11
0
= es s x
D qN zT A dxx
∞ −−∫ (41)
and
( ) 11
0
= e .s s x
D qE zT A dxx
∞ −+ ∫ (42)
The integration in the above expressions is made using the
same approach as applied in Eqs. (27) and (28), consider-
ing the cases of < 1q and > 1q separately and keeping
only first terms in the sum.
So, for < 1q
1( )
11= B , =
1 1(1 ) (1 )
1
s
sD
Ds s
s s
qzT AN s s zT A
qq q
q
Γ Γ − − − − − − Γ −
(43)
and
1
1= B 1,
1(1 )
s
D
s
zT A qE s s
qq
+
+
+ − = −−
1
1
( 1)
1
= .
1(1 )
1
s
D
s
qs s
q
zT A
q
q
+
+
Γ + Γ − −
− Γ −
(44)
Eliminating fugacity between the above two equations we
obtain
= .
( 1)
sE NT
q s s+ −
(45)
For > 1q we have
( )
1
= B , =
1( 1) ( 1)
1
s
sD
Ds s
qs
qzT A qN s zT A
q qq q s
q
Γ Γ −
− − − Γ + −
(46)
and
1
1= B 1,
1( 1)
s
D
s
zT A qE s
qq
+
+
+ = −−
1
1
( 1)
1
= .
( 1) 1
1
s
D
s
qs
q
zT A
qq s
q
+
+
Γ + Γ −
− Γ + + −
(47)
These again yields for energy the same result as for < 1q ,
namely:
= .
( 1)
sE NT
q s s+ −
(48)
In the case of = 1q it reduces to the well-known classical
limit for energy in the form =E sNT . For instance, a gas
of free particles ( = 2b ) in three dimensions ( = 3D and
= / = 3 / 2s D b ) yields the textbook expression:
3= .
2
E NT
The isochoric heat capacity at hight temperatures is thus
= = .
( 1)V
V
E sC N
T q s s
∂
∂ + −
(49)
Note that the nonadditivity parameter must satisfy the con-
dition > / ( 1)q s s + ensuring that this classical limit re-
mains positive.
8. Results in three dimensions
To demonstrate the application of the developed approach
we have performed calculations for two systems. The first
one is the three-dimensional gas of free particles while
the second one might be considered as a rough model for
liquid 4He.
The density of states (8) of the free 3D system is given by
3/2
3/2 1
2( ) =
(3 / 2) 2
V mg −
ε ε Γ π
(50)
with V standing for the 3D volume 3V ≡ . The critical
temperature is defined by
3/2
2
3= .
22
c
q
mTN
V
ζ
π
(51)
The plots for the specific heat /VC N are shown in Fig. 2
for several values of the nonadditivity parameter q.
When comparing the obtained dependences with [18,24]
we can see the discrepancies in the high-temperature be-
havior of the specific heat, especially for < 1q , where the
approach of [18,24] yielded a minimum on the VC curve at
> cT T . Moreover, it can be shown that the asymptotic is
(1 ) /[1 (1 ) ]q s q s
VC T − − −∝ (see Appendix), not the expected
classical constVC → . These differences demonstrate the
importance of the fugacity definition in the deformed sta-
tistics.
The second system was attempted as a rough model
for the specific heat of 4He [26–28], which was historically
the first Bose-system extensively studied both experimen-
tally and theoretically.
The phonon branch of the elementary excitation spec-
trum in 4He leads to the temperature dependence of the iso-
1312 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 10
Ideal Bose-gas in nonadditive statistics
choric heat capacity at low temperatures as 3T , like in the
Debye model of the solid. So, it fixes the value of = 3s .
The classical limit for the heat capacity of a monoatomic
gas, which is the case of helium, equals = 3 / 2VC N . Thus,
from Eq. (48) we obtain
3= =
( 1) 2V
sC N N
q s s+ −
(52)
yielding = 5 / 4q .
Generally, one can expect that in realistic physical sys-
tems the value of q does not significantly deviate from
unity. For instance, an effective accounting for interactions
in a weakly-interacting Bose system can be achieved with
0.978q [21]. Larger deviations, as obtained from (52),
can occur if strong interactions come into play.
The results of calculations are shown in Fig. 3. Note the
discontinuity at the critical temperature occurring both in
real 4He and in the model nonadditive system, but not in
the ordinary 3D ideal Bose-gas.
As one can see, the specific heat curves are plotted in
the relative temperature scale. The reason is that the criti-
cal temperature of the ideal Bose gas with 4He mass and
density is = 3.14cT K versus the lambda-transition tem-
perature in liquid 4He = 2.17Tλ K. Such a discrepancy can
be solved in particular by introducing the notion of an ef-
fective mass [30] calculated within various approaches
[31–35].
It appears that within the proposed nonadditive model
we cannot achieve simultaneous agreement between the cri-
tical temperature and the speed of sound determining the
factor at 3T in the specific heat as both these values are
defined by a single parameter A in the density of states.
The values of the q-generalized zeta function appearing
in calculations are as follows:
5/4
16(3) = (ln8 2) = 0.423688
3
ζ − , (53)
5/4
32(4) = (ln 64 1) = 0.184431 ,
3
ζ − π− (54)
These yield upon putting = 2.17cT K in (31) and (36):
31 65 [K ]N
A V
(55)
and
3( 1)
= ( 1) 0.51
( )
s
qV
q c
sC Ts s T
N s T
ζ +
+ ζ
(56)
with temperature in Kelvins. For the specific heat of a bo-
sonic phonon system with spectrum =p cpε one obtains:
3
5
3
16= ,
15 ( )
VC V T
N N hc
π (57)
where c is the speed of sound and 2h ≡ π. With 4He pa-
rameters, / = 0.02185N V Å 3− and = 237c m/s this gives
30.01 .VC
T
N
(58)
The discrepancy between Eqs. (56) and (58) means that
the speed of sound in the model system is about 64 m/s.
Such a difference is explained by the fact that the 3T be-
havior of the heat capacity holds only at very low tempera-
tures ( < 0.6T K) and for higher temperatures other types
of excitations, in particular the so called rotons, should be
considered as well [26,27]. In order to adjust the proposed
Fig. 2. (Color online) Specific heat of the three-dimensional ideal
Bose-gas with nonadditive statistics compared to the ordinary
Bose-gas. Green lines show the results consistent with the ap-
proach of [18,24]. Red dashed line joins the cusps of the respec-
tive curves corresponding to critical temperatures. For conven-
ience, the units of temperature and energy are fixed by
3/222 = 1N
V m
π
.
Fig. 3. (Color online) Specific heat of 4He (circles, data from [29])
compared to the ordinary 3D ideal Bose-gas (IBG) and the non-
additive model with = 5 / 4q , = 3s . The data are plotted in the
temperature scale relative to the critical temperatures for each
system.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 10 1313
Andrij Rovenchak
nonadditive model for 4He one may also try to define the
value of s from the specific heat behavior at temperatures
closer to the critical one, where 6 7
VC T ÷∝ [27].
9. Conclusions
To summarize, we have proposed an approach to gener-
alize the Bose–Einstein distribution using nonadditive sta-
tistics of Tsallis. Detailed derivations of thermodynamic
functions have been given for a D -dimensional system
with the elementary excitation spectrum = b
p apε . A phe-
nomenon being an analog of the Bose-condensation has
been detected and the corresponding critical temperature
has been calculated. The low- and high-temperature behav-
ior of energy and specific heat has been analyzed.
Both analytical and numerical calculations have been
made for two model systems. From the analysis of the first
one, an ideal 3D gas, the importance of the fugacity defini-
tion has been revealed in comparison with other approaches.
The second system could be a rough model for the specific
heat of 4He. Its limitations have been briefly discussed. In
particular, it became clear that a single nonadditivity pa-
rameter q is not sufficient to agree both the value of the
critical temperature and the speed of sound with experi-
mental data.
The results of the present work would be useful in ap-
plications of unconventional types of statistics as effective
models of real physical systems.
Acknowledgment
This work was partly supported by Project FF-30F
(No. 0116U001539) from the Ministry of Education and
Science of Ukraine.
Appendix A
Consider the expression for the number of particles N and
energy E in the approach of [18,24]:
( )/ ( )/
0 0
( ) ( )= , = .
e 1 e 1T T
q q
g d g dN E
∞ ∞
ε−µ ε−µ
ε ε ε ε ε
− −∫ ∫ (A1)
Note that these authors originally used a mirrored defini-
tion of the q-exponential with the parameter q related to
the parameter q of the present paper via 1 = 1q q− − .
Upon making a numerical analysis one can conclude
that in the limit T →∞ the chemical potential has the fol-
lowing behavior:
| |= .
T T
µ µ
− → −∞ (A2)
In such a limit, the unity in the denominator of the expres-
sion for occupation numbers can be neglected, so
1 1( )/ 1 | |/
0 0
( ) e = e .T s s x T
q D qN d g AT dxx
∞ ∞− −ε−µ − + µ ε ε ∫ ∫
(A3)
For definiteness, we will consider < 1q . The number of
particles is thus
1
11
0
| |1 (1 ) qs s
DN AT dxx q x
T
−∞
−− µ + − + ∫
11
11 1
0
| |(1 ) 1
| |
qqs s
D
TAT q x x
T
−∞
−− − µ − + = µ ∫
1
1| | 1= | | (1 ) B , .
1
qs
D A q s s
T q
− µ µ − − −
(A4)
In the same fashion the energy is
1
11
0
| |1 (1 ) qs s
DE AT dxx q x
T
−∞
−+ µ + − + ∫
1
11 | | 1| | (1 ) B 1, 1 .
1
qs
D A q s s
T q
−+ µ µ − + − − −
(A5)
So,
1B 1, 1
1
= | |,
1B ,
1
s s
qE
N
s s
q
+ − − − µ
− −
(A6)
while
1
1 (1 )1
11 (1 )| | = .
1B ,
1
s q
q
D
N q T
A
s s
q
− −
−
− µ
− −
(A7)
Therefore, the specific heat depends on temperature as
(1 )
1 (1 ) = .
q s
s qVC
T T
N
−
γ− −∝ (A8)
Such a relation can be shown to hold for > 1q as well. The
numerical values are, in particular, for = 3 / 2s (see also
Fig. 4):
0.1763= 0.9 : = , ,
17 Vq C T +γ + ∝ (A9)
0.2303= 1.2 : = , .
13 Vq C T −γ − ∝ (A10)
1314 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 10
Ideal Bose-gas in nonadditive statistics
_______
1. C. Tsallis, J. Stat. Phys. 52, 479 (1988).
2. Nonextensive Statistical Mechanics and Its Applications,
S. Abe and Y. Okamoto (eds.), Springer, Berlin (2001).
3. Nonextensive Entropy: Interdisciplinary Applications,
M. Gell-Mann and C. Tsallis (eds.), Oxford University Press,
New York (2004).
4. G.P. Pavlos, L.P. Karakatsanis, A.C. Iliopoulos, E.G. Pavlos,
M.N. Xenakis, P. Clark, J. Duke, and D.S. Monos, Physica A
438, 188 (2015).
5. G. Ruiz and A.F. de Marcos, Eur. Phys. J. B 91, 1 (2018).
6. A. Rovenchak and S. Buk, J. Quant. Ling. 25, 1 (2018).
7. F. Büyükklç, D. Demirhan, and A. Güleç, Phys. Lett. A 197,
209 (1995).
8. B. Tanatar, Phys. Rev. E 65, 046105 (2002).
9. H.H. Aragão-Rêgo, D.J. Soares, L.S. Lucena, L.R. da Silva,
E.K. Lenzi, and K.S. Fa, Physica A 317, 199 (2003).
10. H. Mohammadzadeh, F. Adli, and S. Nouri, Phys. Rev. E 94,
062118 (2016) .
11. A. Khare, Fractional Statistics and Quantum Theory, World
Scientific, Singapore (2005).
12. L. Salasnich, Intern. J. Mod. Phys. B 14, 405 (2000).
13. C. Tsallis, Química Nova 17, 468 (1994).
14. E.M.F. Curado and C. Tsallis, J. Phys. A: Math. Gen. 24,
L69 (1991).
15. C. Tsallis, R.S. Mendes and A.R. Plastino, Physica A 261,
534 (1998).
16. C. Tsallis, Introduction to Nonextensive Statistical Mechanics:
Approaching a Complex World, Springer, New York (2009).
17. G. Wilk and Z. Włodarczyk, Physica A 413, 53 (2014).
18. J. Chen, Z. Zhang, G. Su, L. Chen, and Y. Shu, Phys. Lett. A
300, 65 (2002).
19. K.-M. Shen, B.-W. Zhang and E.-K. Wang, Physica A 487,
215 (2017).
20. T. Yamano, Physica A 305, 486 (2002).
21. A. Rovenchak, Phys. Rev. A 89, 052116 (2014).
22. A. Rovenchak, Eur. Phys. J. B 87, 175 (2014).
23. M.Y. Hornetska and A.A. Rovenchak, Ukr. J. Phys. 61, 168
(2016).
24. C. Ou and J. Chen, Phys. Rev. E 68, 026123 (2003).
25. L. Chen, Z. Yan, M. Li, and C. Chen, J. Phys. A 31, 8289
(1998).
26. R.A. Hull, K.R. Wilkinson, and J. Wilks, Proc. Phys. Soc.
Sec. A 64, 379 (1951).
27. H.C. Kramers, J.D. Wasscher, and C.J. Gorter, Physica 18,
329 (1952).
28. O.V. Lounasmaa, Cryogenics 1, 212 (1961).
29. V.D. Arp, R.D. McCarty, and D.G. Friend, Natl. Inst. Stand.
Technol. Tech. Note 1334 (revised) (1998).
30. R.P. Feynman, Phys. Rev. 91, 1291 (1953).
31. A. Isihara and T. Samulski, Phys. Rev. B 16, 1969 (1977).
32. A.A. Rovenchak, Fiz. Nizk. Temp. 29, 145 (2003) [Low
Temp. Phys. 29, 105 (2003)].
33. K.A. Gernoth, M. Serhan, and M.L. Ristig, Phys. Rev. B 78,
054513 (2008).
34. I.O. Vakarchuk, V.S. Pastukhov, and R.O. Prytula, Ukr.
Phys. J. 57, 1214 (2012).
35. I.O. Vakarchuk, O.I. Hryhorchak, V.S. Pastukhov, and R.O.
Prytula, Ukr. J. Phys. 61, 29 (2016).
___________________________
Ідеальний бозе-газ з неадитивною статистикою
А. Ровенчак
Проаналізовано підхід до узагальнення традиційної ста-
тистики Бозе–Ейнштейна на основі неадитивної ентропії Цал-
ліса. Подано докладне виведення термодинамічних функцій.
Зроблено розрахунки для питомої теплоємності двох модель-
них систем, а саме: ідеального тривимірного газу, що підкоря-
ється неаддитивній модифікації статистики Бозе–Ейнштейна,
та системи з лінійним спектром елементарних збуджень,
прийнятої за якісне наближення термодинаміки рідкого 4He.
Ключові слова: статистика Бозе–Ейнштейна, ентропія Цаллі-
са, неадитивна статистика, ідеальний бозе-газ, 4He.
Идеальный бозе-газ с неаддитивной статистикой
А. Ровенчак
Проанализирован подход к обобщению традициионной
статистики Бозе–Эйнштейна на основе неаддитивной энтро-
пии Цаллиса. Представлен подробный вывод термодинами-
ческих функций. Сделаны расчеты для удельной теплоемкости
двух модельных систем, а именно: идеального трехмерного
газа, подчиняющегося неаддитивной модификации статисти-
ки Бозе–Эйнштейна, и системы с линейным спектром эле-
ментарных возбуждений, принятой как качественное при-
ближение термодинамики жидкого 4He.
Ключевые слова: статистика Бозе–Эйнштейна, энтропия Цал-
лиса, неаддитивная статистика, идеальный бозе-газ, 4He.
Fig. 4. (Color online) Specific heat of the three-dimensional ideal
Bose-gas with nonadditive statistics according to the approach of
[18,24]. Red dashed lines show the high-temperature behavior
(A8). The units of temperature are fixed as in Fig. 2.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 10 1315
https://doi.org/10.1007/BF01016429
https://doi.org/10.1016/j.physa.2015.06.044
https://doi.org/10.1140/epjb/e2017-80535-3
https://doi.org/10.1080/09296174.2017.1324601
https://doi.org/10.1016/0375-9601(94)00941-H
https://doi.org/10.1103/PhysRevE.65.046105
https://doi.org/10.1016/S0378-4371(02)01330-4
https://doi.org/10.1103/PhysRevE.94.062118
https://doi.org/10.1088/0305-4470/24/2/004
https://doi.org/10.1016/S0378-4371(98)00437-3
https://doi.org/10.1016/j.physa.2014.06.044
https://doi.org/10.1016/S0375-9601(02)00781-8
https://doi.org/10.1016/j.physa.2017.06.030
https://doi.org/10.1016/S0378-4371(01)00567-2
https://doi.org/10.1103/PhysRevA.89.052116
https://doi.org/10.1140/epjb/e2014-50171-8
https://doi.org/10.15407/ujpe61.02.0168
https://doi.org/10.1103/PhysRevE.68.026123
https://doi.org/10.1088/0305-4470/31/41/003
https://doi.org/10.1088/0370-1298/64/4/304
https://doi.org/10.1088/0370-1298/64/4/304
https://doi.org/10.1016/S0031-8914(52)80156-9
https://doi.org/10.1016/S0011-2275(61)80004-0
https://doi.org/10.1103/PhysRev.91.1291
https://doi.org/10.1103/PhysRevB.16.1969
https://doi.org/10.1063/1.1542407
https://doi.org/10.1063/1.1542407
https://doi.org/10.1103/PhysRevB.78.054513
https://doi.org/10.15407/ujpe61.01.0029
1. Introduction
2. Density of states
3. Nonadditive statistics of Tsallis
4. Generalization of the Bose-statistics
5. Critical temperature
6. Low temperature
7. High temperatures and classical limit
8. Results in three dimensions
9. Conclusions
Acknowledgment
Appendix A
|