-Deformed Bosonic Newton Oscillators: Algebra and Thermodynamics

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q
-
Deformed
B
osonic Newton
O
scillators
:

Algebra and
T
hermodynamics


Abdullah Algin
*



*
Department of Physics,
Faculty of Arts and Sciences,

Eskisehir Osmangazi University,

Meselik, 26480
-
Eskisehir, Turkey

(
Tel: 222
-
239 37 50;
e
-
mail:


aalgin@ogu.edu.tr)


Abstract:


In this
work
, w
e discuss the quantum algebraic properties of the multi
-
dimensional
q
-
deformed bosonic Newton oscillators with
U
(
d
)
-
symmetry. Starting with a
q
-
deformed Bose
-
Einstein
distribution function, several thermodynamical functions of t
he system such as the critical temperature,
the internal energy, the specific heat are derived in the thermodynamical limit.
On the basis of the
recently published paper (Algin and Arslan, 2008), p
articular emphasize is given to a discussion of the
Bose
-
Ei
nstein condensation phenomenon for such a system.
Copyright © 2002 IFAC


Keywords:

q
-
oscillators
,
Quantum groups
,
Deformed bosons,
Algebraic approaches, Thermal properties
,

Bose
-
Einstein condensation.



1.

INTRODUCTION


Recently,
statistical and thermodyn
amical consequences of
studying
q
-
deformed physical systems have been
extensively investigated in the literature

(Lee and Yu,
1992; Su and Ge, 1993; Song
et al
., 1993; Tuszynski
et al
.,
1993; Chaichian
et al
., 1993; Monteiro
et al
., 1994;
Lavagno and Swamy
, 2000; 2002a, b; S
h
u
et al
., 200
2
;
Swamy, 2006).

P
ossible connections between quantum
groups and Tsallis nonextensive statistical mechanics have
been investigated

(Tsallis, 1994; Abe, 1997)
.
However, it
was shown in
(Algin, 2002a,b; Algin and Deviren, 200
5;
Jellal, 2002; Ubriaco, 1997; Ubriaco, 1998; Arik and
Kornfilt
, 2002)
that the high
-

and low
-
temperature
behaviours of

“the quantum group symmetric” bosonic
and fermionic
oscillator
gas
models depend radically on
the real deformation parameters.


In thi
s
work
, we first review the
multi
-
dimensional
q
-
deformed
bosonic Newton oscillator

algebra invariant
under the undeformed group
U
(
d
)
.
We
then discuss
the
thermostatistical properties of a gas of the
q
-
deformed
bosonic Newton oscillators
via the
q
-
deformed
Bose
-
Einstein distribution function of the system

(Algin and
Arslan, 2008)
.

In this context,
p
articular emphasize is
given to a discussion of the Bose
-
Einstein condensation
phenomenon for such a system.

Finally, we present
concluding remarks as well as som
e open problems
concerning our studies.


2.
THE
BOSONIC NEWTON OSCILLATOR
AL
GEBRA


In this section, we present the multi
-
dimensional
q
-
deformed bosonic Newton oscillator algebra

invariant
under the undeformed
Lie
group
U
(
d
) (
Arik
et al
., 1999;
Penson and
Solomon, 1999; Algin and Arslan, 2008
)
.


The
U
(
d
)
-
invariant algebra generated by the
bosonic
Newton o
scillators

together with their corresponding
creation operators


is defined by the following
commutation rel
ations
(Arik
et al
., 1999)
:
















(1)





where

is the total boson number operator in
d

dimensions, and

,

.

This system has the
following total deformed boson number operator for
d
-
dimensional case:



,




(
2
)


w
hose

spectrum
is given by






(
3
)


w
here


From (1)
-
(3), the multi
-
dimensional undeformed

bosonic oscillator algebra can be
obtained in the lim
it
q

= 1.

Also, one can check that the
multi
-
dimensional
q
-
deformed bosonic Newton oscillator
algebra in (1) shows
U
(
d
)
-
symmetry.
Under the linear

t
ransformation








(
4
)



the relations given in (1) are invariant.
Here,
the
matrix

, and it satisfies the unitarity condition
,
where the matrix
is the adjoint matrix of

T
. This
property justifies the name New
ton.
Th
e
bosonic Newton
oscillator algebra in (1) can
be derived from the
quantization of the harmonic oscillator through its Newton
equation and its invariance properties
(Arik
et al
.,
1999
)
.

Moreover, from the commutation relations
in (1), it
follows tha
t the
q
-
deformed Newton oscillator
s

have
bosonic degeneracy for all values of the deformation
parameter
q

(Algin
et al.
, 200
0
).



From all above
quantum algebraic
considerations, it
follows that instead of studying a one
-
dimensional system,
the multi
-
dimen
sional
q
-
deformed
bosonic Newton
oscillators deserve a special interest in the framework of
statistical mechanics.

This will be the main focus in the
following section.


3.


THE
RMOSTATISTICS

OF THE
BOSONIC

NEWTON
OSCILLATOR
S


In this section, we
consider
some of the
thermostatistical
properties of
a gas of
the
bosonic
Newton
oscillators
defined in (1)
-
(
3
)

in the thermodynamical limit.
In
the
grand canonical ensemble, the Hamiltonian of such a free

q
-
deformed

bosonic
Newton
oscillator gas has the
following
form

(Algin and Arslan, 2008)
:





,



(
5
)


where

is the kinetic energy of a particle in the state
i
,

is the chemical potential which is considere
d as a
function of
q
, and

is the boson number operator
relative to
. Similar Hamiltonians were also considered
by several other researchers
(
Song
et al
., 1993;
Lee and
Yu, 1992; Su and Ge, 1993;; Tuszynski
et

al
., 1993;
Chaichian
et al
., 1993;
Shu
et al
., 2002;
Lavagno and
Swamy, 2000; 2002a, b
;

Adamska and Gavrilik, 2004;
Swamy, 200
6
).

It should be noted that this Hamiltonian
represents essentially a non
-
interacting system of the
bosonic Newton

oscillators s
i
nce we do have neither
a
specific deformed commutation relation between bosonic
annihilation (or creation) operators
, nor

a quantum group
symmetry structure in (1).

Also, this Hamiltonian does
implicitly incorporate deformation, since the occupation
number

depends on the deformation parameter
q

by means
of (1)
-
(3).


By

following the procedure
proposed
in (Altherr and
Grandou, 1993),
from (1)
-
(3),
we have the following
relation
including the thermal averages
:







(6)


which leads to
the
s
tatistical distribution function

for the bosonic Newton
oscillators as






(
7
)


The function

has the following properties

(Algin
and Arslan, 2008)
:


(i)
T
he usual Bose
-
Einstein distribution

can be recovered
i
n the limit
.

(ii)
The distribution function

should be nonnegative.
This gives the following constraints on the
q
-
deformed

fugacity

, where
,
and the
chemical potential
:




(
8
)


In the
limit
, this equation reduces to





or





(
9
)



as in the case of an undeformed
Bose

gas.
We should
note
that the
q
-
deformed

fugacity

is independent of the
dimension of the
o
scillators.

(iii) The total number of particles i
s given by

the
constraint

.






Using
(7)
, one can find

the logarithm of the bosonic grand
partition function
of the system
as



,

(
10
)


which gives all of the
th
ermo
dynamical
functions in terms
of the deformation parameter

q
.

Referring the reader to
(Algin and Arslan, 2008), we
here
present
the necessary
formulae for
a
discussion
of the low
-
temperature
thermodynamical
behavior of the
q
-
deformed bosonic
Newton osci
llator gas
model
in the thermodynamical limit.
This model will exhibit the Bose
-
Einstein condensation

when the following condition is
satisfied:



,



(11)


where
the thermal wav
elength
is
,
, and
t
he generalized Bose
-
Einstein function

is

defined
by








(
12
)



where
. These generalized fun
ctions reduce
to the standard Bose
-
Einstein functions

in the limit

(Huang, 1987)
. They are also different from

in
(Lavagno and Swamy
, 2002a
)
.

In the
framework of the constraints in (
8
)
, we consider the case

in the rest of the calculations of this study. Since the
function

with

in (
12
)
reveals
the same
results as in the case of an undeformed boson gas.

The
internal e
nergy
U

of the
bosonic Newton
oscillators gas
can be found
by

,

which leads to






(
13
)


where the
q
-
deformed
Bose
-
Einstein function


is de
fined by


. (14)


Hence, t
he specific heat of
the system
can be obtained
from
the relation

.
I
n the limit
, the specific heat of our model is



,

(
1
5
)


where
the
critical temperature

is defined by



.

(
16
)


Furthermore, one can find
a relation between the critical
temperatu
re of the present
bosonic Newton
oscillator gas
and of the undeformed boson gas:







(17)


In
Fig. 1
, we show the plot of (
17
) as a function of the
deformation parameter
q

for the case

(Al
gin and
Arslan, 2008).

Thus, the critical temperature for the
bosonic Newton oscillator gas
is much larger than the
critical temperature

for an undeformed boson gas in
the special region of the deformation parameter
q

close to
zero.







Fig.

1.

The ratio

of the
q
-
deformed critical
temperature

and the undeformed

as a
function of the deformation parameter
q
.


On the other hand, the

specific heat for the bosoni
c
Newton oscil
lator gas in the limit

can be
approximated as



.

(
18
)


In Fig. 2,
we show the plot of the specific heat

as
a function of

for
s
everal
values of the
deformation parameter
q

smaller than one

(Algin and
Arslan, 2008)
.





Fig. 2.
The specific heat

as a function of

for
values of the deformation parameter
q

smaller than one.



From (
15
) a
nd (
18
), we deduce the gap in the specific heat
in the limit

as






(
19
)


We should emphasize that all the thermostatistical
functions given in (5)
-
(19) reduce to the undeformed
boson gas func
tions in the limit
q

= 1 (Huang, 1987).


4
. CONCLUSIONS


In this work, we discussed both the
quantum
algebraic and
the thermostatistical properties of a
system

of the bosonic
Newton oscillators

(Algin and Arslan, 2008)
.
Starting with
a
q
-
deformed Bo
se
-
Einstein distribution function, several
lo
w
-
temperature
thermo
dynamical
functions via the grand
partition function of the system are calculated. Due to
some

algebraic reasons originating from (
7
), (
8
), (
12
), and
(
1
4), we obtained such
thermo
dynamical
fu
nctions in
terms of the deformation parameter
q

for its specific
interval
.


As shown in Fig. 2,
the specific heat of the
bosonic
Newton oscillator gas model
shows a discontinuity at the
critical temperature
, and has
a
-
point transition
behavior
,

which is not exhibited by an undeformed boson gas. An
interesting point is that when the deformation parameter
q

approaches to zero, the discontinuity in the specific heat of
the system increases.
However,
it disappears

in the
limit

, showing an undeformed boson gas behavior.


Furthermore,
it follows from (19)
that
the gap in
the
specific heat of the
bosonic Newton oscillator
gas at the
condensation temperature decreases with the values of the
defo
rmation parameter
q

up to the limit
.
Therefore,
t
he
bosonic Newton oscillator gas with
U
(
d
)
-
symmetry

shows

the Bose
-
Einstein condensation for low
temperatures in the interval

(Algin and Arslan,
2008).


It is
also interesting to remark that
the theoretical critical
temperature

for

,

3.13 K
(Huang, 1987)
,
corresponds to a

q

value of about 0.9.
T
he same value

q
0.9 fits very well the gap in t
he specific heat of a dilute
gas of rubidium atoms
(Ensher
et al
., 1996)
. Such a result
may be physically important, since some recent studies
similarly adduced a value of the deformation parameter
q
.
Hence, one might well view
q
-
deformation as a
phenomeno
logical means of introducing an extra
parameter, “
q
”, to account for non
-
linearity in the system.
Such an approach was considered in
(Katriel and Solomon,
1994)
, where a value of
q

is found to fit the properties of a
real (non
-
ideal) laser.


Finally, some

open problems parallel to this work are as
follows:

It will be interesting to investigate
possible
statistical
consequences of the bosonic Newton oscillator
algebra with the deformation parameter
q

being a root of
unity
, which would hopefully provide some

new insights
in order
to discover new forms of non
-
standard particle
statistics.


A
CKNOWLEDGMENTS


This work is supported by the Scientific and Technological
Research Council of Turkey (TUBITAK) under the project
number 106T593.

I also
thank the
o
rganiz
in
g committee

of
the
I
nternational
W
orkshop on New Trends in Science and
Technology (NTST
08
)

for inviting
me

to present this
work
.


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