On statistical mechanics in noncommutative spaces
We study the formulation of quantum statistical mechanics in noncommutative spaces. We construct microcanonical and canonical ensemble theory in noncommutative spaces and study some basic and important examples in the framework of noncommutative statistical mechanics.
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irk-123456789-1110232017-01-08T03:03:57Z On statistical mechanics in noncommutative spaces Alavi, S.A. Kinetic theory We study the formulation of quantum statistical mechanics in noncommutative spaces. We construct microcanonical and canonical ensemble theory in noncommutative spaces and study some basic and important examples in the framework of noncommutative statistical mechanics. Вивчається формулювання квантової статистичної механіки в некомутативних просторах. Будується теорія мікроканонічного й канонічного ансамблів у некомутативних просторах і вивчаються деякі основні й важливі приклади в рамках некомутативної статистичної механіки. Изучается формулировка квантовой статистической механики в некоммутативных пространствах. Строится теория микроканонического и канонического ансамблей в некоммутативных пространствах и изучаются некоторые основные и важные примеры в рамках некоммутативной статистической механики. 2007 Article On statistical mechanics in noncommutative spaces / S.A. Alavi // Вопросы атомной науки и техники. — 2007. — № 3. — С. 301-304. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 02.40.Gh, 05.30.-d http://dspace.nbuv.gov.ua/handle/123456789/111023 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Kinetic theory Kinetic theory Alavi, S.A. On statistical mechanics in noncommutative spaces Вопросы атомной науки и техники |
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We study the formulation of quantum statistical mechanics in noncommutative spaces. We construct microcanonical and canonical ensemble theory in noncommutative spaces and study some basic and important examples in the framework of noncommutative statistical mechanics. |
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Alavi, S.A. |
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Alavi, S.A. |
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Alavi, S.A. |
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On statistical mechanics in noncommutative spaces |
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On statistical mechanics in noncommutative spaces |
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On statistical mechanics in noncommutative spaces |
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On statistical mechanics in noncommutative spaces |
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On statistical mechanics in noncommutative spaces |
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on statistical mechanics in noncommutative spaces |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2007 |
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Kinetic theory |
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http://dspace.nbuv.gov.ua/handle/123456789/111023 |
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On statistical mechanics in noncommutative spaces / S.A. Alavi // Вопросы атомной науки и техники. — 2007. — № 3. — С. 301-304. — Бібліогр.: 9 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT alavisa onstatisticalmechanicsinnoncommutativespaces |
| first_indexed |
2025-07-08T01:31:21Z |
| last_indexed |
2025-07-08T01:31:21Z |
| _version_ |
1837040435699122176 |
| fulltext |
Section F. KINETIC THEORY
ON STATISTICAL MECHANICS IN NONCOMMUTATIVE SPACES
S.A. Alavi
Department of Physics, Sabzevar university of Tarbiat Moallem, Sabzevar, P.O. Box 397,
Iran and Sabzevar House of Physics, Asrar Physics Laboratory, Laleh Square, Sabzevar, Iran;
e-mail: alavi@sttu.ac.ir; alialavi@fastmail.us
We study the formulation of quantum statistical mechanics in noncommutative spaces. We construct microcan-
onical and canonical ensemble theory in noncommutative spaces and study some basic and important examples in
the framework of noncommutative statistical mechanics.
PACS: 02.40.Gh, 05.30.-d
1. INTRODUCTION
To study quantum mechanical systems composed of
indistinguishable entities, as most physical systems are,
one finds that it is advisable to rewrite the ensemble
theory in a language that is more natural to a quantum-
mechanical treatment, namely the language of the
operators and the wave functions [1]. Once we set out to
study these systems in detail, we encounter a stream of
new and altogether different physical concepts. In par-
ticular, we find that the behavior of even a noninteract-
ing system, such as an ideal gas, departs considerably
from the pattern set by the so-called classical treat-
ments. In the presence of interactions the pattern be-
comes still more complicated.
Recently there have been notable studies on the for-
mulation and possible experimental consequences of
extensions of the usual physical theories in the non-
commutative spaces (see for example [2-6]). The study
on noncommutative spaces is much important for un-
derstanding phenomena at short distances beyond the
present test of different physical theories. For a mani-
fold parameterized by the coordinates , the noncom-
mutative relations can be written as:
ix
[ ] [ ] [ ] ,0,,, =δ=θ= jiijjiijji ppipxixx (1)
where is an antisymmetric tensor which can be de-
fined as
ijθ
kijkθε=
2
1
ijθ .
In this paper we study the formulation of quantum
statistics, namely the quantum-mechanical ensemble
theory, the density matrix, etc., in a noncommutative
space and the new features that arise. We consider for
illustration some basic and important examples in the
framework of noncommutative statistical mechanics:
(i). An electron in a magnetic field. (ii). A free particle
in a box. (iii). A linear harmonic oscillator.
Contrary to the other fields of physics, statistical
mechanics has not been studied extensively in non-
commutative spaces and this work can be a motivation
for more studying this field of physics.
2. PERTURBATION ASPECTS
OF NONCOMMUTATIVE DYNAMICS
NCQM is formulated in the same way as the stan-
dard quantum mechanics SQM (quantum mechanics in
commutative spaces), that is in terms of the same dy-
namical variables represented by operators in a Hilbert
space and a state vector that evolves according to the
Schrödinger equation:
,>ψ>=ψ ncH
dt
di (2)
we have taken in to account . denotes
the Hamiltonian for a given system in the noncommuta-
tive space. In the literatures two approaches have been
considered for constructing the NCQM:
1= θ≡ HHnc
a) , so that the only difference between SQM
and NCQM is the presence of a nonzero θ in the com-
mutator of the position operators, i.e. Eq.(1).
HH =θ
b) By deriving the Hamiltonian from the Moyal analog
of the standard Schrödinger equation:
),,(),(),1(),( txHtxx
i
pHtx
t
i ψ≡ψ∗∇==ψ
∂
∂
θ (3)
where is the same Hamiltonian as in the stan-
dard theory, and as we observe the dependence enters
now through the star product [7]. In [8], it has been
shown that these two approaches lead to the same
physical theory. Since the noncommutativity parameter,
if it is non-zero, should be very small compared to the
length scales of the system, one can always treat the
noncommutativity effects as some perturbations of the
commutative counterpart. For the Hamiltonian of the
type:
),( xpH
θ
)(
2
),(
2
xV
m
pxpH += (4)
the modified Hamiltonian can be obtained by a
shift in the argument of the potential [2-6,9]:
θH
,
2
1
, iijijii pppxx =+= θ (5)
which leads to
).
2
1(
2
2
jiji pxV
m
pH θ−+=θ (6)
The variables and now satisfy the same commu-
tation relations as the usual case:
ix ip
[ ] [ ] [ ] .,,0,, ijjijiji pxppxx δ=== (7)
Now we discuss the perturbation aspects of non-
commutative dynamics. Using
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 301-304. 301
,)(
!
)()()(
1
)(
n
n
n
x
n
xUxUxxU ∆+=∆+ ∑
∞
=
(8)
and Eq.(6) we have:
,)(
!
)()(
2 1
)(2
n
i
n
i
n
inc x
n
xVxV
m
pHH ∆++== ∑
∞
=
θ (9)
where jiji px θε−=∆
2
1 and )(
2
2
xV
m
pH += is the
Hamiltonian in ordinary(commutative) space.
To the first order we have:
.)(
2
2
Ii
i
i
i
inc HHx
x
VHx
x
VxV
m
pH θ+=∆
∂
∂
+=∆
∂
∂
++=
(10) − ∑∑∑ ((aH
n k
l
kWe can use perturbation theory to obtain the eigen-
values and eigenfunctions of : ncH
....)2(2)1(000 +θ+θ+=∆+= nnnnnn EEEEEE (11)
,)( k
nk
nknn C φθφφ ∑
≠
+= (12)
where:
. (13) ...)( )2(2)1( +θ+θ=θ nknknk CCC
To the first order in perturbation theory we have:
,)1( >φθφ=<θ nInn HE (14)
,)1(
k
nk
nknn C φθ+φ=φ ∑
≠
(15)
,00
)1(
kn
nIk
nk EE
H
C
−
>φθφ<
=θ (16) k nn
where and 0
nE nφ are the n-th eigenvalue and eigen-
function of the Hamiltonian . and φ are the n-
th eigenvalue and eigenfunction of .
H nE
H
n
nc
3. THE DENSITY OPERATOR
IN NONCOMMUTATIVE SPACES
Using the orthonormal functions φ , an arbitrary
wave function in a noncommutative space can be writ-
ten as:
n
,)()( ∑=
n
n
k
n
k tat φψ (17) Tr
where: :
∫= .)()( * τψφ dtta k
n
k
n (18)
The time variation of these coefficients will be given
by:
=τψ
∂
∂
φ=
∂
∂
∫ dt
t
ia
t
i k
n
k
n )(*
∑ ∑∫ ∫ ==
m m
k
mnmm
k
nn
k
n taHdtaHdtH ),()()( ** τφφτψφ (19)
where . We now introduce the density
operator
mnnm HH φφ= ∫ *
)(tρ in a noncommutative space by the matrix
elements: :
[∑
=
=ρ
n
k
k
n
k
mnm tata
n
t
1
* )()(1)( ]. (20)
Clearly the matrix element )(tnmρ is the ensemble
average of the quantity , which as a rule
varies from member to member in the ensemble.
)() * tan(tam
We shall now determine the equation of motion for
the density matrix )(tnmρ :
=ρ
∂
∂ )(t
t
i mn
.)()]([)()]([1
1
**
∂
∂
+
∂
∂∑
=
n
k
k
m
k
n
k
n
k
m tata
t
itata
t
i
n
(21)
It can be written in the following form:
=∑
==
)()]([)])[1
1
***
1
tataHtat k
m
n
k
nl
k
n
k
lml
n
mnHH )( ρ−ρ . (22)
Using the commutator notation, it can be written as:
],[ ρ=ρ Hi . (23)
Now, we consider the expectation value of a physi-
cal quantity G in a noncommutative space which is dy-
namically represented by an operator G . This will natu-
rally be determined by the double averaging process:
τψψ>=< ∑∫
=
dG
n
G
n
k
kk
1
*1 (24)
or:
,1
1 ,
*∑∑
=
>=<
n
m
nm
k
m
k
n GaaG (25)
where:
τφφ= ∫ dGG mnnm
* . (26)
Introducing the density matrix ρ , it takes a particularly
neat form:
)()(
,
GTrGGG
m
mm
nm
nmn ρ=ρ=ρ>=< ∑∑ . (27)
We note that if the original wave functions , were
not normalized then the expectation value would
be given by the formula
kψ
>G<
)(
)(
ρ
ρ
>=<
GTrG . (28)
The interesting point is that the equations (23) and (28)
are the same as the commutative case with quantities
replaced by their noncommutative counterparts.
4. STATISTICS OF THE VARIOUS
ENSEMBLE
The microcanonical ensemble. The construction of
the microcanonical ensemble is based on the premise that
the systems constituting the ensemble are characterized
by a fixed number of particles , a fixed volume V and
an energy lying within the interval
N
),
2
,
2
( ∆∆
− EE +
where . The total number of distinct microstates
accessible to a system is then denoted by the symbol
and by assumption, any of these micro-
states is just as likely to occur as any other.
E<<∆
);, ∆EV,(Γ N
302
Accordingly, the density matrix mnρ (which in the
energy representation must be a diagonal matrix) will be
of the form
Γ
1 , for each of the accessible states and 0,
for all other states.
We note that ρ is independent of energy (energy ei-
genstates) and the volume V, so it does not depend on
space coordinates. It means that the noncommutativity
of space has no effects on ρ in microcanonical ensem-
ble.
The dynamics of the system determined by the ex-
pression for its entropy, which in turn is given by:
,Γ= kLnS (29)
which as mentioned above remains unchanged in non-
commutative spaces.
The canonical ensemble. In this ensemble the mac-
rostate of a member system is defined through the pa-
rameters N, V and T; the energy E now becomes a vari-
able quantity. The probability that a system, chosen at
random from the ensemble, possesses an energy E, is
determined by the Boltzmann factor exp( , where )Eβ−
kT
1
=β . The density matrix in the energy representa-
tion is therefore takes as:
,mnnmn δρρ = (30)
where: :
,...2,1,0; ==ρ β− nce nE
n (31)
Here are the energy eigenvalues in noncommutative
space (Eq.11), and the constant c is given by:
nE
:
.
)(
1
)exp(
1
ββ N
n
n QE
c =
−
=
∑
(32)
Where , is the partition function of the sys-
tem in the noncommutative space. The density operator
in the canonical ensemble may be written as:
)(βNQ
:
|
)(
1| n
E
Nn
n ne
Q
φ
β
φρ β 〈〉= −∑
.
)(
||
)(
1
H
H
n
n
n
H
N eTr
ee
Q β
β
β φφ
β −
−
− =〈〉= ∑ (33)
Then the expectation value of a physical quantity G, in
a noncommutative space is given by: :
.
)(
)()(
H
H
N
eTr
eGTrGTrG
β
β
ρ
−
−
==>< (34)
EXAMPLES
(i). An electron in a magnetic field. The Hamiltonian
of the system has the following form:
),.( BH B σµ−= (35) 2
pH =
where
mc
e
B 2
=µ . The Hamiltonian is space independ-
ent, so there is no corrections due to the noncommuta-
tivity of space on the statistical (thermodynamical)
properties of this system.
(ii). A free particle in a box. Let us consider the
motion of a particle with charge e and mass m in the
presence of a magnetic field produced by a vector
potential . The Lagrangian is as follows: A
1 ),,(
2
2 yxVVA
c
emVL −⋅+= (36)
where (no summation, i ) are the com-
ponents of the velocity of the particle and
are the scale factors. V describes additional inter-
actions (impurities). For the case of a free particle
. In the absence of the quantum spectrum the
well-known Landau levels consists of V. In the strong
magnetic field limit only the lowest Landau level is
relevant. But the large B limit corresponds to small m,
so setting the mass to zero effectively projects onto the
lowest Landau level. In the chosen gauge
and in that limit, the Lagrangian (36) takes the follow-
ing form:
iii qhV =
0) =
3,2,1=
)3,2,1( =ihi
),0( 11 Bqh=
),( yx
,( yxV
A
),,(2121 yxVqqhBh
c
eL −=′ (37)
which is of the form , and suggests that ),( qpHqp −
11qBhc
e and are canonical conjugates, so we have
:
22qh
,],[ 2211 eB
ciqhqh −= (38)
which can be written in general form:
,],[ ijjjii iqhqh θ= (39)
which is the fundamental space-space noncommutativ-
ity relation in a general noncommuting curvilinear co-
ordinates. The Cartesian, circular cylindrical and spheri-
cal polar coordinates are three special cases.
So a free particle in a noncommutative space is
equal to a particle in commutative space but in the pres-
ence of a magnetic field. The Hamiltonian of a particle
in a magnetic field is:
2)(
2
1 A
c
qP
m
H −= . (40)
On the other hand let us introduce the noncommutativity
to momentums instead of space coordinates:
,],[],[0],[ ijjiijjiji ippipxxx θ=δ== (41)
one can easily show that there is a transformation:
,,
2
1
iijijii xxxpp =+= θ (42)
where the new variables and satisfy the standard
commutation relations (7). We note that (42) is the same
as
ip ix
c
Aqpp i
ii −→ with jx
q
c
−=
2
1
ijθiA .
Now the Hamiltonian of a free particle in a non-
commutative space is:
21
m
),(
2
1
))((
2
1
2
1
2
1
22
22
2
θθ
θθθ
OLp
Opxp
m
xp
m
i
ijijijiji
+⋅−=
++=
+=
(43)
303
where . Since for a free particle ,
so to the first order there is no corrections to the Hamil-
tonian and therefore there is no corrections due to non-
commutativity of space on the statistical (thermody-
namical) properties of the this system.
jiijkk pxL =∈ 0=L
(iii). A harmonic oscillator. The case of a linear
harmonic oscillator is irrelevant, because there is only
one space variable. In the case of harmonic oscillator in
higher dimensions, for instance spherical harmonic os-
cillator, the corrections on the Hamiltonian due to non-
commutativity of space is given by:
,
2
1
j
i
iji
i
I p
x
Vx
x
VH
∂
∂
−=∆
∂
∂
= θ (44)
where: :
∑
=
ω=
3
1
22
2
1
i
ixmV . (45)
We put and the rest of the components to
zero, which can be done by a rotation or by a redifini-
tion of coordinates. So we have:
θ=θ3 θ
.
4
1
4
1
4
1
)(
2
1
2
2
θθθεω
ωθ
zkjiijk
jiijI
LLpxm
pxmH
=⋅−==
−=
(46)
For a spherical harmonic oscillator the unperturbed
(commutative) eigenfunctions are given by:
,
)()(
)(
21 2
1
2
2
r
YrLer
r lmn
e
mn
r
θφα
αθφφ
α ++ −
(47)
where )( 22
1
rLn α+ are Lagure's function. Using
Eqs.(14) and (31), one can easily derive the energy ei-
genvalues in noncommutative space and so the density
operator ρ and the partition function
. The thermodynamical properties of
the system can be done straightforwardly using partition
function. We have:
∑ β−
n
Ene=βnQ )(
θ>=ϕϕ=< mHE nemInemn 4
1||)1( .
So:
)(4)( β
θ
β
−
=β N
m
N QeQ .
The Helmholtz free energy is given by:
AmkA +θ
β+
=
4
.
Whence we obtain:
Smk
T
AS +θ
β
=
∂
∂
−=
4
,
UmkU +θ
β+
=
2
,
CmkC +θ
β
=
2
,
where and C are the entropy, the internal energy
and the specific heat of the system in Noncommutative
space and S, U and C are their counter part in commuta-
tive case.
US,
REFERENCES
1. R.K. Pathria. Statistical mechanics. Butterworth-
Heinemann, 1996.
2. N. Seiberg, E. Witten //JHEP. 1999, v. 9909, p. 032.
3. I. Avramidi //Phys. Lett. B. 2003, v. 576, p. 195.
4. A. Kokado //Phys. Rev. 2004, D69, 125007.
5. B. Basu, S. Chosh //Phys. Lett. A346. 2005, p. 133.
6. S.A. Alavi //Mod. Phys. LettA. 2006, v. 21, p. 1.
7. L. Mezincescu, hep-th/0007076.
8. O. Espinosa, hep-th/0206066.
9. M. Chaichian, M.M. Sheikh-Jabbari and A. Tureanu
//Phys. Rev. Lett. 2001, v. 86, p. 2716.
К СТАТИСТИЧЕСКОЙ МЕХАНИКЕ В НЕКОММУТАТИВНЫХ ПРОСТРАНСТВАХ
С.А. Алави
Изучается формулировка квантовой статистической механики в некоммутативных пространствах. Стро-
ится теория микроканонического и канонического ансамблей в некоммутативных пространствах и изучают-
ся некоторые основные и важные примеры в рамках некоммутативной статистической механики.
ДО СТАТИСТИЧНОЇ МЕХАНІКИ В НЕКОМУТАТИВНИХ ПРОСТОРАХ
С.А. Алаві
Вивчається формулювання квантової статистичної механіки в некомутативних просторах. Будується тео-
рія мікроканонічного й канонічного ансамблів у некомутативних просторах і вивчаються деякі основні й
важливі приклади в рамках некомутативної статистичної механіки.
304
|