On the possibility of the existence of the spatially periodical Bose condensate
The possibility of constructing the theory relating to a spatially periodical Bose condensate in the model of a weakly nonideal Bose gas is studied. Equations for determining a spatially periodical order parameter at the temperature T = 0 are formulated and their one-periodical solutions are ob...
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Інститут фізики конденсованих систем НАН України
2000
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| Цитувати: | On the possibility of the existence of the spatially periodical Bose condensate / A.S. Peletminsky, S.V. Peletminsky, Yu.V. Slyusarenko // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 227-237. — Бібліогр.: 7 назв. — англ. |
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Peletminsky, A.S. Peletminsky, S.V. Slyusarenko, Yu.V. 2017-06-13T13:09:59Z 2017-06-13T13:09:59Z 2000 On the possibility of the existence of the spatially periodical Bose condensate / A.S. Peletminsky, S.V. Peletminsky, Yu.V. Slyusarenko // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 227-237. — Бібліогр.: 7 назв. — англ. 1607-324X DOI:10.5488/CMP.3.2.227 PACS: 03.75.Fi, 05.30.Jp, 05.70.a https://nasplib.isofts.kiev.ua/handle/123456789/121022 The possibility of constructing the theory relating to a spatially periodical Bose condensate in the model of a weakly nonideal Bose gas is studied. Equations for determining a spatially periodical order parameter at the temperature T = 0 are formulated and their one-periodical solutions are obtained based on the isolated Bose condensate method. Here it is possible to neglect the contribution of quasiparticles to the thermodynamics of a boson system. The problems of thermodynamic stability of the spatially periodical Bose condensate are presented in short. У роботi вивчено можливiсть побудови теoрiї просторово-перiодичного бозе-конденсату в моделi слабконеiдеального бозе-газу. На основi методу видiленого бозе-конденсату сформульовано рiвняння для визначення просторово-перiодичного параметра порядку при температурi T = 0 , коли можна знехтувати внеском квазiчастинок до термодинамiки системи бозонiв, та знайдено їх одноперiодичнi розв’язки. Коротко викладено питання термодинамiчної стiйкостi просторово-перiодичного бозе-конденсату. This work was supported in part by the German Ministry of Science and Technology (BMBF: 411 4006 06ROS02), and also by the Ukrainian State Foundation for Fundamental Studies. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the possibility of the existence of the spatially periodical Bose condensate Про можливIсть Iснування просторово перIодичного бозе-конденсату Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On the possibility of the existence of the spatially periodical Bose condensate |
| spellingShingle |
On the possibility of the existence of the spatially periodical Bose condensate Peletminsky, A.S. Peletminsky, S.V. Slyusarenko, Yu.V. |
| title_short |
On the possibility of the existence of the spatially periodical Bose condensate |
| title_full |
On the possibility of the existence of the spatially periodical Bose condensate |
| title_fullStr |
On the possibility of the existence of the spatially periodical Bose condensate |
| title_full_unstemmed |
On the possibility of the existence of the spatially periodical Bose condensate |
| title_sort |
on the possibility of the existence of the spatially periodical bose condensate |
| author |
Peletminsky, A.S. Peletminsky, S.V. Slyusarenko, Yu.V. |
| author_facet |
Peletminsky, A.S. Peletminsky, S.V. Slyusarenko, Yu.V. |
| publishDate |
2000 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Про можливIсть Iснування просторово перIодичного бозе-конденсату |
| description |
The possibility of constructing the theory relating to a spatially periodical
Bose condensate in the model of a weakly nonideal Bose gas is studied. Equations for determining a spatially periodical order parameter at the
temperature T = 0 are formulated and their one-periodical solutions are
obtained based on the isolated Bose condensate method. Here it is possible to neglect the contribution of quasiparticles to the thermodynamics of
a boson system. The problems of thermodynamic stability of the spatially
periodical Bose condensate are presented in short.
У роботi вивчено можливiсть побудови теoрiї просторово-перiодичного бозе-конденсату в моделi слабконеiдеального бозе-газу. На основi методу видiленого бозе-конденсату сформульовано рiвняння
для визначення просторово-перiодичного параметра порядку при
температурi T = 0 , коли можна знехтувати внеском квазiчастинок до термодинамiки системи бозонiв, та знайдено їх одноперiодичнi розв’язки. Коротко викладено питання термодинамiчної стiйкостi просторово-перiодичного бозе-конденсату.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121022 |
| citation_txt |
On the possibility of the existence of the spatially periodical Bose condensate / A.S. Peletminsky, S.V. Peletminsky, Yu.V. Slyusarenko // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 227-237. — Бібліогр.: 7 назв. — англ. |
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| first_indexed |
2025-11-24T18:43:50Z |
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2025-11-24T18:43:50Z |
| _version_ |
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| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 2(22), pp. 227–237
On the possibility of the existence of
the spatially periodical Bose
condensate
A.S.Peletminsky 1 , S.V.Peletminsky 2 ∗, Yu.V.Slyusarenko 2 †
1 Electrophysical Scientific and Technical Centre
of the National Academy of Sciences of Ukraine,
61002 Kharkiv, Ukraine
2 National Science Centre, “Kharkiv Institute of Physics and Technology”,
61108 Kharkiv, Ukraine
Received March 6, 2000
The possibility of constructing the theory relating to a spatially periodical
Bose condensate in the model of a weakly nonideal Bose gas is stud-
ied. Equations for determining a spatially periodical order parameter at the
temperature T = 0 are formulated and their one-periodical solutions are
obtained based on the isolated Bose condensate method. Here it is possi-
ble to neglect the contribution of quasiparticles to the thermodynamics of
a boson system. The problems of thermodynamic stability of the spatially
periodical Bose condensate are presented in short.
Key words: weakly nonideal Bose-gas, isolated Bose-condensate
method, spatially periodic Bose-condensate, thermodynamic stability
PACS: 03.75.Fi, 05.30.Jp, 05.70.a
1. Introduction
Recently the experiments have been carried out where the existence of the Bose-
Einstein condensation phenomena in an ideal gas at temperature near 2 · 10−8 K
was proved [1]. This fact makes it interesting to turn again to the Bose-Einstein
condensation problem in the model of a weakly nonideal Bose gas. Within this
model the microscopic theory of superfluidity was constructed by N.N.Bogolyubov
with the use of a quasiaverage method. Applying the microscopic approach which
contains the quantum Gibbs distribution as a fundamental aspect it was shown
[4] how the basic equations of motion for a quantum crystal can be obtained. The
phenomenological theory of the quantum crystal was described in [5].
∗E-mail: spelet@kipt.kharkov.ua
†E-mail: slusarenko@kipt.kharkov.ua
c© A.S.Peletminsky, S.V.Peletminsky, Yu.V.Slyusarenko 227
A.S.Peletminsky, S.V.Peletminsky, Yu.V.Slyusarenko
In the present work we want to touch upon the problem of constructing the
thermodynamics of the one-periodical quantum crystal in the model of a weakly
nonideal Bose gas. Here we consider the questions connected with the description
of spatially periodical structures at T = 0 which correspond to the existence of the
spatially periodical Bose condensate. The problems of thermodynamic stability of
the states studied are also briefly considered.
It would be interesting to construct the hydrodynamics equations for the one-
periodical solutions, to generalize the theory for the case of two and three-periodical
states, as well as to clarify the effect of the external magnetic field on these states.
The latter task was considered for the spatially homogeneous Bose condensate in [6].
We also want to emphasize that spatially periodical states in the fermion systems
were studied in [7] with the use of the fermi-liquid approach proposed by Landau.
2. Spatially periodical Bose condensation in the model of a
weakly nonideal Bose gas
In this section we consider the generalization of the Bogolyubov theory of a
weakly nonideal Bose gas for the case when the Bose condensate consists of particles
with the infinite set of momenta ~p + ~τ , where vectors ~τ form a certain reciprocal
lattice with respect to a direct one with vectors ~ai, i = 1, 2, 3 (~τ =
∑3
i=1 ni
~bi, ni are
the integer numbers, ~bi~aj = 2πδij) and the vector ~p belongs to the first Brillouin
zone. To study this problem we use the quasiaverage method leading to the model
with an isolated condensate.
According to this method, an average value of arbitrary function Â(ψ̂) of the
annihilation operator ψ̂(~x) can be defined by the formula
{Â(ψ̂)} = Sp ŵ(ψ)Â(ψ), (2.1)
where the operator Â(ψ) can be obtained from the operator Â(ψ̂) by the substitution
ψ̂(~x) → ψ(~x) + ψ̂′(~x), (2.2)
where
ψ̂
′
(~x) =
1√
V
∑
~q 6=~p+~τ
â~q e
i~q~x (2.3)
and the quantity ψ(~x) represents a C-numerical function
ψ(~x) =
1√
V
∑
~q=~p+~τ
a~q e
i~q~x, (a~q ∼
√
V ) (2.4)
and
ŵ(ψ) = exp
(
Ω(ψ)− β(Ĥ(ψ)− ~v ~̂P (ψ)− µN̂(ψ))
)
,
Ω(ψ) = − ln Sp exp β
(
Ĥ(ψ)− ~v ~̂P (ψ)− µN̂(ψ)
)
. (2.5)
228
On the existence of periodical Bose-condensate
Here Ĥ(ψ) is the Hamilton operator, ~̂P (ψ) is the operator of momenta, and N̂(ψ)
is the operator of number of particles. The function ψ(~x) (or quantity a~p+~τ ) can be
determined from a minimum condition of the potential Ω(ψ):
δΩ
δψ
= 0,
δΩ
δψ∗
= 0. (2.6)
In the weakly nonideal gas approximation we can write
Ĥ(ψ) ≈ H(ψ) =
1
2m
∫
d~x∇ψ∗(~x)∇ψ(~x)
+
1
2
∫
d~x1
∫
d~x2ψ
∗(~x1)ψ
∗(~x2)ν(~x1 − ~x2)ψ(~x1)ψ(~x2),
~̂P (ψ) ≈ ~P (ψ) = − i
2
∫
d~x
(
ψ∗(~x)∇ψ(~x)−∇ψ∗(~x)ψ(~x)
)
,
N̂(ψ) ≈ N(ψ) =
∫
d~xψ∗(~x)ψ(~x). (2.7)
Then, according to (2.6) the equation for determining the quantity ψ(~x) has the
form {
− 1
2m
∆+ i~v∇− µ+ ν|ψ(~x)|2
}
ψ(~x) = 0 (2.8)
(we suppose that the function ψ(x) slightly changes in the course of the interaction
radius r0). If we introduce the function u(~x) instead of ψ(~x)
ψ(~x) = ei~p~xu(~x), (2.9)
where u(~x) is the periodical function of ~x
u(~x+ ~a) = u(~x), (2.10)
which can be represented in the form
u(~x) = η(~x)eiϕ(~x), (2.11)
then we obtain the following equations for determining the functions η, ϕ:
− 1
2m
∆η +
η
2m
(∇ϕ)2 − η~̃v∇ϕ− µ̃η + νη3 = 0,
− 1
m
∇η∇ϕ− η
2m
∆ϕ+ ~̃v∇η = 0, (2.12)
where
~̃v = ~v − ~p
m
, µ̃ = µ+ ~p~v − ~p 2
2m
. (2.13)
Further, it will be supposed that the condensate has got only one period X along
the axis x and the system is spatially homogeneous with respect to the axes y and z.
229
A.S.Peletminsky, S.V.Peletminsky, Yu.V.Slyusarenko
Bearing in mind this supposition, the solution of the last equation from (2.12) can
be written as
ϕ(x) = mṽ||x+ C
∫ x
a
dx
ζ
, (2.14)
where ζ(x) = η2(x), ṽ|| = ṽx. The integration constant C, equalled to
C = (2πn−mṽX)
/ ∫ X
0
dx
ζ
(2.15)
is defined from the “periodical” condition of the function ϕ(x)
ϕ(x+X) = ϕ(x) + 2πn,
where n is an arbitrary integer and a fixed number. This number determines different
solutions (or phases) of equations (2.12). Substituting the function ϕ(x) into the first
equation from (2.12) we get:
− 1
2m
d2η
dx2
+
C2
2m
1
η3
− ηµ∗ + νη3 = 0, (2.16)
where
µ∗ = µ+
mv2
2
− m~̃v
2
⊥
2
, ~̃v⊥ = ~̃v − ~̃v||. (2.17)
Multiplying (2.16) by dη/dx, we find the first integral:
(
dζ
dx
)2
= f(ζ), (2.18)
where
f(ζ) = 4µν
(
ζ3 − 2
µ∗
ν
ζ2 + 2
E
ν
ζ − C2
mν
)
≡ 4µν(ζ − ζ1)(ζ − ζ2)(ζ − ζ3), (2.19)
moreover
ζ1 + ζ2 + ζ3 =
2µ∗
ν
, ζ1ζ2 + ζ1ζ3 + ζ2ζ3 =
2E
ν
, ζ1ζ2ζ3 =
C2
mν
. (2.20)
(E is the integration constant).
Next, it will be shown that for achieving stability of the periodical Bose conden-
sate with respect to a normal state, the following conditions
ν > 0, µ∗ > 0 (2.21)
should be fulfilled (see (3.4)). The first inequality leads to
f(0) < 0.
230
On the existence of periodical Bose-condensate
It is necessary to have all the three roots ζi real and positive for the existence of the
periodical solutions ζ(x) > 0 of equation (2.18). Moreover, the positive roots ζ i can
be always chosen so that the following inequalities will be valid
0 < ζ1 < ζ2 < ζ3. (2.22)
Let us also note that the inequality E > 0 holds true. It is evident that the
periodical solution ζ(x) of equation (2.18) lies between ζ1 and ζ2
ζ1 < ζ(x) < ζ2.
Solution of equation (2.18) for ζ(x) in terms of Jacobi’s elliptic functions has the
form
ζ(x) = ζ1 + (ζ2 − ζ1)sn
2(u, k), (2.23)
where
u =
2K(k)
X
x, k2 =
ζ2 − ζ1
ζ3 − ζ1
(2.24)
and K(k) is the full elliptic integral of the first type
K(k) =
∫ π/2
0
dϕ
√
1− k2 sin2 ϕ
.
We want to emphasize that the function ζ(x) has a period which equals to X due
to the formula
sn(u+ 2K(k), k) = −sn(u, k).
Let us turn to determining the phase ϕ(x). According to (2.14) we have
ϕ(x) = mṽ||x+ (2πn−mṽ||X)Z(x), (2.25)
where
Z(x) =
∫ x
0
dx
ζ
/∫ X
0
dx
ζ
.
Proceeding from the last formula for the function Z(x) it is easy to conclude that
Z(x) = l + Z0(x− lX), lX < x < (l + 1)X. (2.26)
Here
Z0(x) =
{
1
2
Y (x), 0 < x < X
2
,
1− 1
2
Y (x), X
2
< x < X,
(2.27)
where
Y (x) =
(∫ ζ2
ζ1
dζ
ζ
√
f
)−1 ∫ ζ(x)
ζ1
dζ
ζ
√
f
=
G(θ(x), ζ2
ζ1
− 1, k)
G(π
2
, ζ2
ζ1
− 1, k)
,
θ(x) ≡ θ(ζ(x)) = arcsin
√
ζ(x)− ζ1
ζ2 − ζ1
and G(θ, h, k) is the elliptic integral of the third type
G(θ, h, k) =
∫ x
0
dx
(1 + h sin2 x)
√
1− k2 sin2 x
.
Expressions (2.24)–(2.27) along with formula (2.11) determine the structure of the
function u(x).
231
A.S.Peletminsky, S.V.Peletminsky, Yu.V.Slyusarenko
3. Thermodynamics of the spatially periodical Bose
condensate
According to (2.5), the density of the Gibbs thermodynamic potential ω in a
model with the isolated condensate is defined by the formula
ω =
Ω
V β
= − 1
V β
ln Sp exp
[
−β
(
Ĥ(ψ)− ~v ~̂P (ψ)− µN̂(ψ)
)]
, (3.1)
or in the approximation of a weakly nonideal Bose gas we can write
ω =
1
V
(
H(ψ)− ~v ~P (ψ)− µN(ψ)
)
, (3.2)
where H(ψ), ~P (ψ), N(ψ) are defined by (2.7). Therefore, using (2.24)–(2.27), (2.11)
we find
ω =
1
X
∫ X
0
dx
{
− η
2m
d2η
dx2
+
1
2m
C2
η2
+
1
2
νη4−µ∗η2
}
, µ∗ = µ+
m~v 2
2
−m~̃v
2
⊥
2
. (3.3)
Excluding d2η/dx2 with the help of (2.16) we get
ω = − ν
2X
∫ X
0
dxη4 = − ν
2X
∫ X
0
dxζ2. (3.4)
Since the potential of a normal state is equal to zero (see formula (3.2), where ψ = 0),
the condition of the thermodynamic stability of the considered spatially-periodical
Bose condensate has the form
ω < 0. (3.5)
Therefore, from (2.24) it follows that ν > 0, and from (2.23) that µ∗ > 0.
Let us obtain an explicit form for the thermodynamic potential ω. Changing
from the integration variable x to ζ in the integrand of formula (3.4) and using
(2.18), (2.19) we find
ω = − ν
X
∫ ζ2
ζ1
dζ
dx
dζ
ζ2 = − ν
X
∫ ζ2
ζ1
dζ
ζ2
√
f(ζ)
. (3.6)
Calculating the integral entering this formula, it can be shown that
ω(ζ) =
ν
6
{
ζ1ζ2 + ζ1ζ3 + ζ2ζ3 − 2ζ3(ζ1 + ζ2 + ζ3) + 2(ζ3 − ζ1)(ζ1 + ζ2 + ζ3)
E(k)
K(k)
}
,
k2 =
ζ2 − ζ1
ζ3 − ζ1
. (3.7)
where E(k) is the full elliptic integral of the second type.
E(k) =
∫ π/2
0
dϕ
√
1− k2 sin2 ϕ. (3.8)
232
On the existence of periodical Bose-condensate
Density of the condensate particles is given by the formula
1
X
∫ X
0
dx|ψ(x)|2 = 1
V
∑
~τ
|a~p+~τ |2 ≡ n0 =
1
X
∫ X
0
dxζ(x) =
2
X
∫ ζ2
ζ1
dζ
ζ
√
f(ζ)
,
or
n0 = ζ3 − (ζ3 − ζ1)
E(k)
K(k)
. (3.9)
Thus, we can use the variables ζ1, ζ2, ζ3 instead of the thermodynamic variables
µ∗, ~̃v||, X . The obtained quantities ω, n0 are the functions of these new variables
ζ1, ζ2, ζ3. It can be easily seen that the variables ζ1, ζ2, ζ3 and µ∗, ṽ||, X related by
K(k)√
ζ3 − ζ1
=
1
2
√
mνX, ζ1 + ζ2 + ζ3 =
2µ∗
ν
,
2
ζ1
√
ζ3 − ζ1
√
ζ1ζ2ζ3G
(
ζ2
ζ1
− 1, k
)
= |2πn−mṽ||X|, (3.10)
where G(h, k) is the full elliptic integral of the third type
G(h, k) =
∫ π/2
0
dϕ
(1 + h sin2 ϕ)
√
1− k2 sin2 ϕ
.
Using expression (3.6) for the thermodynamic potential ω one can obtain a thermo-
dynamic identity
δω = − 1
X
(〈
1
m
(
∂η
∂x
)2〉
+
2πnC
mX
)
δX − Cδṽ|| − n0δµ
∗, (3.11)
which represents the second thermodynamic law for reversible processes.
We also want to write a following formula for the average value of the density of
momenta
~π = − i
2X
∫ X
0
dx
{
ψ∗∇ψ −∇ψ∗ψ
}
.
It yields
π|| = n0mv|| + C, ~π⊥ = n0~v⊥.
Thus, a physical meaning of the constant C (see (2.15)) determining the component
of momenta along the x axis is shown. As we can see, the transverse component of
momenta ~π⊥ coincides with the density of momenta for a superfluid movement (we
take into account that the density of the normal component is equal to zero in the
approximation considered).
The above mentioned formulae can be simplified for the limit ζ 1 → 0. In this
case according to (2.24) k2 = ζ2/ζ3 and according to (2.23) the function ζ(x) has
the form:
ζ(x) = ζ2sn
2
(
x
X
2K(k)
)
,
233
A.S.Peletminsky, S.V.Peletminsky, Yu.V.Slyusarenko
and the phase ϕ(x) is given by
ϕ(x) = mṽ||x+ (2πn−mṽ||X)
(
l +
1
2
)
, lX < x < (l + 1)X.
Bearing in mind (2.20) we have
ζ2 + ζ3 =
2µ∗
ν
, ζ2ζ3 =
2E
ν
,
wherefrom one can find
ζ2 =
µ∗
ν
(
1−
√
1− E
U0
)
, ζ3 =
µ∗
ν
(
1 +
√
1− E
U0
)
,
where U0 = µ∗2/2ν. These formulae lead to
k2 =
ζ2
ζ3
=
1−
√
1− E
U0
1 +
√
1− E
U0
and, consequently,
ζ2 =
µ∗
ν
2k2
1 + k2
, ζ3 =
µ∗
ν
2
1 + k2
. (3.12)
Therefore, formulae (3.7), (3.12) imply that
ω(µ∗, k) =
2
3
µ∗2
ν
1
1 + k2
(
2
E(k)
K(k)
− 2 + k2
1 + k2
)
, X =
2√
2mµ∗
√
1 + k2K(k).
The quantity ṽ|| entering ϕ(x) should be associated with the infinitesimal quantity
ζ1. In order to set this connection let us return to formula (3.10) and find the
asymptotes of the function G(ζ2/ζ1 − 1, k) at ζ1 → 0. To this end we note that by
means of an operation of integration by part it is easy to prove the validity of the
relation
∫ π/2
0
dϕ
(ζ1 + χ2 sin
2 ϕ)
√
1− k2 sin2 ϕ
=
=
∫ π/2
0
dϕ
ζ1 + χ2 sin
2 ϕ
+
∫ π/2
0
dϕ
k2 sinϕ cosϕ
(1− k2 sin2 ϕ)3/2
∫ π/2
ϕ
dϕ
ζ1 + χ2 sin
2 ϕ
,
here (χ2 = ζ2 − ζ1). Using this formula we have
ζ−1
1 G
(
ζ2
ζ1
− 1; k
)
−−→
ζ1→0
π
2
1√
ζ1
+
K(k)− E(k)
ζ2
.
This yields
2
√
ζ1ζ2ζ3
ζ1
√
ζ3 − ζ1
G
(
ζ2
ζ1
− 1; k
)
−−→
ζ1→0
π + 2
K(k)−E(k)√
ζ2
√
ζ1.
234
On the existence of periodical Bose-condensate
Thus, formula (3.10) takes the form
2
K(k)−E(k)√
ζ2
√
ζ1 = 2π(n− 1/2)−mṽ||X > 0
(the case 2π(n−1/2)−mṽ||X < 0 is not realized). In such a waymṽ||X → 2π(n−1/2)
at ζ1 → 0 and consequently (see (2.27))
ϕ(x) = mṽx+ πZ0(x) = 2π(n− 1/2)
x
X
+ π
(
l +
1
2
)
, lX < x < (l + 1)X.
Let us find now the function u(x) ≡ eiϕ(x)
√
ζ(x). According to (2.11), (2.23) we can
write
u(x) =
√
ζ2
∣∣∣∣sn(
x
X
K(k))
∣∣∣∣ e
iϕ(x) = i
√
ζ2 sn(
x
X
2K(k)) exp (−iπx/X) exp (2πnix/X),
where sn(x2K(k)/X) exp (−iπx/X) is the periodical function of x with the periodX
due to sn(u+2K(k)) = −snu. The periodX is given by the formulaX = (2/
√
2mµ∗)
×
√
1 + k2K(k). Expanding the periodical function sn(x2K(k)/X) exp (−iπx/X)
into the Fourier series
exp
(
−iπ
x
X
)
sn
(
2x
X
K(k)
)
=
∑
m
bm exp
(
2πim
x
X
)
,
where
2πbm =
∫ π
−π
dte−i(m+1/2)tsn
(
t
π
K(k)
)
= − iπ2
kK
(
sh
(
n+
1
2
)
π
K
′
K
)−1
,
K ′ = K|k2→1−k2
we obtain
u(x) = i
√
ζ2 exp
(
2πin
x
X
)∑
m
bm exp
(
2πim
x
X
)
.
The last formula with (2.4) indicate that a population of the states by condensate
particles with the momenta ~p+~i2π(m+ n)/X is determined by the formula
∣∣∣∣a~p+~i2πm/X
∣∣∣∣
2
= V
2µ∗
ν
k2|bm|2.
Now let us study the problem concerning the stability of the Bose condensate.
As it was mentioned the inequality
ω(ζ) < 0
takes place at ν > 0. Since the potential ω = 0 corresponds to a normal state (the
order parameter ψ turns to zero for the normal state), the spatially periodical Bose
condensate is thermodynamically stable with respect to the normal state.
235
A.S.Peletminsky, S.V.Peletminsky, Yu.V.Slyusarenko
Let us consider the phase with n = 0. Then according to (3.11)
∂ω
∂X
= − 1
mX
〈(
∂η
∂x
)2〉
< 0.
It means that ω is the decrease function of the period X . Thus, at n = 0 the
spatially homogeneous state with X = ∞ is an absolutely stable state. This state
corresponds to the Bogolyubov solution. However, the mentioned fact does not mean
that the spatially periodical solutions do not realize even in the case n = 0. Really,
the considered spatially periodical solutions correspond to the statistical operator ρ̂
which is invariant with respect to the unitary transformation U = exp iX(P̂x − pxN̂)
UρU+ = ρ
and, consequently, in virtue of the conservation laws for P̂x and N̂ , the period X
is the motion constant and cannot change if the conservation law for P̂x is not
violated. But if the conservation law for P̂x is weakly violated, then the studied
spatially periodical states become quasistationary ones and parameter X will be
slowly turned to infinity (in virtue of the weakly violation of the conservation law for
P̂x). That is why the considered spatially periodical solution turns to the Bogolyubov
spatially homogeneous solution at t→ ∞.
This is similar to the situation for the normal state when the term −~v ~̂P is added
to the Gibbs exponent. The quantity ~v is the motion constant and the state is stable
until the conservation law is violated. But the state becomes quasistationary and
parameter ~v turns to zero at the weakly violation of the momenta conservation law.
Even a more complicated situation can arise for phases with n 6= 0. In this case
the condition ∂ω/∂X < 0 becomes not evident and, hence, the possibility of the
existence of absolutely stable and spatially periodical solutions with a definite value
of the period X can appear. The period X is not an independent thermodynamic
parameter (as in quantum crystals) anymore, but it can be treated as a certain
function of µ∗, ṽ;X = X(µ∗, ṽ||).
4. Acknowledgements
This work was supported in part by the German Ministry of Science and Tech-
nology (BMBF: 411 4006 06ROS02), and also by the Ukrainian State Foundation
for Fundamental Studies.
References
1. Anderson M.H., Ensher I.R., Mattews M.R. et al. Observation of Bose-Einstein con-
densation in a dilute atomic vapors. // Science, 1995, vol. 269, p. 198–201.
2. Bogolyubov N.N. On the theory of superfluidity. // Izv. Akad. Nauk. SSSR, ser. fiz.,
1947, vol. 11, No. 1, p. 77–90 (in Russian).
3. Bogolyubov N.N. Quasiaverages in the problems of statistical mechanics. Preprint of
Joint Institute of Nuclear Researches, JINR–1451, Dubna, 1963, 123 p. (in Russian).
236
On the existence of periodical Bose-condensate
4. Lavrinenko N.M., Peletminsky S.V. Thermodynamics and motion equations of quantum
crystals. // Teor. Mat. Fiz., 1986, vol. 66, No. 2, p. 314–325 (in Russian).
5. Andreev A.F., Lifshitz I.M. Quantum theory of crystal defects. // Zh. Eksp. Teor. Fiz.,
1969, vol. 56, No. 6, p. 2057–2068 (in Russian).
6. Akhiezer A.I., Peletminskii S.V., Slyusarenko Yu.V. Theory of a weakly nonideal Bose
gas in a magnetic field. // JETP, 1998, vol. 86, No. 3, p. 501–506.
7. Peletminsky A.S., Peletminsky S.V., Slyusarenko Yu.V. On phase transitions in a Fermi
liquid. II. Transition associated with translational symmetry breaking. // Low Tem-
perature Physics, 1999, vol. 25, No. 5, p. 303–313.
Про можливIсть Iснування просторово перIодичного
бозе-конденсату
О.С.Пелетминський 1 , С.В.Пелетминський 2 ,
Ю.В.Слюсаренко 2
1 Науково-технiчний центр електрофiзичної обробки НАН України,
61002 Харкiв
2 Нацiональний науковий центр
“Харкiвський фiзико-технiчний iнститут”,
61108 Харкiв
Отримано 6 березня 2000 р.
У роботi вивчено можливiсть побудови теoрiї просторово-перiодич-
ного бозе-конденсату в моделi слабконеiдеального бозе-газу. На ос-
новi методу видiленого бозе-конденсату сформульовано рiвняння
для визначення просторово-перiодичного параметра порядку при
температурi T = 0 , коли можна знехтувати внеском квазiчасти-
нок до термодинамiки системи бозонiв, та знайдено їх одноперiо-
дичнi розв’язки. Коротко викладено питання термодинамiчної стiй-
костi просторово-перiодичного бозе-конденсату.
Ключові слова: слабко неiдеальний бозе-газ, метод видiленого
бозе-конденсату, просторово перiодичний бозе-конденсат,
термодинамiчна стiйкiсть
PACS: 03.75.Fi, 05.30.Jp, 05.70.a
237
238
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