Bogolyubov’s Approximation for Bosons
We analyze the approximating Hamiltonian method for Bose systems. Within the framework of this method, the pressure for the mean field model of an imperfect boson gas is calculated. The problem is considered by the systematic application of the Bogolyubov–Ginibre approximation. Проаналiзовано метод...
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| Cite this: | Bogolyubov’s Approximation for Bosons / N.N. Bogolyubov (jr.), D.P. Sankovich // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 104-108. — Бібліогр.: 14 назв. — англ. |
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| citation_txt | Bogolyubov’s Approximation for Bosons / N.N. Bogolyubov (jr.), D.P. Sankovich // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 104-108. — Бібліогр.: 14 назв. — англ. |
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| description | We analyze the approximating Hamiltonian method for Bose systems. Within the framework of this method, the pressure for the mean field model of an imperfect boson gas is calculated. The problem is considered by the systematic application of the Bogolyubov–Ginibre approximation.
Проаналiзовано метод апроксимуючого гамiльтонiана для бозе-систем. У межах цього методу знайдено тиск для моделi середнього поля неiдеального бозе-газу. Задачу розглянуто за допомогою послiдовного застосування апроксимацiї Боголюбова–Жiнiбра.
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GENERAL PROBLEMS OF THEORETICAL PHYSICS
104 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
BOGOLYUBOV’S APPROXIMATION FOR BOSONS
N.N. BOGOLYUBOV, JR., D.P. SANKOVICH
V.A. Steklov Institute of Mathematics, RAS
(8, Gubkin Str., Moscow 119991, Russia)
PACS 05.30.Jp, 03.75.Hh,
03.75.Gg, 67.40.-w
c©2010
We analyze the approximating Hamiltonian method for Bose sys-
tems. Within the framework of this method, the pressure for
the mean field model of an imperfect boson gas is calculated.
The problem is considered by the systematic application of the
Bogolyubov–Ginibre approximation.
1. Introduction
The approximating Hamiltonian method [1], having its
roots in the work of Bogolyubov [2], gives possibility to
provide an elegant way of proving the thermodynami-
cal equivalence of some Hamiltonians. In the main, this
method was applied for fermion models. In boson sys-
tems exhibiting the Bose condensation, we meet certain
difficulties.
Pulé and Zagrebnov [3] considered the mean field bo-
son gas by the approximating Hamiltonian technique
and found the pressure of this model in the thermody-
namic limit. The essential feature of their proof is the
addition of sources, i.e. they used the Bogolyubov con-
cept of quasiaverages. Here, we suggest an alternative
way of applying the approximating Hamiltonian method
for boson models.
Consider a system of identical bosons of mass m con-
fined to a d-dimensional cubic box Λ ⊂ Rd of volume V
centered around the origin. Let EΛ
0 < EΛ
1 ≤ EΛ
2 ≤ . . .
be the eigenvalues of the operator hΛ
.= −Δ/(2m) (we
suppose ~ = 1) on Λ with some linear boundary con-
ditions, and let {ΦΛ
l } with l = 0, 1, 2, . . . be the cor-
responding eigenfunctions. Let FΛ be the symmetric
Fock space constructed from L2(Λ). Let al
.= a(ΦΛ
l ) and
a†l
.= a†(ΦΛ
l ) be the boson annihilation and creation op-
erators on FΛ. Denote, by TΛ, the Hamiltonian of the
free Bose gas on FΛ constructed from hΛ in the usual
manner, that is TΛ =
∑∞
l=0E
Λ
l Nl, where Nl = a†l al. Let
NΛ =
∑∞
l=0Nl be the operator of the number of particles
on FΛ.
The Hamiltonian we consider is
HΛ = TΛ +
a
2V
N2
Λ, (1)
where a is a positive coupling constant. Hamiltonian (1)
is known as the mean field Bose gas model [4].
Let µ0
.= limΛ↑Rd EΛ
0 . Denote, by p0(µ) and ρ0(µ), the
grand-canonical pressure and the density, respectively,
for the free Bose gas at chemical potential µ < µ0, that
is
p0(µ) = −
∫
ln[1− exp(−β(ν − µ))]F (dν),
ρ0(µ) =
∫
1
exp(β(ν − µ))− 1
F (dν),
F being the integrated density of states of hΛ in the limit
Λ ↑ Rd. Let ρc
.= limµ→µ0 ρ0(µ). The grand-canonical
pressure of the mean field Bose gas model is
pΛ(µ) =
1
βV
ln Tr exp[−β(HΛ − µNΛ)]. (2)
We put p(µ) = limΛ↑Rd pΛ(µ).
The main result of work [3] is the following:
Proposition 1. Pressure (2) in the thermodynamic
limit exists and is given by
p(µ) =
1
2
aρ2(µ) + p0(µ− aρ(µ)) if µ ≤ µc,
(µ− µ0)2
2a
+ p0(µ0) if µ > µc,
(3)
where µc = µ0 + aρc, and ρ(µ) is the unique solution of
the equation ρ = ρ0(µ− aρ).
BOGOLYUBOV’S APPROXIMATION
This result was obtained by many authors for special
boundary conditions (see [3] for references). The inno-
vation of work [3] is in proving Proposition 1 by the
approximating Hamiltonian method.
The main technical ingredient of the approximating
Hamiltonian method is an estimate for the difference of
the appropriate pressures of model and approximating
Hamiltonians obtained by using the Bogolyubov convex-
ity inequality [5]. But, in the case of model (1), the ob-
vious intention to use this inequality for the “natural”
approximating Hamiltonian
Happr
Λ (ρ) = TΛ + aρNΛ −
aρ2
2
V, (4)
where ρ is the self-consistency parameter, meets the fail-
ure. Really, by Bogolyubov’s inequality,
0 ≤ pappr
Λ (ρ̄Λ, µ)− pΛ(µ) ≤
≤ 1
V
〈HΛ −Happr
Λ (ρ̄Λ)〉Happr
Λ (ρ̄Λ), (5)
where
pappr
Λ (ρ, µ) =
1
βV
ln Tr exp[−β(Happr
Λ (ρ)− µNΛ)],
〈. . . 〉Γ denotes the appropriate grand-canonical average
with respect to the Hamiltonian Γ, and ρ̄Λ satisfies the
self-consistency equation
ρ̄Λ =
1
V
〈NΛ〉Happr
Λ (ρ̄Λ).
One can see that the right-hand side of (5) does not tend
to zero as V →∞ for all µ ∈ R. It tends to aρ2
0/2, where
ρ0 is the Bose condensate density. Thus, the thermody-
namical equivalence of the model Hamiltonian (1) and
the approximating Hamiltonian (4) takes place only in
the domain, where there is no Bose condensate. One
easily obtains that this domain is µ ≤ µc, and
p(µ) = inf
α<µ0
[
(µ− α)2
2a
+ p0(α)
]
. (6)
Curiously, that expressions (3) and (6) are equal for
any µ. This fact holds out a hope for the chance to use,
nevertheless, the approximating Hamiltonian method for
the mean field model. It is clearly necessary to construct
a different approximating Hamiltonian. This idea was
realized in [3].
First, the authors of [3] constructed the auxiliary
Hamiltonian for η ∈ C,
HP−Z
Λ (η) = HΛ +
√
V (ηa†0 + η∗a0). (7)
A convenient approximating Hamiltonian has the form
HP−Z
Λ (ρ, η) = TΛ + aρNΛ −
aρ2
2
V+
+
√
V (ηa†0 + η∗a0), (8)
where the self-consistency parameter ρ ∈ R.
Next, to prove Proposition 1, the authors showed that,
for η 6= 0, the pressure
pP−Z
Λ (η, µ) =
1
βV
ln Tr exp[−β(HP−Z
Λ (η)− µNΛ)]
in the thermodynamic limit coincides with the pressure
pP−Z
Λ (ρ, η, µ) =
1
βV
ln Tr exp[−β(HP−Z
Λ (ρ, η)− µNΛ)]
minimized with respect to ρ (Lemmata 1 and 2 in [3])
and that, in turn, this minimization can be performed
after the passage to the limit (Lemma 3 in [3]).
Finally, the authors switched off the source η (η → 0)
in Lemma 4 to obtain the limiting pressure p(µ).
For Hamiltonians (7) and (8), the Bogolyubov’s con-
vexity inequality is
0 ≤ pP−Z
Λ (ρ̄Λ, η, µ)− pP−Z
Λ (η, µ) ≤ 1
2V 2
ΔΛ(η),
where
ΔΛ(η) = a〈(NΛ − V ρ̄Λ)2〉HP−Z
Λ (ρ̄Λ,η)
,
and ρ̄Λ satisfies the equation
ρ̄Λ =
1
V
〈NΛ〉HP−Z
Λ (ρ̄Λ,η)
.
In this case, as distinct from case (5), the authors proved
that
lim
V→∞
1
V 2
ΔΛ(η) = 0
for η 6= 0. Therefore (Lemma 2 of [3]),
lim
Λ↑Rd
pP−Z
Λ (η, µ) = lim
Λ↑Rd
pP−Z
Λ (ρ̄Λ, η, µ) (9)
for η 6= 0. This is the benefit of the Pulé–Zagrebnov
approach.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 105
N.N. BOGOLYUBOV, jr., D.P. SANKOVICH
2. Application of the Approximating
Hamiltonian Method to the Mean
Field Model
Let us first restrict our discussion to the cases of peri-
odic, Dirichlet, Neumann, and repulsive-wall boundary
conditions. In the case of attractive walls, there are two
negative eigenvalues for the free Bose gas. Owing to
this fact, we have some features in the thermodynam-
ical properties of the model. We consider the case of
attractive boundary conditions separately.
We suggest to take advantage of the macroscopic oc-
cupation of the zero momentum one-particle state to re-
place the corresponding operators a0, a
†
0 by c-numbers
[6–8]. Therefore, we define
HΛ(c) =
∞∑
l=1
[
EΛ
l − µ+ |c|2a
]
Nl +
a
2V
N
′2
Λ +
+
a|c|2
2
+
a|c|4
2
V + (EΛ
0 − µ)|c|2V, (10)
where c ∈ C, N ′Λ =
∑∞
l=1Nl, and the HamiltonianHΛ(c)
contains the term −µNΛ. This is the Bogolyubov ap-
proximation which was proposed as early as 1947 [2, 9].
The replacement is exact in the thermodynamic limit
[10]. Therefore, we have
lim
Λ↑Rd
pΛ(µ) = lim
Λ↑Rd
sup
c
pΛ(c, µ) ≡ p(c̄, µ), (11)
where
pΛ(c, µ) =
1
βV
ln Tr′ exp[−βHΛ(c)],
and |c̄|2 is the density of the Bose condensate. Tr′
means trace in F ′Λ, where F ′Λ is the boson Fock space
constructed on the orthogonal complement of the one-
dimensional subspace of L2(Λ) generated by ΦΛ
0 .
Starting from Hamiltonian (10), we get the following
approximating Hamiltonian for ρ′ ∈ R:
HΛ(c̄, ρ′) =
∞∑
l=1
(
EΛ
l − µ+ |c̄|2a
)
Nl+
+
a|c̄|2
2
+
a|c̄|4
2
V +(EΛ
0 −µ)|c̄|2V +aρ′N ′Λ−
aρ′2
2
V. (12)
Now, we are going to apply the Bogolyubov’s convexity
inequality
0 ≤ pΛ(ρ̄′Λ, c̄, µ)− pΛ(c̄, µ) ≤ 1
2V 2
Δ′Λ(c̄), (13)
where
pΛ(ρ′, c, µ) =
1
βV
ln Tr′ exp [−βHΛ(c, ρ′)] ,
Δ′Λ(c̄) = a〈(N ′Λ − V ρ̄′Λ)2〉HΛ(c̄,ρ̄′Λ),
and ρ̄′Λ satisfies the equation
ρ̄′Λ =
1
V
〈N ′Λ〉HΛ(c̄,ρ̄′Λ).
One calculates
Δ′Λ(c̄) = a
∞∑
l=1
exp(βεΛl )
[exp(βεΛl )− 1]2
,
where εΛl = EΛ
l − µ + a(ρ̄′Λ + |c̄|2). Referring to the in-
equality cothx < 1 + x−1 with x ≥ 0, we get
Δ′Λ(c̄) < a
∞∑
l=1
1
exp(βεΛl )− 1
(
1 +
2
βεΛl
)
. (14)
Use inequality [11]
〈
[A, [H,A†]]
〉
≥ 0 which is valid for
any operator A and for any self-conjugate Hamiltonian
H. Take A = a†l , H = HΛ(c, ρ′). One gets EΛ
0 + a(ρ̄′Λ +
|c̄|2)− µ ≥ 0 for large V . So we obtain that
εΛl > EΛ
1 − EΛ
0 ≥
π2
2m
V −2/3 for l = 1, 2, 3, . . . . (15)
Inserting this estimate into (14), we have
Δ′Λ(c̄) < aρ̄′ΛV
(
1 +
4m
π2β
V 2/3
)
.
Therefore,
lim
V→∞
Δ′Λ(c̄)
V 2
= 0,
and we conclude that
lim
Λ↑Rd
pΛ(µ) = lim
Λ↑Rd
pΛ(ρ̄′Λ, c̄, µ).
Hence, the grand-canonical pressure of the mean field
model is
p(µ) = −a|c̄|
4
2
+ (µ− µ0)|c̄|2+
+ inf
ρ≥0
[
aρ2
2
+ p0(µ− a(|c̄|2 + ρ))
]
.
106 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
BOGOLYUBOV’S APPROXIMATION
Finally, for µ ≤ µc, we have |c̄| = 0, and ρ′(µ) is the
unique solution of ρ′ = ρ0(µ− aρ′); for µ > µc, we have
µ− a(|c̄|2 + ρ′) = µ0. Proposition 1 is proved.
From the very beginning, we consider the case of a
cubic box Λ. If we take a parallelepiped of the same
volume with sides of length Lj = V αj , j = 1, 2, 3,
such that α1 ≥ α2 ≥ α3 > 0 and α1 + α2 + α3 = 1,
the situation changes. We can prove estimate (15) for
α1 < 1/2 only. For a long parallelepiped in the direc-
tion j = 1, there is the generalized condensation in Gi-
rardeau’s sense [12], and the proposed method must be
improved.
Consider the case of attractive walls in the dimension
d = 1. The d > 1 generalization becomes more tedious
but is not hard. The spectrum of the one-dimensional
Schrödinger equation − 1
2mΦ′′ = εΛΦ with attractive
boundary conditions Φ′(−L/2) = σΦ(−L/2),Φ′(L/2) =
−σΦ(L/2), where σ < 0, consists of two negative eigen-
values tending to the same limit (when L→∞) and an
infinite number of positive eigenvalues (for L|σ| > 2),
namely
εΛ0 = − σ
2
2m
−O(e−L|σ|),
εΛ1 = − σ
2
2m
+O(e−L|σ|),
1
2m
(
(k − 1)π
L
)2
< εΛk <
1
2m
(
kπ
L
)2
for k ≥ 2.
We take this fact into account in the Bogolyubov approx-
imation, replacing the operators a#
0 , as well as a#
1 , by
c-numbers. Thus, instead of the approximating Hamil-
tonian (10), we can write
HΛ(c) = (εΛ0 − µ)|c|2L+
a|c|4
2
L+
∞∑
l=2
(
εΛl − µ +
+|c|2a
)
Nl +
a
2L
Ñ2
Λ +
a|c|2
2
+ ΔεΛ|c1|2L, (16)
where ΔεΛ .= εΛ1 − εΛ0 ∼ O(e−L|σ|), ÑΛ =
∑∞
l=2Nl and
|c|2 = |c0|2 + |c1|2 with |ci|2 = 〈a†iai/L〉HΛ(c), i = 0, 1.
Obviously, the last term in (16) is inessential in the
thermodynamic limit. Reiterating our previous con-
sideration, we prove Proposition 1 in the case of at-
tractive walls. The only difference in comparison with
the above-stated cases of boundary conditions is con-
nected with the replacement of the operator N ′Λ by
ÑΛ. Thus, we must begin to sum over l from 2 in
the appropriate formulae, and the main estimate (15)
is
εΛl > εΛ2 − εΛ0 >
σ2
2m
for l = 2, 3, . . . .
In [13], Vandevenne and Verbeure rigorously studied
the imperfect Bose gas with attractive boundary condi-
tions, where
H̃Λ = TΛ +
a
2V
Ñ2
Λ. (17)
The authors gave a proof of the occurrence of Bose
condensation in the one-dimensional case. The con-
densation is equally distributed over the two nega-
tive energy levels and is localized in the same area
as that for the free Bose gas with attractive bound-
ary conditions [14]. Remark that the interaction in
(17) is not of the usual form (1). The reason for
choosing (17) instead of (1) is the breaking of the
spatial translation invariance in the terms with N0
and N1.
The model Hamiltonian (17) can be treated by the
stereotyped approximating Hamiltonian method, so long
as we have
lim
V→∞
〈H̃Λ − H̃appr
Λ (ρ̄Λ)〉H̃appr
Λ (ρ̄Λ) = 0,
where
H̃appr
Λ (ρ) = TΛ + aρÑΛ −
aρ2
2
V,
and ρ̄Λ satisfies the self-consistency equation
ρ̄Λ =
1
V
〈ÑΛ〉H̃appr
Λ (ρ̄Λ).
As distinct from the usual Hamiltonian (1), Hamil-
tonian (17) is not superstable, and µ ≤ µ0 =
−dσ2/(2m). Consequently, one must take this restric-
tion into account in the formula for the pressure of model
(17),
p̃(µ) = lim
Λ↑Rd
p̃appr
Λ (ρ̄, µ) = inf
ρ≥0
[
aρ2
2
+ p0(µ− aρ)
]
,
where
p̃appr
Λ (ρ, µ) =
1
βV
ln Tr exp
[
−β
(
H̃appr
Λ (ρ)− µNΛ
)]
.
We conclude by remarking that, in the case of
attractive boundary conditions, the condensate has
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 107
N.N. BOGOLYUBOV, jr., D.P. SANKOVICH
essentially infinite density and occupies essentially
zero volume near the walls as for the free Bose
gas.
1. N.N. Bogolyubov, jr., A Method for Studying Model
Hamiltonians (Pergamon Press, Oxford, New York,
1972).
2. N.N. Bogolyubov, J. Phys. (USSR) 11, 23 (1947).
3. J.V. Pulé and V.A. Zagrebnov, J. Phys. A: Math. Gen.
37, 8929 (2004).
4. K. Huang, Statistical Mechanics (Wiley, New York,
1963).
5. N.N. Bogolyubov, Phys. Abh. Sow. Un. 6, 113 (1962).
6. M. Corgini and D.P. Sankovich, Theor. Math. Phys. 108,
421 (1996).
7. M. Corgini and D.P. Sankovich, Int. J. Mod. Phys. B 11,
3329 (1997).
8. M. Corgini, D.P. Sankovich, and N.I. Tanaka, Theor.
Math. Phys. 120, 130 (1999).
9. J. Ginibre, Commun. Math. Phys. 8, 26 (1968).
10. D.P. Sankovich, J. Math. Phys. 45, 4288 (2004).
11. N.N. Bogolyubov, jr., Physica (Amsterdam) 32, 933
(1966).
12. M. Girardeau, J. Math. Phys. 1, 516 (1960).
13. L. Vandevenne and A. Verbeure, Rep. Math. Phys. 56,
109 (2005).
14. D.W. Robinson, Commun. Math. Phys. 50, 53 (1976).
Received 02.09.09
БОГОЛЮБОВСЬКА АПРОКСИМАЦIЯ ДЛЯ БОЗОНОВ
М.М. Боголюбов (мол.), Д.П. Санковiч
Р е з ю м е
Проаналiзовано метод апроксимуючого гамiльтонiана для
бозе-систем. У межах цього методу знайдено тиск для мо-
делi середнього поля неiдеального бозе-газу. Задачу розгля-
нуто за допомогою послiдовного застосування апроксимацiї
Боголюбова–Жiнiбра.
108 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
|
| id | nasplib_isofts_kiev_ua-123456789-13291 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-12-07T18:40:03Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Bogolyubov (jr.), N.N. Sankovich, D.P. 2010-11-04T10:18:52Z 2010-11-04T10:18:52Z 2010 Bogolyubov’s Approximation for Bosons / N.N. Bogolyubov (jr.), D.P. Sankovich // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 104-108. — Бібліогр.: 14 назв. — англ. 2071-0194 PACS 05.30.Jp, 03.75.Hh, 03.75.Gg, 67.40.-w https://nasplib.isofts.kiev.ua/handle/123456789/13291 We analyze the approximating Hamiltonian method for Bose systems. Within the framework of this method, the pressure for the mean field model of an imperfect boson gas is calculated. The problem is considered by the systematic application of the Bogolyubov–Ginibre approximation. Проаналiзовано метод апроксимуючого гамiльтонiана для бозе-систем. У межах цього методу знайдено тиск для моделi середнього поля неiдеального бозе-газу. Задачу розглянуто за допомогою послiдовного застосування апроксимацiї Боголюбова–Жiнiбра. en Відділення фізики і астрономії НАН України Загальні питання теоретичної фізики Bogolyubov’s Approximation for Bosons Боголюбовська апроксимація для бозонів Article published earlier |
| spellingShingle | Bogolyubov’s Approximation for Bosons Bogolyubov (jr.), N.N. Sankovich, D.P. Загальні питання теоретичної фізики |
| title | Bogolyubov’s Approximation for Bosons |
| title_alt | Боголюбовська апроксимація для бозонів |
| title_full | Bogolyubov’s Approximation for Bosons |
| title_fullStr | Bogolyubov’s Approximation for Bosons |
| title_full_unstemmed | Bogolyubov’s Approximation for Bosons |
| title_short | Bogolyubov’s Approximation for Bosons |
| title_sort | bogolyubov’s approximation for bosons |
| topic | Загальні питання теоретичної фізики |
| topic_facet | Загальні питання теоретичної фізики |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/13291 |
| work_keys_str_mv | AT bogolyubovjrnn bogolyubovsapproximationforbosons AT sankovichdp bogolyubovsapproximationforbosons AT bogolyubovjrnn bogolûbovsʹkaaproksimacíâdlâbozonív AT sankovichdp bogolûbovsʹkaaproksimacíâdlâbozonív |