The Bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble
We show that the Bogolubov generating functional method is a very effective tool for studying distribution functions of both equilibrium and nonequilibrium states of classical many-particle dynamical systems. In some cases the Bogolubov generating functionals can be represented by means of infinite...
Saved in:
| Date: | 2007 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2007
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/7241 |
| 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: | The Bogolubov generating functional method in statistical physics and "collective" variables transform within the grand canonical ensemble / N.N. Bogolubov (jr.), A.K. Prykarpatsky // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 37-50. — Бібліогр.: 10 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859518962349375488 |
|---|---|
| author | Bogolubov (jr.), N.N. Prykarpatsky, A.K. |
| author_facet | Bogolubov (jr.), N.N. Prykarpatsky, A.K. |
| citation_txt | The Bogolubov generating functional method in statistical physics and "collective" variables transform within the grand canonical ensemble / N.N. Bogolubov (jr.), A.K. Prykarpatsky // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 37-50. — Бібліогр.: 10 назв. — англ. |
| collection | DSpace DC |
| description | We show that the Bogolubov generating functional method is a very effective tool for studying distribution functions of both equilibrium and nonequilibrium states of classical many-particle dynamical systems. In some cases the Bogolubov generating functionals can be represented by means of infinite Ursell –Mayer diagram expansions, whose convergence holds under some additional constraints on the statistical system under consideration. The classical Bogolubov idea to use the Wigner density operator transformation for studying the nonequilibrium distribution functions is developed, a new analytic nonstationary solution to the classical Bogolubov evolution functional equation is constructed.
Доведено, що метод породжуючих функціоналів Боголюбова є досить ефективним для вивчення функцій розподілу рівноважних та нерівноважних станів класичних багаточастинкових динамічних систем. У деяких випадках породжуючі функціонали Боголюбова можна виразити через нескінченні розвинення діаграм Урселла - Мартіна, які збігаються при накладанні додаткових умов на розглядувані статистичні системи. Розвинуто класичну ідею Боголюбова про використання перетворення Вігнера оператора щільності для вивчення нерівноважних функцій розподілу та побудовано новий нестаціонарний розв'язок класичного рівняння еволюції функціонала Боголюбова.
|
| first_indexed | 2025-11-25T20:53:14Z |
| format | Article |
| fulltext |
UDC 517 . 9
THE BOGOLUBOV GENERATING FUNCTIONAL METHOD
IN STATISTICAL PHYSICS AND “COLLECTIVE” VARIABLES
TRANSFORM WITHIN THE GRAND CANONICAL ENSEMBLE
МЕТОД ПОРОДЖУЮЧИХ ФУНКЦIОНАЛIВ БОГОЛЮБОВА
В СТАТИСТИЧНIЙ ФIЗИЦI ТА ПЕРЕТВОРЕННЯ
ДО „КОЛЕКТИВНИХ” ЗМIННИХ
У ВЕЛИКОМУ КАНОНIЧНОМУ АНСАМБЛI
N. N. Bogolubov (jr.)
V. A. Steklov Math. Inst. Rus. Acad. Sci.
Gubkina Str., 8, Moscow, 119991, Russia
e-mail: nickolai_bogolubov@hotmail.com
A. K. Prykarpatsky
AGH Univ. Sci. and Technol.
Krakow, 30059, Poland, and Inst. Appl. Problems Mech.
and Math. Nat. Acad. Sci. Ukraine, Lviv
Naukova Str., 3 B, Lviv, Ukraine
e-mail: prykanat@cybergal.com
pryk.anat@ua.fm
We show that the Bogolubov generating functional method is a very effective tool for studying distribution
functions of both equilibrium and nonequilibrium states of classical many-particle dynamical systems. In
some cases the Bogolubov generating functionals can be represented by means of infinite Ursell – Mayer
diagram expansions, whose convergence holds under some additional constraints on the statistical system
under consideration. The classical Bogolubov idea to use the Wigner density operator transformation for
studying the nonequilibrium distribution functions is developed, a new analytic nonstationary solution to
the classical Bogolubov evolution functional equation is constructed.
Доведено, що метод породжуючих функцiоналiв Боголюбова є досить ефективним для вивчен-
ня функцiй розподiлу рiвноважних та нерiвноважних станiв класичних багаточастинкових ди-
намiчних систем. У деяких випадках породжуючi функцiонали Боголюбова можна виразити че-
рез нескiнченнi розвинення дiаграм Урселла – Мартiна, якi збiгаються при накладаннi додат-
кових умов на розглядуванi статистичнi системи. Розвинуто класичну iдею Боголюбова про
використання перетворення Вiгнера оператора щiльностi для вивчення нерiвноважних функ-
цiй розподiлу та побудовано новий нестацiонарний розв’язок класичного рiвняння еволюцiї
функцiонала Боголюбова.
1. Introduction. Bogolubov functional equation and distribution functional. Let a large system
of N ∈ Z+ (one-atomic and spinless) bose-particles with a fixed density ρ̄ := N/Λ in a volu-
me Λ ⊂ R3 be specified by a quantum-mechanical Hamiltonian operator Ĥ : L(sym)
2 (R3N ;
C) → L
(sym)
2 (R3N ; C) of the form
Ĥ := − ~2
2m
N∑
j=1
∇2
j +
N∑
j<k
V (xj − xk), (1.1)
c© N. N. Bogolubov (jr.), A. K. Prykarpatsky, 2007
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1 37
38 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
where∇j := ∂/∂xj , j = 1, N, ~ is the Plank constant,m ∈ R+ a particle mass and V (x−y) :=
:= V (|x − y|), x, y ∈ Λ, a two-particle potential energy allowing for a partition V = V (l) +
+V (s) with V (s) being a short range potential of the Lennard – Johns type and V (l) a long range
potential of the Coulomb type. Making use of the second quantization representation [1, 2], the
Hamiltonian (1.1), as Λ → R3 and N → ∞, can be written as a sum H = H0+V, where
H0 := − ~2
2m
∫
R3
d3xψ+∇2
xψ,
(1.2)
V :=
1
2
∫
R3
d3x
∫
R3
d3yV (x− y)ψ+(x)ψ+(y)ψ(y)ψ(x),
and the operator H : Φ → Φ acts on a suitable Fock space [1, 2] with the standard scalar
product (·, ·), and ψ+(x), ψ(y): Φ → Φ are creation and annihilation operators defined, corres-
pondingly, at points x ∈ R3 and y ∈ R3.
Assume now that our particle system is under the thermodynamic equilibrium at an “inverse”
temperature R+ 3 β → ∞. Then the corresponding Bogolubov n-particles distribution functi-
ons can be written [1, 3] as
Fn(x1, x2, . . . , xn) := (Ω, : ρ(x1)ρ(x1) . . . ρ(xn) : Ω), (1.3)
where n ∈ Z+, ρ(x) := ψ+(x)ψ(x) is the density operator at x ∈ R3, : : the usual [1, 2] Wick
normal ordering over the creation and annihilation operators, and Ω ∈ Φ is the ground state of
the Hamiltonian (1.2) at the temperature β → ∞, normalized by the condition (Ω,Ω) = 1. If
we introduce the Bogolubov generating functional
L(f) := (Ω, exp[iρ(f)]Ω) (1.4)
for any “test” Schwartz function f ∈ S(R3; R), where ρ(f) :=
∫
R3
d3xf(x)ρ(x), then for n-
particle distribution functions we can get the expression
Fn(x1, x2, . . . , xn) =:
1
i
δ
δf(x1)
1
i
δ
δf(x2)
. . .
1
i
δ
δf(xn)
: L(f)|f=0. (1.5)
Here xj ∈ R3, j = 1, n, n ∈ Z+, and the symbol :
1
i
δ
δf(x1)
1
i
δ
δf(x2)
. . .
1
i
δ
δf(xn)
: imitates the
application of the symbol : : to the operator expressions ρ(x1)ρ(x1) . . . ρ(xn), that is,
:
1
i
δ
δf(x1)
:=
1
i
δ
δf(x1)
,
(1.6)
:
1
i
δ
δf(x1)
1
i
δ
δf(x2)
:=
1
i
δ
δf(x1)
[
1
i
δ
δf(x2)
− δ(x1 − x2)
]
,
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE BOGOLUBOV GENERATING FUNCTIONAL METHOD IN STATISTICAL PHYSICS . . . 39
and so on. Consider now the expression (1.4) at some β ∈ R+, making use of the statistical
operator P: Φ → Φ and the “shifted” Hamiltonian H(µ) := H−µ
∫
R3
d3xρ(x) with µ ∈ R,
which give a suitable “chemical” potential,
L(f) := tr (P exp[iρ(f)]), P :=
exp(−βH(µ))
tr exp(−βH(µ))
, (1.7)
where “tr” means the operator trace-operation in the Fock space Φ. Posing within this work the
problem of studying distribution functions (1.3) in the classical statistical mechanics case, we
need to calculate the trace in (1.7) as ~ → 0. The latter gives rise to the following expressions:
L(f) =
Z(f)
Z(0)
, Z(f) := exp[−βV (δ)]L0(f),
(1.8)
L0(f) = exp
z ∫
R3
d3x{exp[if(x)]− 1}
,
where z := exp(βµ)(2π~2βm− 1)−3/2 is the system “activity” [1], and
V (δ) :=
1
2
∫
R3
d3x
∫
R3
d3yV (x− y) :
1
i
δ
δf(x)
1
i
δ
δf(y)
: . (1.9)
Based now on expressions (1.8) and (1.9) we can formulate the following proposition.
Proposition 1.1. The functional (1.4) satisfies [1, 3, 4] the following functional Bogolubov
type equation:
[∇x − i∇xf(x)]
1
i
δL(f)
δf(x)
= −β
∫
R3
d3y∇xV (x− y) :
1
i
δ
δf(x)
1
i
δ
δf(y)
: L(f), (1.10)
with the expression (1.8) being its exact functional-analytic solution.
Below we proceed to construct effective analytic tools allowing to find exact functional-
analytic solutions to the Bogolubov functional equation (1.10), describing equilibrium many-
particle dynamical systems, as well as, we will generalize the obtained results to the case of
nonequilibrium dynamical many particle systems.
2. The Bogolubov – Zubarev “collective” variables transform. Taking into account two-
particle potential energy partition V = V (s) + V (l), owing to representation (1.8), one can
easily write the following expression for the generating functional Z(f), f ∈ S(R3; R) :
Z(f) = exp[−βV (s)(δ)]L(l)(f), L(l)(f) := exp[−βV (l)(δ)]L0(f), (2.1)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
40 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
where we put
V (l)(δ) :=
1
2
∫
R3
d3x
∫
R3
d3yV (l)(x− y) :
1
i
δ
δf(x)
1
i
δ
δf(y)
: ,
(2.2)
V (s)(δ) :=
1
2
∫
R3
d3x
∫
R3
d3yV (s)(x− y) :
1
i
δ
δf(x)
1
i
δ
δf(y)
: .
To calculate the functional L(l)(f), f ∈ S(R3; R), corresponding to the long range part V (l)
of the full potential energy V : Φ → Φ, we will apply the analog of Bogolubov – Zubarev [5]
“collective” variables transform within the grand canonical ensemble, suggested before in [4,
6, 7]. Namely, denote by L(l)
(n)(f), n ∈ Z+, a partial solution to the functional equation (1.10),
possessing exactly n ∈ Z+ particles. Then, owing to the results of [3], for L(l)(f), n ∈ Z+, the
following exact expression holds:
L(l)
(n)(f) =
∫
R3
d3x1
∫
R3
d3x2 . . .
∫
R3
d3xn
n∏
j=1
exp[if(xj)] exp(−βV (l)
n ), (2.3)
where V (l)
n is the long term part potential energy of an n-particle group of the system. Then we
get that
L(l)(f) :=
∑
n∈Z+
zn
n!
L(l)
(n)(f)Q−1
0, Q0 :=
∑
n∈Z+
zn
n!
L(l)
(n)(0)
−1
. (2.4)
The sum in (2.3) can be calculated exactly, giving rise to the expression
L(l)
(n)(f) =
∫
D(ω)
z
∫
R3
d3x exp[if(x)]g(x;ω)
n
J(ω), (2.5)
where D(ω) :=
∏
k∈R3
i
2
(dωk ∧ dωk), ω∗k := ω−k ∈ C, k ∈ R3,
g(x;ω) := exp
−2πi
∫
R3
d3kωk exp(ikx) +
β
2
∫
R3
d3kν(k)
,
(2.6)
J(ω) := exp
−∫
R3
d3k
2π2
βν(k)
ωkω−k +
∫
R3
d3k ln
π
βν(k)
,
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE BOGOLUBOV GENERATING FUNCTIONAL METHOD IN STATISTICAL PHYSICS . . . 41
and ν(k) := (2π)−3
∫
R3
d3xV (l) exp(−ikx), k ∈ R3. Now from (2.4), (2.5) and (2.6) one easily
finds that
L(l)(f) =
∫
D(ω) exp
z̄ ∫
R3
d3x{exp[if(x)− 1]}g(x;ω)
J (l)(ω)Q−1, (2.7)
where z̄ := z exp
β
2
∫
R3
d3kν(k)
= z exp
[
β
2
V (l)(0)
]
and the function J (l)(ω), ω ∈ R3, allows
for the following series expansion:
J (l)(ω) := J(ω) exp
∫
R3
d3xg(x;ω)
= J(ω) exp
−(2π)2
2!
(2π)3
∫
R3
d3kωkω−k +
+
∑
n6=2
(−2πi)n
n!
(2π)3
∫
R3
d3k1
∫
R3
d3k2 . . .
∫
R3
d3kn
n∏
j=1
ωkj
δ
(
N∑
J=1
kj
) . (2.8)
The expression (2.7) can be represented now [6] in the following cluster Ursell form:
L(l)(f) = exp
∞∑
n=1
z̄n
n!
∫
R3
d3x1
∫
R3
d3x2 . . .
∫
R3
d3xn
n∏
j=1
{exp[if(x)− 1]}gn(x1, x2, . . . , xn)
.
(2.9)
Here, for any n ∈ Z+,
gn(x1, x2, . . . , xn) :=
∑
σ[n]
(−1)m+1(m− 1)!
m∏
j=1
Rσ[j](xk ∈ σ[j]),
(2.10)
Rn(x1, x2, . . . , xn) :=
∑
σ[n]
m∏
j=1
gσ[j](xk ∈ σ[j]),
where gn(x1, x2, . . . , xn), n ∈ Z+, are called the n-particle Ursell cluster functions,Rn(x1, x2, . . .
. . . , xn), n ∈ Z+, are suitable “correlation” functions [1, 4, 6] and σ[n] denotes a partition of
the set {1, 2, . . . , n} into nonintersecting subsets {σ[j] : j = 1,m}, that is, σ[j] ∩ σ[k] = ∅ for
j 6= k = 1,m, and σ[n] = ∪m
j=1σ[j]. Having separated from the function J (l)(ω), ω ∈ C3, the
natural “Gaussian” part J (l)
0 (ω), ω ∈ R3, one can write down that
g1(x1) =
G(ξ(1)
k )
G(0)
, g2(x1, x2) =
G(ξ(2)k )
G(0)
− g1(x1)g1(x2), . . . , (2.11)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
42 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
where ξ(n)
k := −2πi
∑n
s=1 exp(ikxs), k ∈ R3, n ∈ Z+,
G(ξ(n)
k ) := exp[M(ξ(n)
k )]
∫
D(ω)g(l)(ξ(n)
k ;ω)J0(ω),
M(ξ(n)
k ) :=
∑
m6=2
(−2πi)m
m!
(2π)3
∫
R3
d3k1
∫
R3
d3k2 . . .
∫
R3
d3kmδ
(
m∑
s=1
ks
)
m∏
s=1
δ
δξ
(n)
ks
, (2.12)
g(l)(ξ(n)
k ;ω) :=
n∏
j=1
g(xj ;ω).
Since the integrals
∫
D(ω)g(l)(ξ(n)
k ;ω)J (l)(ω), n ∈ Z+, are calculated exactly, the formulae (2.9)
and (2.10) are sources of the so-called “virial” variables for the Ursell – Mayer “cluster” correla-
tion functions gn(x1, x2, . . . xn), n ∈ Z+,which have important applications. In particular, from
the function J (l)(ω), ω ∈ C3, one gets right away that the cluster expansion for the functions
gn(x1, x2, . . . xn), n ∈ Z+, are fulfilled by means of the “screened” potential function V̄ (l)(x−
−y), x, y ∈ R3, where
V̄ (l)(x− y) :=
∫
R3
d3k
ν(k) exp[ik(x− y)]
1 + ν(k)βz̄(2π)3
. (2.13)
In particular, from (1.5) and (2.9) one easily finds that
F1(x1) = z
∫
D(ω)g(x;ω)J (l)(ω)
[∫
D(ω)J (l)(ω)
]−1
=
= ρ̄ ' z̄ exp
[
β
2
∫
d3k
βν2(k)(2π)3z̄
1 + ν(k)βz̄(2π)3
]
,
(2.14)
F2(x1, x2) = z2
∫
D(ω)g(x1;ω)g(x2;ω)J (l)(ω)
[∫
D(ω)J (l)(ω)
]−1
'
' ρ̄2 exp[−βV̄ (l)(x2 − x1)]
1 + ρ̄
∫
R3
d3x3
[
exp
(
−βV̄ (l)(x1 − x3)
)
− 1 +
+ βV̄ (l)(x1 − x3)
][
exp
(
−βV̄ (l)(x2 − x3)
)
− 1 + βV̄ (l)(x2 − x3)
]
+
+ ρ̄
∫
R3
d3x3[−βV̄ (l)(x1 − x3)][exp(−βV̄ (l)(x2 − x3))− 1 + βV̄ (l)(x2 − x3)]+
+ρ̄
∫
R3
d3x3[−βV̄ (l)(x2 − x3)][exp(−βV̄ (l)(x1 − x3))− 1 + βV (l)(x1 − x3)]+. . .
. . . ,
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE BOGOLUBOV GENERATING FUNCTIONAL METHOD IN STATISTICAL PHYSICS . . . 43
and so on. The result, presented above, can be obtained by means of a little formal calculations,
based on generalized functions and operator theories [4, 8]. Indeed, as ~ → 0, one has that
L(l)(f) = exp[−βV (l)]L0(f)Q−1 =
= tr
exp(−βH(µ)
0 ) exp
−β
2
∫
R3
d3kν(k) : ρkρ−k :
exp[i(ρ(f)]
=
= tr
exp(−βH(µ)
0 ) exp
β
2
∫
R3
d3kν(k)
∫
R3
d3xρ(x)
×
×
∫
D(ω) exp
−∫
R3
d3k
2π2
βν(k)
ωkω−k −
∫
R3
d3k2πiωkρk
exp[i(ρ, f)]
Q−1 =
=
∫
D(ω)J(ω)tr
exp(−βH(µ)
0 ) exp
i
ρ, f − 2π
∫
R3
d3kωk exp(ikx)−
− iβ
2
∫
R3
d3kν(k)
Q−1 =
=
∫
D(ω)J(ω)L0(f − 2π
∫
R3
d3kωk exp(ikx)− iβ
2
∫
R3
d3kν(k))Q−1 =
=
∫
D(ω)J (l)(ω) exp
∫
R3
d3k{exp[if(x)]− 1}g(x;ω)
, (2.15)
where H(µ)
0 := H0 − µ
∫
R3
d3xρ(x), ρk :=
∫
R3
d3xρ(x) exp(ikx), k ∈ R3. The expression
(2.15) coincides exactly with that of (2.9), thereby proving the validity of our expressions (1.8)
and (2.1) for the N. N. Bogolubov type generating functional L(f), f ∈ S( R3; R), satisfying the
functional equation (1.10) in Proposition 1.1.
3. The Ursell – Mayer type diagram expansion. Having considered expressions (2.1) and
(2.7) as starting ones, with known functions gn(x1, x2, . . . xn), n ∈ Z+, for the functional L(f),
f ∈ S(R3; R), one can obtain the following result:
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
44 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
L(f) =
Z(f)
Z(0)
, Z(f) = exp[−βV (s)(δ)]L(l)(f) =
= exp[−βV (s)(δ)] exp
∞∑
n=1
zn
n!
∫
R3
d3x1
∫
R3
d3x2 . . .
∫
R3
d3xn ×
×
n∏
j=1
{exp[if(xj)]− 1}gn(x1, x2, . . . , xn)
=
= exp
[ ∞∑
N=1
1
N !
W (G(c)
N )
]
, (3.1)
where the functionals W (G(c)
N ), N = 1,∞, are calculated via the following rule. Denote by
G
(c)
N , N = 1,∞, a connected graph and that consists of exactlyN generalized vertices of [γ(nj)]
type, j = 1, N, and
∑N
j=1 nj ordinary vertices of [α] type. Let it each vertex [y(n)] is necessari-
ly connected with n vertices of type [α] by means of dashed lines, but between themselves
[α] vertices can be connected arbitrarily by means of solid lines. If now we assign the factor
gn(x1, x2, . . . , xn) to each generalized [γ(n)]-vertex the factor z
∫
R3
d3x exp[if(x)] to each simple
[α]-vertex, and the factor {exp [−βV (s)(xl1 − xl2)] − 1} to the line connecting them, then the
obtained resulting expression will be exactly equal to the functional W (G(c)
N ). The final sum-
ming up over all such connected graphs gives the expression (2.15), where the factor 1/N !
counts for the symmetry order of the graphG(c)
N under permutations of the generalized vertices.
It is evident that, by representing the factor exp[if(x)] that enters the vertex [α] as {exp[i×
×f(x)]−1}+1, the expression (2.15) can be easily resumed into Ursell – Mayer type expressions
but already with suitably another functions gn, replacing the former ones and giving rise to
expansions similar to (2.14), based already on the “screened” potential (2.13), which we will
not discuss here in more details.
Thereby, taking into account the results of [4, 6] we can formulate the next proposition,
characterizing the Bogolubov type generating functional L(f), f ∈ S( R3; R), satisfying the
functional equation (1.10).
Proposition 3.1. Let the Bogolubov type generating functional L(f), f ∈ S( R3; R), represen-
ted analytically as a series (3.1) of graph-generated functionals, satisfy the following conditions:
i) continuity with respect to the natural topology on S( R3; R), |L(f)| ≤ 1, f ∈ S(R3; R);
ii) positivity,
∑n
j,k=1 cjc
∗
kL(fj − fk) ≥ 0 for any f ∈ S( R3; R) and all cj ∈ C, j = 1, n,
n ∈ Z+;
iii) symmetry and normalization conditions, L∗(f) = L(−f) for all f ∈ S( R3; R) and
L(0) = 1;
iv) translational-invariance, L(f) = L(fa), where fa(x) := f(x − a), x, a ∈ R3, for any
f ∈ S(R3; R);
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE BOGOLUBOV GENERATING FUNCTIONAL METHOD IN STATISTICAL PHYSICS . . . 45
v) cluster condition or, equivalently, the Bogolubov correlation decay, limλ→∞[L(f + gλa)−
−L(fa)L(gλa)] = 0, a ∈ R3, for any f, g ∈ S( R3; R);
vi) density condition,
1
i
δL(f)
δf(x)
∣∣∣
f=0
= ρ̄ ∈ R+.
Then the functional (3.1) solves the Bogolubov type functional equation (1.10), giving the
positive measure dµ̄ exp Fourier representation on the adjoint tempered generalized function
space S ′( R3; R) as
L(f) =
∫
S′(R3;R)
dµ̄(ξ) exp[i(ξ, f), (3.2)
where (ξ, f) :=
∫
R3
d3xξ(x)f(x) for ξ ∈ S ′(R3; R) and f ∈ S(R3; R).
The obtained result makes it possible to find the many-particle distribution functions (1.5)
and apply them for constructing different thermodynamic functions that are important [1, 9] in
applications.
Below, following the Bogolubov method [3], we obtain, based on the expression (2.3), the
important Kirkwood – Saltzbourg – Simansic functional equation for the Bogolubov generating
functional L(f), f ∈ S( R3; R). Namely, making use of the expression (2.3) we can write the
following relationship:
1
i
δL(N+1)(f)
δf(x)
= exp[if(x)]
(N + 1)ZN
ZN+1
L(N)(f(·) + iβV (· − x)) (3.3)
for any x ∈ R3. Since, by definition,
lim
N→∞
L(N)(f) = L(f), f ∈ S(R3; R), lim
N→∞
(N + 1)ZN
ZN+1
:= z ∈ R+,
from (3.3) one gets right away that
exp[−if(x)]
1
i
δL(f)
δf(x)
= zL(f(·) + iβV (· − x)), (3.4)
which is called the Kirkwood – Saltzbourg – Symansic functional equation, which is very impor-
tant for proving Proposition 3.1 by means of the classical Leray – Schauder fixed point theorem
[1, 2, 10] in some suitably defined Banach space. In particular, at f = 0 from (3.4) one finds the
following important relationship:
ρ̄ = zL(iβV (· − x)) (3.5)
for any x ∈ R3.
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
46 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
4. The quantized Wigner operator and the N. N. Bogolubov generating functional method
in nonequilibrium statistical mechanics. For studying nonequilibrium properties of a many-
particle classical statistical system it was proposed [4, 6] to use the quasiclassical quantized
Wigner density operator
w(x; p) :=
1
(2π)3
∫
R3
d3α exp (iαp)ψ+
(
x+
~α
2
)
ψ(x− ~α
2
), (4.1)
where the one-particle phase space variables satisfy (x; p) ∈ T ∗(R3). By means of simple
calculations one can see that the Hamilton operator H : Φ → Φ can be written as
H =
∫
T ∗(R3)
d3pd3x
p2
2m
w(x; p) +
∫
T ∗(R3)
d3pd3x
∫
T ∗(R3)
d3ξd3yV (x− y) : w(x; p)w(y; ξ) : , (4.2)
where the symbol : : as before, denotes the usual Wick ordering of creation and annihilati-
on operators on the Fock space Φ. With regard to the following applications, let us mention
formulae for Wigner density operators (4.1) in the Wick sence, ∫
R3
d3xψ+(x)∇2
xψ(x), w(z;ϑ)
~→0' ~
i
{
ϑ2
2m
,w(z;ϑ)
}
,
∫
R3
d3x
∫
R3
d3yV (x− y) : ρ(x)ρ(y) :, w(z;ϑ)
~→0' 2~
i
∫
R3
d3y {V (z − y), : ρ(y)w(z;ϑ) :} ,
(4.3)
w(x; p)w(y, ξ)
~→0' : w(x; p)w(y, ξ) : +w(x; p)δ(x− y)δ(p− ξ),
where the bracket [·, ·] means the usual commutator of operators in the Fock space Φ and
{·, ·} means the classical canonical Poisson bracket on the phase space T ∗(R3). Following the
Bogolubov ideas, we will define a Bogolubov generating functional L(f), f ∈ S(T (R3); R), as
L(f) := tr (P exp[i(w, f)], (4.4)
where, by definition, (w, f) :=
∫
T (R3)
d3xd3pw(x; p)f(x; p) and P : Φ → Φ is the statistical
operator satisfying the following [1, 3, 4, 7] evolution equation with respect to the time variable
t ∈ R+ :
∂P
∂t
=
i
~
[P,H], trP = 1, P|t=0 = P̄, (4.5)
where the initial operator P̄ : Φ → Φ is assumed to be given a priori.
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE BOGOLUBOV GENERATING FUNCTIONAL METHOD IN STATISTICAL PHYSICS . . . 47
Concerning the n-particle distribution functions Fn(x1, x2, . . . , xn; p1, p2, . . . , pn|t), n ∈ Z+,
the expressions
Fn(x1, x2, . . . , xn; p1, p2, . . . , pn|t) : =
= tr (P : w(x1; p1)w(x2; p2) . . . w(xn; pn) :) =
=:
1
i
δ
δf(x1; p1)
1
i
δ
δf(x2; p2)
. . .
1
i
δ
δf(xn; pn)
: L(f)|f=0,
(4.6)
hold as ~ → 0, where
:
1
i
δ
δf(x1; p1)
:=
1
i
δ
δf(x1; p1)
,
:
1
i
δ
δf(x1; p1)
1
i
δ
δf(x2; p2)
:=
1
i
δ
δf(x1; p1)
(
1
i
δ
δf(x2; p2)
− δ(x1 − x2)δ(p1 − p2)
)
, (4.7)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
and so on, owing to the last expression of (4.3).
For finding the distribution functions (4.6) we will derive, following N. N. Bogolubov [1, 3],
the corresponding evolution functional equation on the N. N. Bogolubov generating functional
(4.4). Making use of the relationship (4.4), one obtains easily that
∂L(f)
∂t
= tr
(
∂P
∂t
exp[i(w, f)]
)
= tr
(
P i
~
[H, exp[i(w, f)]]
)
=
= tr
P ∫
T (R3)
d3xd3p
{
p2
2m
,w(x; p) exp[i(w, f)]
}+
+
1
2
tr
P ∫
T (R3)
d3xd3p
∫
T (R3)
d3yd3ξ {V (x− y), : w(x; p)w(y; ξ) : exp[i(w, f)]}
.
(4.8)
Now, based on relationships (4.3), we finally obtain the following Bogolubov type evolution
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
48 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
functional equation:
∂L(f)
∂t
=
∫
T (R3)
d3xd3p
{
T (p),
1
i
δL(f)
δf(x; p
}
+
+
1
2
∫
T (R3)
d3xd3p
∫
T (R3)
d3yd3ξ
{
V (x− y), :
1
i
δ
δf(x; p)
1
i
δ
δf(y; ξ)
: L(f
}
, (4.9)
where, by definition, T (p) :=
p2
2m
, p ∈ R3, is the kinetic free particle energy.
Having analyzed the Bogolubov generating functional (4.4) within the quasiclassical Wigner
density operator representation (4.1), one can obtain an exact functional-operator solution to
the evolution Bogolubov functional equation (4.9):
L(f) =
Z(f)
Z(0)
, Z(f) = exp[Φ(δ)]L0(f) (4.10)
for f ∈ S(T (R3); R). Here we denoted
Φ(δ) =
∑
n∈Z+
1
n!
∫
T (R3)
d3x1d
3p1
∫
T (R3)
d3x2d
3p2 . . .×
×
∫
T (R3)
d3xnd
3pnΦn(x1, x2, . . . , xn; p1, p2, . . . , pn|t)×
× :
1
i
δ
δf(x1; p1)
1
i
δ
δf(x2; p2)
. . .
1
i
δ
δf(xn; pn)
: , (4.11)
L0(f) =
∑
n∈Z+
1
n!
∫
T (R3)
d3x1d
3p1
∫
T (R3)
d3x2d
3p2 . . .
∫
T (R3)
d3xnd
3pn×
× F̄n
(
x1 −
p1
m
t, x2 −
p2
m
t, . . . , xn −
pn
m
t; p1, p2, . . . , pn
) n∏
j=1
{exp[if(xj ; pj)]− 1},
where F̄n(x1, x2, . . . , xn; p1, p2, . . . , pn), n ∈ Z+, are given n-particle distribution functions at
t = 0, that is, owing to the definition (4.6),
F̄n(x1, x2, . . . , xn; p1, p2, . . . , pn) : = tr (P̄ : w(x1; p1)w(x2; p2) . . . w(xn; pn) :) =
=:
1
i
δ
δf(x1; p1)
1
i
δ
δf(x2; p2)
. . .
1
i
δ
δf(xn; pn)
: L(f)|t=0,f =0,
(4.12)
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
THE BOGOLUBOV GENERATING FUNCTIONAL METHOD IN STATISTICAL PHYSICS . . . 49
and Φn(x1, x2, . . . , xn; p1, p2, . . . , pn|t), n ∈ Z+, are so called cluster potential functions determi-
ned recursively by means of the following functional-operator relationships:
log(P−1
0 P) : =
∑
n∈Z+
1
n!
∫
T (R3)
d3x1d
3p1
∫
T (R3)
d3x2d
3p2 . . .
∫
T (R3)
d3xnd
3pn×
× Φn(x1, x2, . . . , xn; p1, p2, . . . , pn|t) : w(x1; p1)w(x2; p2) . . . w(xn; pn) : (4.13)
with
P0 = exp
(
− it
~
H0
)
P̄ exp
(
it
~
H0
)
(4.14)
being the statistical operator of a noninteracting particle system.
If the initial distribution at t = 0 is “chaotic”, that is, for all n ∈ Z+ the relationships
F̄n(x1, x2, . . . , xn; p1, p2, . . . , pn) =
n∏
j=1
F̄1(xj ; pj) (4.15)
hold, one gets easily from (4.11) and (4.15) that
L0(f) = exp
∫
T (R3)
d3xd3pF̄1
(
x− p
m
t; p
)
{exp[if(x; p)]− 1}
. (4.16)
If the “chaotic” condition is not fulfilled, we can proceed to the usual cluster Ursell – Mayer
type representation [4, 6] for the Bogolubov generating functional (4.10),
L0(f) = exp
∑
n∈Z+
1
n!
∫
T (R3)
d3x1d
3p1
∫
T (R3)
d3x2d
3p2 . . .
∫
T (R3)
d3xnd
3pn ×
×ḡn
(
x1 −
p1
m
t, x2 −
p2
m
t, . . . , xn −
pn
m
t; p1, p2, . . . , pn
) n∏
j=1
{exp[if(xj ; pj)]− 1}
,
(4.17)
where the “cluster” distribution functions ḡn(x1, x2, . . . , xn; p1, p2, . . . , pn), n ∈ Z+, have the
form
ḡn(x1, x2, . . . , xn; p1, p2, . . . , pn) :=
∑
σ[n]
(−1)m+1(m− 1)!
m∏
j=1
F̄σ[j]((xk; pk) ∈ σ[j]),
F̄n(x1, x2, . . . , xn; p1, p2, . . . , pn) :=
∑
σ[n]
m∏
j=1
ḡσ[j]((xk; pk) ∈ σ[j]),
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
50 N. N. BOGOLUBOV (JR.), A. K. PRYKARPATSKY
and σ[n] denotes a partition of the set {1, 2, . . . , n} into nonintersecting subsets {σ[j] : j =
= 1,m}, that is, σ[j] ∩ σ[k] = ∅ for j 6= k = 1,m, and σ[n] = ∪m
j=1σ[j]. In particular,
ḡ1(x1; p1) = F̄1(x1; p1),
ḡ2(x1, x2; p1,p2) = F̄2(x1, x2; p1, p2)− F̄1(x1; p1)F̄1(x2; p2), . . . ,
and so on. The N. N. Bogolubov generating functional (4.10), owing to (4.11) and (4.17), allows
a natural infinite series expansion whose coefficients can be represented as above by means of
the usual Ursell – Mayer type diagram expressions, which can be effectively used for studying
kinetic properties of our many-particle statistical system.
5. Conclusions. In the article we showed that the N. N. Bogolubov generating functional
method is a very effective tool for studying distribution functions of both equilibrium and
nonequilibrium states of classical many-particle dynamical systems. In some cases the
N. N. Bogolubov generating functionals can be represented by means of infinite Ursell – Mayer
diagram expansions that converge under some additional constraints on the statistical system
under consideration. We show for the first time that the Bogolubov idea [1] to use the Wigner
density operator transformation for studying the nonequilibrium distribution functions proved
to be very effective, having proposed a new analytic form of a nonstationary solution to the
classical N. N. Bogoliubov evolution functional equation.
1. Bogolubov N. N., Bogolubov N .N. (jr.) Introduction to quantum statistical mechanics. — New York: World
Sci., 1986.
2. Bogolubov N. N., Shirkov D. V. Introduction to the theory of quantizerd fields. — New York: Intersci., 1959.
3. Bogolubov N. N. Dynamical problems of statistical physics (in Russian). — Moskow: Gostekhizdat, 1946.
4. Bogolubov N. N. (jr.), Prykarpatsky A. K. The N. N. Bogolubov generating functional method in statistical
mechanics and a collective variables transform analog // Theor. and Math. Phys. — 1986. — 66, № 3. —
P. 463 – 480.
5. Bohm D. The general collective variables theory (in Russian). — M.: Mir, 1964.
6. Bogolubov N. N. (jr.), Prykarpatsky A. K. The Wigner quantized operator and N. N. Bogolubov generating
functional method in non-equilibrium statistical physics // Dokl. AN SSSR. — 1985. — 285, № 6. — P. 1365 –
1370.
7. Prykarpatsky A. K. The N. N. Bogolubov generating functional method in statistical mechanics and a collecti-
ve variables transform analog within the grand canonical ensemble // Ibid. — № 5. — P. 1096 – 1101.
8. Vladimirov V. S. Generalized functions in mathematical physics (in Russian). — Moscow: Nauka, 1979.
9. Prykarpatsky A. K., Taneri U., Bogolubov N. N. (jr.) Quantum field theory with application to quantum
nonlinear optics.— New York: World Sci., 2002.
10. Granas A., Dugunji J. Fixed point theory. — New York: Springer, 2003.
Received 19.09.2006
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 1
|
| id | nasplib_isofts_kiev_ua-123456789-7241 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-11-25T20:53:14Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bogolubov (jr.), N.N. Prykarpatsky, A.K. 2010-03-26T10:01:20Z 2010-03-26T10:01:20Z 2007 The Bogolubov generating functional method in statistical physics and "collective" variables transform within the grand canonical ensemble / N.N. Bogolubov (jr.), A.K. Prykarpatsky // Нелінійні коливання. — 2007. — Т. 10, № 1. — С. 37-50. — Бібліогр.: 10 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/7241 517.9 We show that the Bogolubov generating functional method is a very effective tool for studying distribution functions of both equilibrium and nonequilibrium states of classical many-particle dynamical systems. In some cases the Bogolubov generating functionals can be represented by means of infinite Ursell –Mayer diagram expansions, whose convergence holds under some additional constraints on the statistical system under consideration. The classical Bogolubov idea to use the Wigner density operator transformation for studying the nonequilibrium distribution functions is developed, a new analytic nonstationary solution to the classical Bogolubov evolution functional equation is constructed. Доведено, що метод породжуючих функціоналів Боголюбова є досить ефективним для вивчення функцій розподілу рівноважних та нерівноважних станів класичних багаточастинкових динамічних систем. У деяких випадках породжуючі функціонали Боголюбова можна виразити через нескінченні розвинення діаграм Урселла - Мартіна, які збігаються при накладанні додаткових умов на розглядувані статистичні системи. Розвинуто класичну ідею Боголюбова про використання перетворення Вігнера оператора щільності для вивчення нерівноважних функцій розподілу та побудовано новий нестаціонарний розв'язок класичного рівняння еволюції функціонала Боголюбова. en Інститут математики НАН України The Bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble Метод породжуючих функціоналів Боголюбова в статистичній фізиці та перетворення до "колективних" змінних у великому канонічному ансамблі Article published earlier |
| spellingShingle | The Bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble Bogolubov (jr.), N.N. Prykarpatsky, A.K. |
| title | The Bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble |
| title_alt | Метод породжуючих функціоналів Боголюбова в статистичній фізиці та перетворення до "колективних" змінних у великому канонічному ансамблі |
| title_full | The Bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble |
| title_fullStr | The Bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble |
| title_full_unstemmed | The Bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble |
| title_short | The Bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble |
| title_sort | bogolubov generating functional method in statistical physics and “collective” variables transform within the grand canonical ensemble |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/7241 |
| work_keys_str_mv | AT bogolubovjrnn thebogolubovgeneratingfunctionalmethodinstatisticalphysicsandcollectivevariablestransformwithinthegrandcanonicalensemble AT prykarpatskyak thebogolubovgeneratingfunctionalmethodinstatisticalphysicsandcollectivevariablestransformwithinthegrandcanonicalensemble AT bogolubovjrnn metodporodžuûčihfunkcíonalívbogolûbovavstatističníifízicítaperetvorennâdokolektivnihzmínnihuvelikomukanoníčnomuansamblí AT prykarpatskyak metodporodžuûčihfunkcíonalívbogolûbovavstatističníifízicítaperetvorennâdokolektivnihzmínnihuvelikomukanoníčnomuansamblí AT bogolubovjrnn bogolubovgeneratingfunctionalmethodinstatisticalphysicsandcollectivevariablestransformwithinthegrandcanonicalensemble AT prykarpatskyak bogolubovgeneratingfunctionalmethodinstatisticalphysicsandcollectivevariablestransformwithinthegrandcanonicalensemble |