On evolution equations for marginal correlation operators
This paper is devoted to the problem of the description of nonequilibrium correlations of quantum many-particle systems. A non-perturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed as an expansion over particle clusters...
Збережено в:
| Дата: | 2011 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут кібернетики ім. В.М. Глушкова НАН України
2011
|
| Назва видання: | Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/48794 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On evolution equations for marginal correlation operators / V.I. Gerasimenko, D.O. Polishchuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2011. — Вип. 5. — С. 44-60. — Бібліогр.: 17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-48794 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-487942013-09-04T03:02:29Z On evolution equations for marginal correlation operators Gerasimenko, V.I. Polishchuk, D.O. This paper is devoted to the problem of the description of nonequilibrium correlations of quantum many-particle systems. A non-perturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed as an expansion over particle clusters which evolution is governed by the corresponding-order cumulant of the nonlinear groups of operators generated by the von Neumann hierarchy. Робота присвячена проблемі опису нерівноважних кореляцій квантових багаточастинкових систем. Побудовано розв'язок задачі Коші нелінійної квантової ієрархії рівнянь ББГКІ у формі розкладу по групах частинок, еволюція яких описується відповідного порядку кумулянтом груп нелінійних операторів ієрархії рівнянь фон Неймана. 2011 Article On evolution equations for marginal correlation operators / V.I. Gerasimenko, D.O. Polishchuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2011. — Вип. 5. — С. 44-60. — Бібліогр.: 17 назв. — англ. XXXX-0059 http://dspace.nbuv.gov.ua/handle/123456789/48794 517.9+531.19+530.145 en Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки Інститут кібернетики ім. В.М. Глушкова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
This paper is devoted to the problem of the description of nonequilibrium correlations of quantum many-particle systems. A non-perturbative solution of the Cauchy problem of the nonlinear quantum BBGKY hierarchy for marginal correlation operators is constructed as an expansion over particle clusters which evolution is governed by the corresponding-order cumulant of the nonlinear groups of operators generated by the von Neumann hierarchy. |
| format |
Article |
| author |
Gerasimenko, V.I. Polishchuk, D.O. |
| spellingShingle |
Gerasimenko, V.I. Polishchuk, D.O. On evolution equations for marginal correlation operators Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| author_facet |
Gerasimenko, V.I. Polishchuk, D.O. |
| author_sort |
Gerasimenko, V.I. |
| title |
On evolution equations for marginal correlation operators |
| title_short |
On evolution equations for marginal correlation operators |
| title_full |
On evolution equations for marginal correlation operators |
| title_fullStr |
On evolution equations for marginal correlation operators |
| title_full_unstemmed |
On evolution equations for marginal correlation operators |
| title_sort |
on evolution equations for marginal correlation operators |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/48794 |
| citation_txt |
On evolution equations for marginal correlation operators / V.I. Gerasimenko, D.O. Polishchuk // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2011. — Вип. 5. — С. 44-60. — Бібліогр.: 17 назв. — англ. |
| series |
Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| work_keys_str_mv |
AT gerasimenkovi onevolutionequationsformarginalcorrelationoperators AT polishchukdo onevolutionequationsformarginalcorrelationoperators |
| first_indexed |
2025-07-04T09:31:45Z |
| last_indexed |
2025-07-04T09:31:45Z |
| _version_ |
1836708276896530432 |
| fulltext |
Математичне та комп’ютерне моделювання
44
5. Яковлев С. В. Биологическая очистка производственных сточных вод: Про-
цессы, аппараты и сооружения / С. В. Яковлев, И. В. Скирдов, В. Н. Швецов
и др. ; под ред. С. В. Яковлева. — М. : Стройиздат, 1985. — 208 с.
6. Огняник Н. С. Основы изучения загрязнения геологической среды легки-
ми нефтепродуктами / Н. С. Огняник, Н. К. Парамонов, А. Л. Брикс,
И. С. Пашковский, Д. В. Коннов. — К. : А.П.Н., 2006. — 278 с.
7. Бомба А. Я. Моделювання процесів очищення стічної води на каркасно-
засипних фільтрах з урахуванням зворотного впливу / А. Я. Бомба,
І. М. Присяжнюк, А. П. Сафоник // Фізико-математичне моделювання та
інформаційні технології. — 2007. — Вип. 6. — C. 101—108.
8. Бомба А. Я. Нелінійні сингулярно-збурені задачі типу "конвекція — ди-
фузія" / А. Я. Бомба, С. В. Барановський, І. М. Присяжнюк. — Рівне :
НУВГП, 2008. — 252 с.
The mathematical model of process of aerobic sewage treatment in the
porous environment which considers interaction of bacteria, organic and
biologically not oxidising substances is constructed. The offered algorithm
of the decision corresponding modelling nonlinear singular the indignant
problem of type «convection-diffusion-mass exchange» with time delay
Key words: nonlinear problems, process of aerobic clearing, filtering,
singular indignations, asymptotic, time delay.
Отримано: 06.03.2011
УДК 517.9+531.19+530.145
V. I. Gerasimenko*, Doctor of phys.-math. sci.,
D. O. Polishchuk**, PhD student
*Institute of Mathematics of NAS of Ukraine, Kyiv,
**Taras Shevchenko National University of Kyiv
ON EVOLUTION EQUATIONSFOR MARGINAL
CORRELATION OPERATORS
This paper is devoted to the problem of the description of
nonequilibrium correlations of quantum many-particle systems. A
non-perturbative solution of the Cauchy problem of the nonlinear
quantum BBGKY hierarchy for marginal correlation operators is
constructed as an expansion over particle clusters which evolution
is governed by the corresponding-order cumulant of the nonlinear
groups of operators generated by the von Neumann hierarchy.
Key words: nonlinear quantum BBGKY hierarchy, von Neumann
hierarchy, correlation operator, quantum many-particle system.
Introduction. The importance of the mathematical description of
correlations in numerous problems of the modern statistical mechanics is
well-known. Among them in particular, we refer to such fundamental
© V. I. Gerasimenko, D. O. Polishchuk, 2011
Серія: Фізико-математичні науки. Випуск 5
45
problem as the problem of a description of collective behavior of interacting
particles by quantum kinetic equations [1—8]. Owing to the intrinsic
complexity and richness of these problems, primarily it is necessary to
develop an adequate mathematical theory of underlying evolution
equations.The goal of the paper is to derive rigorously the evolution
equations for marginal correlation operators that give an equivalent
approach to the description of the evolution of states in comparison with
marginal density operators governed by the quantum BBGKY hierarchy and
to construct a solution of the corresponding Cauchy problem.
The von Neumann hierarchy. We consider a quantum system of a
non-fixed, i.e. arbitrary but finite, number of identical (spinless) particles
with unit mass 1m in the space , 1 , that obey the Maxwell-
Boltzmann statistics. Let 0= nn
be the Fock space over the
Hilbert space , where the n -particle Hilbert space n
n
is a
tensor product of n Hilbert spaces and we adopt the usual convention
that 0 = . The Hamiltonian nH of the n -particle system is a self-
adjoint operator with the domain n nH
1 2
1 2
1 1
, ,
= =
=
n n
n
i i i
H K i i i
(1)
where K i is the operator of a kinetic energy of the i particle and
1 2,i i is the operator of a two-body interaction potential. In particular
on functions n that belong to the subspace
2 2
0
n n
nL H L of infinitely differentiable symmetric
functions with compact supports the operator K i acts according to the
formula:
2
2
=n q ni
K i
, where 2 is a Planck constant, and for
the operator we have: 1 21 2, ,=n i i ni i q q , respectively. We
assume that the function 1 2
,i iq q is symmetric with respect to
permutations of arguments and it is translation-invariant bounded function.
States of a system of the Maxwell-Boltzmann particles belong to the
space 1 1
0= = nn
L L of sequences 0 1, , , ,= nf f f f of
trace-class operators 11, ,n n nf f n L and 0f , that satisfy
Математичне та комп’ютерне моделювання
46
the symmetry condition: 11, , = , ,n n nf n f i i for arbitrary
1, , 1, ,ni i n , equipped with the norm
1 1 1, ,
0 0
1, , ,
= =
= Trn n n
nn n
f f f n
L L
where 1, ,Tr n are partial traces over 1, ,n particles [12]. We denote by
1
0 L the everywhere dense set in 1
L of finite sequences of
degenerate operators with infinitely differentiable kernels with compact
supports.
We describe states of a system by means of sequences
0 1= ( , ,1 , , ,1, ,ng t g g t g t n , 1) L of the correlation
operators , 1ng t n . The evolution of all possible states is determined
by the initial-value problem of the von Neumann hierarchy
, ,s
d
g t Y Y g t
dt
(2)
0
, 0, , 1,
=s st
g t Y g Y s (3)
where the following notations are used:
1 2 1 1 2 2 1 2: int 1 2 1 2
,
, , , ,P | | | |
= s s
Y X X i X i X X X
Y g t Y g t Y
i i g t X g t X
(4)
1 2:P =Y X X is the sum over all possible partitions P of the set
1, ,Y s into two nonempty mutually disjoint subsets 1X Y and
2X Y , the operator s defined on 1
0 sL by the formula
,s s s s s
i
Y H f f H
(5)
is the generator of the von Neumann equation [13] and the operator
int is defined by
int 1 2 1 2 1 2, , , .s s s
i
i i f i i f f i i
(6)
Hereafter we use the following notations: 1 |, , |PX X is a set,
elements of which are ||P mutually disjoint subsets 1, ,iX Y s of
the partition |
1: |
=P P
iiY X , i.e. 1 |, , |PX X P . In view of these
notations we state that Y is the set consisting of one element
Серія: Фізико-математичні науки. Випуск 5
47
1, ,=Y s of the partition = 1P P and = 1Y . We introduce the
declasterization mapping 1 |: , , |PX X Y , by the following
formula: 1 |: , , |PX X Y . For example, let 1, ,X s n ,
then for the set , \Y X Y it holds: , \ =Y X Y X .
On the space 1
nL we also introduce the mapping:
n nt t f , which is generated by the solution of the von
Neumann equation of n particles [13]
= .
i i
tH tHn n
n n nt f e f e
(7)
This mapping is an isometric strongly continuous group that
preserves positivity and self-adjointness of operators [12]. On
1
0 n n L the infinitesimal generator of group (7) is
determined by operator (5).
A solution of the Cauchy problem (2)—(3) is given by the following
expansion [9—10]
1
:
, ; 0
, , , 0, , 1,P P
P = P
=
i
i i
i
s
iX
Y X X
g t Y t Y g
t X X g X s
A
(8)
where :P = i iY X is the sum over all possible partitions P of the set
1, ,Y s into ||P nonempty mutually disjoint subsets iX Y , the
evolution operator ||P tA is the ||P th -order cumulant of groups of
operators (7) defined by the formula
1
1
: , , =|
, , , 1
1 ! ,
'|P |
1P P
'P |
'
'P
=
P
k
k
k
k
X X ZP
kZ
Z
t X X
t Z
A
(9)
Here
1 , , =|
'P : | k kP
X X Z
is the sum over all possible partitions
'P of the set 1 |, , |PX X into '|P | nonempty mutually disjoint
subsets |, ,1 |k PZ X X . For operators (8) the estimate holds
Математичне та комп’ютерне моделювання
48
1
3! ,cs s
s
s
g t s e
L
(10)
where 11
11
|
: |
max 0 , , 0
|
| |
= | | |
c
i P
i
P
X XP Y X X
X
g g
L
L
.
If 1 1
00 , 1n n ng n L L , expansion (8) is a strong
(classical) solution of the Cauchy problem (2)—(3) and for arbitrary initial
data 10 , 1n ng n L , it is a weak (generalized) solution.
In case of the absence of correlations between particles at initial time,
i.e. the initial data satisfying the a chaos condition, the sequence of
correlation operators has the form
10 = 0, 0,1 ,0, .g g (11)
The corresponding solution of the initial-value problem of the von
Neumann hierarchy is given by the expansion
1
1
, = , 0, ,
=
s
s s
i
g t Y t Y g iA (12)
where s tA is the sth -order cumulant defined by
| 1
:
, = 1 | | 1 ! , .
|
| |
=
i
i
i
P
s X i
P Y X X Pi
t Y P t X
A (13)
We note that correlations created in evolutionary process of a system
are described by formula (12) and determined by the corresponding-order
cumulant of the groups of operators (7) of the von Neumann equations.
Marginal correlation operators and marginal density operators.
We introduce the marginal correlation operators by the series
1, ,
0
,1, , ,1, , , 1,
!=
1
= rs s s n s n
n
G t s T g t s n s
n
(14)
where the sequence ,1, , , 0s ng t s n n , is a solution of the Cauchy
problem of the von Neumann hierarchy (2).
Traditionally marginal correlation operators are introduced by means
of the cluster expansions of the marginal density operators , 1sF t s ,
governed by the quantum BBGKY hierarchy [11]
:
, = , , 1,| |
=
i
i i
i
s X i
P Y X X P
F t Y G t X s
(15)
Серія: Фізико-математичні науки. Випуск 5
49
where : = i iP Y X is the sum over all possible partitions P of the set
1, ,Y s into ||P nonempty mutually disjoint subsets iX Y .
Hereupon solutions of cluster expansions (15)
| 1
:
, = 1 | | 1 ! , ,
|
| |
=
i
i i
i
P
s X i
P Y X X P
G t Y P F t X
(16)
are interpreted as the operators that describe correlations of many-particle
systems. Thus, marginal correlation operators (16) are cumulants (semi-
invariants) of the marginal density operators.
The marginal ( s -particle) density operators (15) are determined by
the Cauchy problem of the quantum BBGKY hierarchy [11]
1 int 1, , , 1 ,= Trs s s s s
i Y
d
F t Y Y F t Y i s F t
dt
(17)
0
0 , 1.
=
=s st
F t F s (18)
If 1 1
00 = =
n
nnF L L and e , then for t a
unique solution of the Cauchy problem (17)—(18) of the quantum
BBGKY hierarchy exists and is given by the expansion [9; 15]
1, , 1
0
1
, = ,{ }, \ 0, , 1,
!=
rs s s n n s n
n
F t Y T t Y X Y F X s
n
A (19)
where the 1 n th -order cumulant 1 n tA of groups of operators (7) is
defined by
1
1
: , \
|
,{ }, \ = 1 1 !
, ,
P
=
|
P
P
i i
i
i
n
P Y X Y X
iX
X
t Y X Y
t X
A
(20)
where P is the sum over all possible partitions P of the set , \Y X Y
into ||P nonempty mutually disjoint subsets , \iX Y X Y .
Formally, the evolution equations for marginal correlation operators are
derived from the quantum BBGKY hierarchy for marginal density operators
(15) on basis of expression (16). Then the evolution of all possible states of
quantum many-particle systems obeying the Maxwell-Boltzmann statistics
with the Hamiltonian (1) can be described within the framework of marginal
correlation operators governed by the nonlinear quantum BBGKY hierarchy
1 int, , 1= rs s
i Y
d
G t Y Y G t T i s
dt
Математичне та комп’ютерне моделювання
50
1 2
1 2
1 2
1 1 2
: , 1 = ,
; 1
( , , 1 , , ),| | | |s X X
P Y s X X
i X s X
G t Y s G t X G t X
(21)
0
, 0, , 1.
=
=s st
G t Y G Y s (22)
In equation (21) the operator |Y G t is generator of the von
Neumann hierarchy (2) defined by formula (4), i.e.
1 2
1 2 1 1 2 2
int 1 2 1 2
:
,
, , ,| | | |
=
s s
X X
P Y X X i X i X
Y G t Y G t Y
i i G t X G t X
(23)
where the operators s and int are defined by (5) and (6)
respectively,
1 2: =P Y X X
is the sum over all possible partitions P of the
set 1, ,Y s into two nonempty mutually disjoint subsets 1X Y and
2X Y , and
1 2
1 2
:( , 1)= ,
; 1
P Y s X X
i X s X
is the sum over all possible partitions of
the set , 1Y s into two mutually disjoint subsets 1X and 2X such that
ith particle belongs to the subset 1X and 1s th particle belongs to
2X . As far as we know hierarchy (21) was introduced by
M. M. Bogolyubov [11] and in the papers of J. Yvon [16] and
M. S. Green [17] for systems of classical particles.
Another method of the justification of evolution equations for marginal
correlation operators consists in their derivation from the von Neumann
hierarchy for correlation operators (2) on basis of definition (14).
We emphasize that the evolution of marginal correlation operators of
both finitely and infinitely many quantum particles is described by initial-
value problem of the nonlinear quantum BBGKY hierarchy (21). For
finitely many particles the nonlinear quantum BBGKY hierarchy is
equivalent to the von Neumann hierarchy (2).
A non-perturbative solution of the nonlinear quantum BBGKY
hierarchy. To construct a non-perturbative solution of the Cauchy
problem (21)—(22) of the nonlinear quantum BBGKY hierarchy we first
consider its structure for physically motivated example of initial data,
namely, initial data satisfying a chaos property
1 ,10
, 0,1 , 1,
=s st
G t Y G s (24)
where ,1s is a Kronecker symbol. Chaos property (24) means the absence
of state correlations in a system at the initial time.
Серія: Фізико-математичні науки. Випуск 5
51
According to definition (14) and solution expansion (12), in the case
under consideration the following relation between the marginal
correlation operators and correlation operators is true
1 10, = 0, .G i g i (25)
Taking into account the form (12) of a solution of the initial-value
problem of the von Neumann hierarchy (2) in case of initial data (11), for
expansion (14) we obtain
1, , 1
0 1
1
, = ,1, , 0, ,
!= =
r
s n
s s s n s n
n i
G t Y T t s n g i
n
A (26)
where s n tA is the s n th -order cumulant (13). In consequence of
relation (25) we finally derive
1, , 1
0 1
1
, , 0, .
!= =
= r
s n
s s s n s n
n i
G t Y T t X G i
n
A (27)
If 1
1
1 ( )
0 2G e
L
, series (27) converges, since for cumulants
(13) the estimate holds [9]
1 1! .n
n
n n
t f n e f
L L
A
From the structure of series (27) it is clear that in case of absence of
correlations at initial instant in a system the correlations generated by the
dynamics of quantum many-particle systems are completely governed by
cumulants (9) of groups of operators (7).
Thus, the cumulant structure of solution (8) of the von Neumann
hierarchy (2) induces the cumulant structure of solution expansion (27) of
the initial-value problem of the quantum nonlinear BBGKY hierarchy for
marginal correlation operators.
The evolution equations which satisfy expression (27) are derived
similarly to the derivation of hierarchy (21).
We point out that in case of chaos initial data solution expansion (19) of
the quantum BBGKY hierarchy (17) for marginal density operators differs
from solution expansion (27) of the nonlinear quantum BBGKY hierarchy
(21) for marginal correlation operators only by the order of the cumulants of
the groups of operators of the von Neumann equations [10; 15]
1, , 1 1
0 1
1
, = ,{ }, \ 0, ,
!= =
r
s n
s s s n n
n i
F t Y T t Y X Y F i
n
A (28)
where 1 n tA is the 1 n th -order cumulant [20].
The structure of a solution expansion. The direct method of the
construction of a solution of the nonlinear quantum BBGKY hierarchy (21) in
the form of non-perturbative expansion consists in its derivation on the basis
Математичне та комп’ютерне моделювання
52
of expansions (16) from non-perturbative solution (19) of initial-value
problem of the quantum BBGKY hierarchy (17)—(18). Following stated
above approach, we derive a formula for a solution of the quantum nonlinear
BBGKY hierarchy for marginal correlation operators in case of general initial
data on the basis of definition (14) and non-perturbative solution (8) of initial-
value problem of the von Neumann hierarchy (2)—(3). With this aim on
1
n nf L we introduce an analogue of the annihilation operator
1 11, , 1, , , 1 , 1,=Trs ss
f s f s s s a (29)
and, therefore we have
1, ,
0
1
1, , = 1, , .
!=
Tr
n
s s n s n
s n
e f s f s n
n
a
According to definition (14) of the marginal correlation operators, i.e.
= ,G t e g ta
where the sequence g t is a solution of the von Neumann hierarchy for
correlation operators defined by group (8), i.e.
( ) = 0 ,|g t t g
and to the equality: 0 = 0g e Ga , we finally derive
= 0 .|G t e t e Ga a (30)
To set down formula (29) in componentwise form we observe, that
the following equality holds
| 2
1 | 1
1
1
0
| 2
1, ,
0 01 1 | 1 | 2 | 1
1 1
1
1
(0) =
!
!!
! ! ! !
0, , 1, ,
0, , 1, , .
|
|
1
P P
| | =
|
= = | | |
| |
Tr
i
P
P
k
i
Xi kX P
kk
P
s n s n k
k k P P P
X k k
P PX k
e G X
k
kk
k k k k k k
G X s n s n k k
G X s n k k s n k
a
(31)
Then according to formulas (29) and (8), for 1,s we have
1, ,
0 : (1, , =
| 1 |
1
, =
!
, , , 0 ,
= )
| |
| |
r
i
i
i
i
s s s n
n P s n X
P P i
XX P
G t Y T
n
t X X e G X
aA
Серія: Фізико-математичні науки. Випуск 5
53
where ||P tA is ||P th -order cumulant (9), and as a result for sequence
(29) we obtain
1, ,
0 0
1 !
,1, , = 1
! ! != =
Tr
n
k
s s s n
n k
n
G t s
n k n k
| 1 |
: (1, , =
, , ,| |
) i
i
P P
P s n k X
t X X
A
| 2
1 | 1
| 2
0 01 1 | 1 | 2 | 1
!!
! ! ! !
|
|
|
= = | | |
P
P
kk
P
k k P P P
kk
k k k k k k
(32)
1 1
1
| 1 1
1
0, , 1, ,
0, , 1, , .
|
PP
X k k
P PX k
G X s n k s n k
G X s n k s n
Consequently the solution expansion of the nonlinear quantum
BBGKY hierarchy has the following structure
1, , 1
0
1
, ; , 1, , 0 ,
!=
= Trs s s n n
n
G t Y U t Y s s n G
n
(33)
where we introduce the notion of the 1n th -order reduced cumulant
1 nU t of nonlinear groups of operators (8)
| 2
11 | 1
1 1
1
1
0 : ( 1, , , 1, , =
| 1 |
| 2
0 01 | 1 | 2 | 1
1 1
1
; , 1, , 0
!
1
! !
, , ,
!!
! ! !
0, , 1, ,
0, ,
|
|
= )
| |
|
= = | | |
| |
=
!
i
i
P
P
P P
n
n
k
k P s s s n k X
P P
kk
P
k k P P P
X k k
P PX k
U t Y s s n G
n
k n k
t X X
kk
k k k k k k
G X s n k s n k
G X s n k
A
1, , s n
(34)
We give simplest examples of reduced nonlinear cumulants (33):
1
| |
: (1, , =
; 0 = ; 0 =
, , , 0, ,| 1 | | |
) P
| |
i i
i
P P X ii
P s X X
U t Y G t Y G
t X X G X
A
Математичне та комп’ютерне моделювання
54
2
| 1 |
: ( , 1)=
; , 1 0
, , , 0,| | | |i
i i
i
P P X i
P Y s X X P
U t Y s G
t X X G X
A
| 1 |
: 1, , =
|
1
,1
, , ,
0, 0, , 1 .
| |
|
| | | |
=
i
i
i j
i
i j
P P
P s X
P
X i X j
X Pj
X X
t X X
G X G X s
A
We remark that in case of solution expansion (19) of the quantum
BBGKY hierarchy, an analog of reduced cumulant (33) is the reduced
cumulant of groups of operators (7) defined by formula [12]
1
0
!
; , 1, , 1 .
! !=
n
k
n s n k
k
n
U t Y s s n t
k n k
Reduced cumulants of nonlinear groups of operators. We indicate
some properties of reduced nonlinear cumulants (33) of groups of
operators (8). According to formula (32) and properties of cumulants (9),
namely ,10 =n nIA , the following equality holds
1
1
0
,0
0; , 1, , | 0 =
!
1 0, 1, , 0,1, , =
! !
0,1, , ,
=
n
n
k
s n
k
s n n
U Y s s n G
n
s n k G s n
k n k
G s n
A (35)
and hence the marginal correlation operators determined by series (32)
satisfy initial data (22).
In case of 0n for 1
0f L in the sense of the norm
convergence of the space 1
sL the infinitesimal generator of first-order
reduced cumulant (33) coincides with generator (23) of the von Neumann
hierarchy (2)
1
0
1
; | = | , 1,lim s
t
U t Y f f Y Y f s
t
where the operator |Y f is defined by formula (23). In case of 1=n
for second-order reduced cumulant (33) in the same sense we obtain the
following equality
1 2
0
1
; , 1limTrs
t
U t Y s f
t
Серія: Фізико-математичні науки. Випуск 5
55
1 int 1, 1 , , 1Trs s
i Y
i s f t Y s
1 2
1 2
1 2
1 2
:( , 1)= ,
; 1
, , ,| | | |X X
P Y s X X
i X s X
f t X f t X
where notations are used as above for hierarchy (21), and for 2n as a
consequence of the fact that we consider a system of particles interacting
by a two-body potential, it holds
1, , 1
0
1
; , 1, , | = 0.limrs s n n
t
T U t Y s s n f
t
In case of initial data satisfying a chaos property, i.e.
1(0) 0, (0,1),0,G G , for the 1 n th -order reduced cumulant we
have
1)
1 1
1
; , 1, , | 0 ,1, , 0, ,(
=
s n
n s n
i
U t Y s s n G t s n G i
A
i.e. the only summand that gives contribution to the result is the one with
0=k and |=|P s n , since otherwise there is at least one operator 0sG
with 2s in the last product.
For the 1 n th -order reduced cumulant (33) the following
inequality holds
1
3
1 ; , 1, , | 2 ! ! 2 ,
s n s n
n
s n
U t Y s s n f n s e c
L
(36)
where
| 1: 1, , = , , , 1, ,1
max max
|
c
i P
i
P s n k X k k k s n k s n
1 1 1
1 1 | | | | 1
1 | | 1
| |
, , .| | | || |
P P
X k k X kP P
X k k X k
f f
L L
To prove this inequality we first remark that for cumulant (9) the
following estimate holds
1
1
|
| 1 |, , , | ! .|
| | | P
P P n n
nn
t X X f P e f
LL
A
(37)
Indeed, we have
1
1 |
'
| 1 |
': , , =
, , , | 1 !
|
| | |
P k
k
P P n
n P X X Z
t X X f P
L
A
Математичне та комп’ютерне моделювання
56
1
1
|
' 1
, | |, 1 !,
|
=
k
P
k n nZ nk lZ P
n
t Z f f s P l l
L
L
where | |,s P l are the Stirling numbers of second kind and we use the
isometric property of the groups , 1n t n . Estimate (36) holds as a
consequence of the inequality
|
|
1
| |, 1 ! | | ! .
|
|
=
s
P
P
l
P l l P e
Then owing to estimate (36), for the (1 )n th -order reduced
cumulant (33) we have
1
1
| 2
11 1 | | 1 1
| 1 1 1
| | 1
1
|
0 :(1, , = 0 1 1
| 2
0 | 1 | 2 | 1
0
; , 1, , |
! !
| |!
!( )! ! !
!
...
! !
!
|
| |
|
| |
|
= ) =
|
= | | |
=
i
i
P
P P
P
P P
n
s n
n k
P
k P s n k X k
k
P
X k k X k
k X k kP P P X k
n
k
U t Y s s n f
n k
P e
k n k k k k
k
f f
k k k
n
L
L L
| | 1 |
: 1, , =
| | ! .
!
2 |c
i
i
P P
P s n k X
P e
n k
As result of using of the definition of the Stirling numbers of second
kind ,s s n k l and the inequalities
| | 1
0 : (1, , =
2 1
0 1
3( 3
0
!
| | !
!
!
,
!
! !
2 ! ! 2 ,
!
2
= )
= =
)
=
s !
i
i
n
P
k P s n k X
n s n k
l
k l
n s ns n k
k
n
P e
n k
n
s n k l l e
n k
n s n k
e n s e
n k
we obtain estimate (36).
Thus, according to estimate (36), for initial data from the space 1
nL
series (32) converges provided that 1
13
1
max 0 2n
nn
G e
L
, and the
following inequality holds
Серія: Фізико-математичні науки. Випуск 5
57
1
32 ! 2 .
s
s
s
G t s e
L
(38)
A solution of the Cauchy problem of the nonlinear quantum BBGKY
hierarchy for marginal correlation operators (21) is determined by the
following one-parametric mapping
| = | ,t t f e t e f a a (39)
which is defined on the space 1
L owing estimate (38), and has the
group property
1 2 2 1 1 2| | = | | = | .t t f t t f t t f
Indeed, according to definition (29) and taking to attention the group
property of the mapping ( | )t , we obtain
1 2 1 2 1 2
1 2 1 2 1 2
| = | = | | =
| | = | | = | | .
t t f e t t e f e t t e f
e t e e t e f e t e t f t t f
a a a a
a a a a a a
To construct the generator of the strong continuous group , |t Y we
differentiate it in the sense of the norm convergence on the space 1
sL
0 0
1, ,
00
; | |
1
| | |
!
= =
==
Tr
st t
s s n
stn
d d
t Y f e t e f Y
dt dt
X t e f e e f Y
n
a a
a a a
where | f is a generator of the von Neumann hierarchy (2) defined
by formula (4) on the subspaces 1 1
0 , 1s s s L L , or in the
componentwise form in case of a two-body interaction potential
1 int 1, 1 , 1
=
Tr
s
s s
i Y
e e f Y Y f
i s f Y s
a a
(40)
1 2
1 2
1 2
1 2
:( , 1)= ,
; 1
,X X
P Y s X X
i X s X
f X f X
where we use notations as above for formula (21). Formula (40) describes
the structure of the infinitesimal generator of mapping (39) in the general
case of many-body interaction potentials.
Thus, for abstract initial-value problem for hierarchy (22) in the
space 1
L the following theorem is true.
Математичне та комп’ютерне моделювання
58
If 1
13
1
max (0) < 2n
n n
G e
L
, then for t a solution of the
initial-value problem (22)—(23) of the nonlinear quantum BBGKY
hierarchy is determined by expansion (32). If 1 1
0(0)n n nG L L
it is a strong (classical) solution and for arbitrary initial data
1(0)n nG L it is a weak (generalized) solution.
The proof of the theorem is similarly to the prove of an existence
theorem for the von Neumann hierarchy [10].
Conclusion. In the paper the origin of the microscopic description of
non-equilibrium correlations of quantum many-particle systems obeying
the Maxwell-Boltzmann statistics has been considered. The nonlinear
quantum BBGKY hierarchy (22) for marginal correlation operators was
introduced. It gives an alternative approach to the description of the state
evolution of quantum infinite-particle systems in comparison with
quantum BBGKY hierarchy for marginal density operators. The evolution
of both finitely and infinitely many quantum particles is described by
initial-value problem of the nonlinear quantum BBGKY hierarchy (21)
and for finitely many particles the nonlinear quantum BBGKY hierarchy is
equivalent to the von Neumann hierarchy(2).
A non-perturbative solution of the nonlinear quantum BBGKY
hierarchy is constructed in the form of expansion (32) over particle
clusters which evolution is governed by corresponding-order cumulant
(33) of the nonlinear groups of operators generated by solution (8) of the
von Neumann hierarchy (2). We established that in case of absence of
correlations at initial time the correlations generated by the dynamics of
quantum many-particle systems (27) are completely determined by
cumulants (9) of groups of operators (7).
Thus, the cumulant structure of solution (8) of the von Neumann
hierarchy (2) induces the cumulant structure of solution expansion (32) of
initial-value problem of the nonlinear quantum BBGKY hierarchy (22).
We emphasize that intensional Banach spaces for the description of
states of infinite-particle systems, which are suitable for the description of
the kinetic evolution or equilibrium states, are different from the exploit
spaces [12; 14]. Therefore marginal correlation operators from the space
1
L describe finitely many quantum particles. In order to describe the
evolution of infinitely many particles we have to construct solutions for
initial data from more general Banach spaces than the space of sequences
of trace class operators. For example, it can be the space of sequences of
bounded translation invariant operators which contains the marginal
density operators of equilibrium states. In that case every term of the
Серія: Фізико-математичні науки. Випуск 5
59
solution expansion of the nonlinear quantum BBGKY hierarchy (22)
contains the divergent traces, which can be renormalized due to the
cumulant structure of solution expansion (33).
References:
1. Arnold A. Mathematical properties of quantum evolution equations /
A. Arnold // Lecture Notes in Math. — 2008. — Vol. 1946. — P. 45—100.
2. Erdös L. Derivation of the cubic nonlinear Schrodinger equation from quantum
dynamics of many-body systems / L. Erdös, B. Schlein, H.-T. Yau // Invent.
Math. — 2007. — Vol. 167. — P. 515—614.
3. Erdös L. Derivation of the Gross-Pitaevskii Equation for the Dynamics of
Bose-Einstein Condensate / L. Erdös, B. Schlein, H.-T. Yau // Ann. Math. —
2010. — Vol. 172. — P. 291—370.
4. Fröhlich J. Mean-field and classical limit of many-body Schrodinger dynamics
for bosons / J. Fröhlich, S. Graffi, S. Schwarz // Commun. Math. Phys. —
2007. — Vol. 271. — P. 681—697.
5. Pezzotti F. Mean-field limit and semiclassical expansion of quantum particle
system / F. Pezzotti, M. Pulvirenti // Ann. Henri Poincar. — 2009. —
Vol. 10. — P. 145—187.
6. Saint-Raymond L. Kinetic models for superuids: a review of mathematical results /
L. Saint-Raymond // C. R. Physique. — 2004. — Vol. 5. — P. 65—75.
7. Gerasimenko V. I. A description of the evolution of quantum states bymeans
of the kinetic equation / V. I. Gerasimenko, Zh. A. Tsvir //. J. Phys. A: Math.
Theor. — 2010. — Vol. 43.
8. Gerasimenko V. I. Heisenberg picture of quantum kinetic evolution in mean-
field limit / V. I. Gerasimenko // Kinet. Relat. Models. — 2011. — Vol. 4. —
P. 385—399.
9. Gerasimenko V. I. Evolution of correlations of quantum many-particle systems
/ V. I. Gerasimenko, V. O. Shtyk // J. Stat. Mech. Theory Exp. — 2010. —
Vol. 3. — P. 24.
10. Gerasimenko V. I. Dynamics of correlations of Bose and Fermi particles /
V. I. Gerasimenko, D. O. Polishchuk // Math. Meth. Appl. Sci. — 2011. —
Vol. 34. — P. 76—93.
11. Боголюбов M. M. Лекції з квантової статистики. Проблеми статистичної
механіки квантових систем / M. M. Боголюбов. — К. : Рад. школа, 1949.
12. Petrina D. Ya. Mathematical Foundations of Quantum Statistical Mechanics.
Continuous Systems / D. Ya. Petrina. — Dordrecht : Kluwer, 1995.
13. Dautray R. Mathematical Analysis and Numerical Methods for Science and
Technology / R. Dautray, J. L. Lions. — Berlin : Springer, 1992.
14. Cercignani C. Many-Particle Dynamics and Kinetic Equations / C. Cercignani,
V. I. Gerasimenko, D. Ya. Petrina. — Dordrecht : Kluwer, 1997.
15. Polishchuk D. O. BBGKY hierarchy and dynamics of correlations / D. O. Po-
lishchuk // Ukrainian J. Phys. — 2010. — Vol. 55. — P. 593-598.
16. Yvon J. Actualites Scientifiques et Industrielles / J. Yvon. — Paris : Hermann,
1935. — Vol. 49.
17. Green M. S. Boltzmann equation from the statistical mechanical point of view
/ M. S. Green // J. Chem. Phys. — 1956. — Vol. 25. — P. 836—855.
Математичне та комп’ютерне моделювання
60
Робота присвячена проблемі опису нерівноважних кореляцій
квантових багаточастинкових систем. Побудовано розв'язок задачі Коші
нелінійної квантової ієрархії рівнянь ББГКІ у формі розкладу по групах
частинок, еволюція яких описується відповідного порядку кумулянтом
груп нелінійних операторів ієрархії рівнянь фон Неймана.
Ключові слова: нелінійна квантова ієрархія ББГКІ, ієрархія фон
Неймана, кореляційний оператор, квантові багаточастинкові системи.
Отримано: 06.05.2011
УДК 517.5
В. О. Гнатюк, канд. фіз-мат. наук,
Ю. В. Гнатюк, канд. фіз-мат. наук
Кам’янець-Подільський національний університет
імені Івана Огієнка, м. Кам’нець-Подільський
ТЕОРЕМИ ІСНУВАННЯ ЕКСТРЕМАЛЬНОГО ЕЛЕМЕНТА ДЛЯ
ЗАДАЧІ НАЙКРАЩОЇ У РОЗУМІННІ ОПУКЛОЇ НЕПЕРЕРВНОЇ
ФУНКЦІЇ РІВНОМІРНОЇ АПРОКСИМАЦІЇ НЕПЕРЕРВНОГО
КОМПАКНОЗНАЧНОГО ВІДОБРАЖЕННЯ
Доведено деякі теореми існування екстремального елемента
для задачі найкращої у розумінні опуклої неперервної функції
рівномірної апроксимації неперервного компакнозначного відо-
браження множинами неперервних однозначних відображень.
Ключові слова: найкраща у розумінні опуклої неперервної
функції рівномірна апроксимація, компакнозначне відобра-
ження, екстремальний елемент, теореми існування.
Вступ. У статті для задачі найкращої у розумінні опуклої непе-
рервної функції рівномірної апроксимації неперервного компактноз-
начного відображення множинами неперервних однозначних відо-
бражень доведено деякі теореми існування екстремального елемента,
які узагальнюють на випадок цієї задачі відповідні теореми існування
екстремального елемента для задачі найкращого у розумінні опуклої
функції наближення елемента лінійного нормованого простору опук-
лою множиною цього простору, встановлені у праці [1], розглянуто
допоміжні твердження, які представляють і самостійний інтерес.
Постановка задачі. Нехай S -компакт, X -лінійний над полем дій-
сних чисел нормований простір, ,C S X — лінійний над полем дійс-
них чисел нормований простір однозначних відображень g компакта
S в X , неперервних на S , з нормою max
s S
g g s
, K X — су-
© В. О. Гнатюк, Ю. В. Гнатюк, 2011
<<
/ASCII85EncodePages false
/AllowTransparency false
/AutoPositionEPSFiles true
/AutoRotatePages /All
/Binding /Left
/CalGrayProfile (Gray Gamma 2.2)
/CalRGBProfile (sRGB IEC61966-2.1)
/CalCMYKProfile (Coated FOGRA27 \050ISO 12647-2:2004\051)
/sRGBProfile (sRGB IEC61966-2.1)
/CannotEmbedFontPolicy /Warning
/CompatibilityLevel 1.3
/CompressObjects /Tags
/CompressPages true
/ConvertImagesToIndexed true
/PassThroughJPEGImages true
/CreateJobTicket false
/DefaultRenderingIntent /Default
/DetectBlends true
/DetectCurves 0.1000
/ColorConversionStrategy /sRGB
/DoThumbnails false
/EmbedAllFonts true
/EmbedOpenType false
/ParseICCProfilesInComments true
/EmbedJobOptions true
/DSCReportingLevel 0
/EmitDSCWarnings false
/EndPage -1
/ImageMemory 1048576
/LockDistillerParams false
/MaxSubsetPct 100
/Optimize true
/OPM 1
/ParseDSCComments true
/ParseDSCCommentsForDocInfo true
/PreserveCopyPage true
/PreserveDICMYKValues true
/PreserveEPSInfo false
/PreserveFlatness false
/PreserveHalftoneInfo false
/PreserveOPIComments false
/PreserveOverprintSettings true
/StartPage 1
/SubsetFonts true
/TransferFunctionInfo /Apply
/UCRandBGInfo /Remove
/UsePrologue false
/ColorSettingsFile ()
/AlwaysEmbed [ true
]
/NeverEmbed [ true
/Arial-Black
/Arial-BlackItalic
/Arial-BoldItalicMT
/Arial-BoldMT
/Arial-ItalicMT
/ArialMT
/ArialNarrow
/ArialNarrow-Bold
/ArialNarrow-BoldItalic
/ArialNarrow-Italic
/ArialUnicodeMS
/CenturyGothic
/CenturyGothic-Bold
/CenturyGothic-BoldItalic
/CenturyGothic-Italic
/CourierNewPS-BoldItalicMT
/CourierNewPS-BoldMT
/CourierNewPS-ItalicMT
/CourierNewPSMT
/Georgia
/Georgia-Bold
/Georgia-BoldItalic
/Georgia-Italic
/Impact
/LucidaConsole
/Tahoma
/Tahoma-Bold
/TimesNewRomanMT-ExtraBold
/TimesNewRomanPS-BoldItalicMT
/TimesNewRomanPS-BoldMT
/TimesNewRomanPS-ItalicMT
/TimesNewRomanPSMT
/Trebuchet-BoldItalic
/TrebuchetMS
/TrebuchetMS-Bold
/TrebuchetMS-Italic
/Verdana
/Verdana-Bold
/Verdana-BoldItalic
/Verdana-Italic
]
/AntiAliasColorImages false
/CropColorImages false
/ColorImageMinResolution 150
/ColorImageMinResolutionPolicy /OK
/DownsampleColorImages true
/ColorImageDownsampleType /Bicubic
/ColorImageResolution 150
/ColorImageDepth -1
/ColorImageMinDownsampleDepth 1
/ColorImageDownsampleThreshold 1.50000
/EncodeColorImages true
/ColorImageFilter /DCTEncode
/AutoFilterColorImages true
/ColorImageAutoFilterStrategy /JPEG
/ColorACSImageDict <<
/QFactor 0.76
/HSamples [2 1 1 2] /VSamples [2 1 1 2]
>>
/ColorImageDict <<
/QFactor 0.76
/HSamples [2 1 1 2] /VSamples [2 1 1 2]
>>
/JPEG2000ColorACSImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 15
>>
/JPEG2000ColorImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 15
>>
/AntiAliasGrayImages false
/CropGrayImages false
/GrayImageMinResolution 150
/GrayImageMinResolutionPolicy /OK
/DownsampleGrayImages true
/GrayImageDownsampleType /Bicubic
/GrayImageResolution 150
/GrayImageDepth -1
/GrayImageMinDownsampleDepth 2
/GrayImageDownsampleThreshold 1.50000
/EncodeGrayImages true
/GrayImageFilter /DCTEncode
/AutoFilterGrayImages true
/GrayImageAutoFilterStrategy /JPEG
/GrayACSImageDict <<
/QFactor 0.76
/HSamples [2 1 1 2] /VSamples [2 1 1 2]
>>
/GrayImageDict <<
/QFactor 0.76
/HSamples [2 1 1 2] /VSamples [2 1 1 2]
>>
/JPEG2000GrayACSImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 15
>>
/JPEG2000GrayImageDict <<
/TileWidth 256
/TileHeight 256
/Quality 15
>>
/AntiAliasMonoImages false
/CropMonoImages false
/MonoImageMinResolution 1200
/MonoImageMinResolutionPolicy /OK
/DownsampleMonoImages true
/MonoImageDownsampleType /Bicubic
/MonoImageResolution 1200
/MonoImageDepth -1
/MonoImageDownsampleThreshold 1.50000
/EncodeMonoImages true
/MonoImageFilter /CCITTFaxEncode
/MonoImageDict <<
/K -1
>>
/AllowPSXObjects true
/CheckCompliance [
/PDFX1a:2001
]
/PDFX1aCheck false
/PDFX3Check false
/PDFXCompliantPDFOnly false
/PDFXNoTrimBoxError true
/PDFXTrimBoxToMediaBoxOffset [
0.00000
0.00000
0.00000
0.00000
]
/PDFXSetBleedBoxToMediaBox true
/PDFXBleedBoxToTrimBoxOffset [
0.00000
0.00000
0.00000
0.00000
]
/PDFXOutputIntentProfile (None)
/PDFXOutputConditionIdentifier ()
/PDFXOutputCondition ()
/PDFXRegistryName ()
/PDFXTrapped /False
/CreateJDFFile false
/Description <<
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
/BGR <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>
/CHS <FEFF4f7f75288fd94e9b8bbe5b9a521b5efa7684002000410064006f006200650020005000440046002065876863900275284e8e55464e1a65876863768467e5770b548c62535370300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c676562535f00521b5efa768400200050004400460020658768633002>
/CHT <FEFF4f7f752890194e9b8a2d7f6e5efa7acb7684002000410064006f006200650020005000440046002065874ef69069752865bc666e901a554652d965874ef6768467e5770b548c52175370300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c4f86958b555f5df25efa7acb76840020005000440046002065874ef63002>
/CZE <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>
/DAN <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>
/DEU <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>
/ENU (Use these settings to create Adobe PDF documents suitable for reliable viewing and printing of business documents. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
/ESP <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>
/ETI <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>
/FRA <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>
/GRE <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>
/HEB <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>
/HRV (Za stvaranje Adobe PDF dokumenata pogodnih za pouzdani prikaz i ispis poslovnih dokumenata koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
/HUN <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>
/ITA (Utilizzare queste impostazioni per creare documenti Adobe PDF adatti per visualizzare e stampare documenti aziendali in modo affidabile. I documenti PDF creati possono essere aperti con Acrobat e Adobe Reader 5.0 e versioni successive.)
/JPN <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>
/KOR <FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020be44c988b2c8c2a40020bb38c11cb97c0020c548c815c801c73cb85c0020bcf4ace00020c778c1c4d558b2940020b3700020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>
/LTH <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>
/LVI <FEFF0049007a006d0061006e0074006f006a00690065007400200161006f00730020006900650073007400610074012b006a0075006d00750073002c0020006c0061006900200076006500690064006f00740075002000410064006f00620065002000500044004600200064006f006b0075006d0065006e007400750073002c0020006b006100730020006900720020007000690065006d01130072006f00740069002000640072006f016100610069002000620069007a006e00650073006100200064006f006b0075006d0065006e007400750020006100700073006b006100740065006900200075006e0020006400720075006b010101610061006e00610069002e00200049007a0076006500690064006f006a006900650074002000500044004600200064006f006b0075006d0065006e007400750073002c0020006b006f002000760061007200200061007400760113007200740020006100720020004100630072006f00620061007400200075006e002000410064006f00620065002000520065006100640065007200200035002e0030002c0020006b0101002000610072012b00200074006f0020006a00610075006e0101006b0101006d002000760065007200730069006a0101006d002e>
/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken waarmee zakelijke documenten betrouwbaar kunnen worden weergegeven en afgedrukt. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
/NOR <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>
/POL <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>
/PTB <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>
/RUM <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>
/SKY <FEFF0054006900650074006f0020006e006100730074006100760065006e0069006100200070006f0075017e0069007400650020006e00610020007600790074007600e100720061006e0069006500200064006f006b0075006d0065006e0074006f0076002000410064006f006200650020005000440046002000760068006f0064006e00fd006300680020006e0061002000730070006f013e00610068006c0069007600e90020007a006f006200720061007a006f00760061006e006900650020006100200074006c0061010d0020006f006200630068006f0064006e00fd0063006800200064006f006b0075006d0065006e0074006f0076002e00200056007900740076006f00720065006e00e900200064006f006b0075006d0065006e007400790020005000440046002000620075006400650020006d006f017e006e00e90020006f00740076006f00720069016500200076002000700072006f006700720061006d006f006300680020004100630072006f00620061007400200061002000410064006f00620065002000520065006100640065007200200035002e0030002000610020006e006f0076016100ed00630068002e>
/SLV <FEFF005400650020006e006100730074006100760069007400760065002000750070006f0072006100620069007400650020007a00610020007500730074007600610072006a0061006e006a006500200064006f006b0075006d0065006e0074006f0076002000410064006f006200650020005000440046002c0020007000720069006d00650072006e006900680020007a00610020007a0061006e00650073006c006a00690076006f0020006f0067006c00650064006f00760061006e006a006500200069006e0020007400690073006b0061006e006a006500200070006f0073006c006f0076006e0069006800200064006f006b0075006d0065006e0074006f0076002e00200020005500730074007600610072006a0065006e006500200064006f006b0075006d0065006e0074006500200050004400460020006a00650020006d006f0067006f010d00650020006f0064007000720065007400690020007a0020004100630072006f00620061007400200069006e002000410064006f00620065002000520065006100640065007200200035002e003000200069006e0020006e006f00760065006a01610069006d002e>
/SUO <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>
/SVE <FEFF0041006e007600e4006e00640020006400650020006800e4007200200069006e0073007400e4006c006c006e0069006e006700610072006e00610020006f006d002000640075002000760069006c006c00200073006b006100700061002000410064006f006200650020005000440046002d0064006f006b0075006d0065006e007400200073006f006d00200070006100730073006100720020006600f60072002000740069006c006c006600f60072006c00690074006c006900670020007600690073006e0069006e00670020006f006300680020007500740073006b007200690066007400650072002000610076002000610066006600e4007200730064006f006b0075006d0065006e0074002e002000200053006b006100700061006400650020005000440046002d0064006f006b0075006d0065006e00740020006b0061006e002000f600700070006e00610073002000690020004100630072006f0062006100740020006f00630068002000410064006f00620065002000520065006100640065007200200035002e00300020006f00630068002000730065006e006100720065002e>
/TUR <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>
/UKR <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>
/RUS <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>
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AllowImageBreaks true
/AllowTableBreaks true
/ExpandPage false
/HonorBaseURL true
/HonorRolloverEffect false
/IgnoreHTMLPageBreaks false
/IncludeHeaderFooter false
/MarginOffset [
0
0
0
0
]
/MetadataAuthor ()
/MetadataKeywords ()
/MetadataSubject ()
/MetadataTitle ()
/MetricPageSize [
0
0
]
/MetricUnit /inch
/MobileCompatible 0
/Namespace [
(Adobe)
(GoLive)
(8.0)
]
/OpenZoomToHTMLFontSize false
/PageOrientation /Portrait
/RemoveBackground false
/ShrinkContent true
/TreatColorsAs /MainMonitorColors
/UseEmbeddedProfiles false
/UseHTMLTitleAsMetadata true
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/BleedOffset [
0
0
0
0
]
/ConvertColors /ConvertToRGB
/DestinationProfileName (sRGB IEC61966-2.1)
/DestinationProfileSelector /UseName
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements true
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MarksOffset 6
/MarksWeight 0.250000
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PageMarksFile /RomanDefault
/PreserveEditing true
/UntaggedCMYKHandling /UseDocumentProfile
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [600 600]
/PageSize [419.528 595.276]
>> setpagedevice
|