Using of numerical modeling for verification of the functional hypothesis
One-dimensional harmonic oscillator in a quasi-equilibrium medium which consists of non-interacting harmonic oscillators has been considered. Kinetic equation for this Brownian particle has been derived on the basis of the Bogolyubov functional hypothesis. Solution of the kinetic equation was numeri...
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| Cite this: | Using of numerical modeling for verification of the functional hypothesis / A.I. Sokolovsky, Z.Yu. Chelbaevsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 331-334. — Бібліогр.: 3 назв. — англ. |
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Sokolovsky, A.I. Chelbaevsky, Z.Yu. 2017-01-07T17:33:17Z 2017-01-07T17:33:17Z 2007 Using of numerical modeling for verification of the functional hypothesis / A.I. Sokolovsky, Z.Yu. Chelbaevsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 331-334. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 02.70.-c, 05.20.Dd https://nasplib.isofts.kiev.ua/handle/123456789/111016 One-dimensional harmonic oscillator in a quasi-equilibrium medium which consists of non-interacting harmonic oscillators has been considered. Kinetic equation for this Brownian particle has been derived on the basis of the Bogolyubov functional hypothesis. Solution of the kinetic equation was numerically compared with an exact solution obtained by Bogolyubov. The results of this comparison are presented in a simple graphic form. Розглянуто одновимірний гармонічний осцилятор у квазірівноважному середовищі, яке складається з гармонічних осциляторів, що не взаємодіють. На основі функціональної гіпотези Боголюбова одержано кінетичне рівняння для цієї броунівської частинки. Розв’язок кінетичного рівняння чисельно порівняно з точним розв’язком, одержаним Боголюбовим. Підсумки порівняння представлені в простій графічній формі. Рассмотрен одномерный гармонический осциллятор в квазиравновесной среде, которая состоит из невзаимодействующих гармонических осцилляторов. На основе функциональной гипотезы Боголюбова получено кинетическое уравнение для этой броуновской частицы. Решение кинетического уравнения численно сравнено с точным решением, полученным Боголюбовым. Результаты сравнения представлены в простой графической форме. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory Using of numerical modeling for verification of the functional hypothesis Використання чисельного моделювання для перевірки функціональної гіпотези Использование численного моделирования для проверки функциональной гипотезы Article published earlier |
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Using of numerical modeling for verification of the functional hypothesis |
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Using of numerical modeling for verification of the functional hypothesis Sokolovsky, A.I. Chelbaevsky, Z.Yu. Kinetic theory |
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Using of numerical modeling for verification of the functional hypothesis |
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Using of numerical modeling for verification of the functional hypothesis |
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Using of numerical modeling for verification of the functional hypothesis |
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Using of numerical modeling for verification of the functional hypothesis |
| title_sort |
using of numerical modeling for verification of the functional hypothesis |
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Sokolovsky, A.I. Chelbaevsky, Z.Yu. |
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Sokolovsky, A.I. Chelbaevsky, Z.Yu. |
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Kinetic theory |
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Kinetic theory |
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Вопросы атомной науки и техники |
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Використання чисельного моделювання для перевірки функціональної гіпотези Использование численного моделирования для проверки функциональной гипотезы |
| description |
One-dimensional harmonic oscillator in a quasi-equilibrium medium which consists of non-interacting harmonic oscillators has been considered. Kinetic equation for this Brownian particle has been derived on the basis of the Bogolyubov functional hypothesis. Solution of the kinetic equation was numerically compared with an exact solution obtained by Bogolyubov. The results of this comparison are presented in a simple graphic form.
Розглянуто одновимірний гармонічний осцилятор у квазірівноважному середовищі, яке складається з гармонічних осциляторів, що не взаємодіють. На основі функціональної гіпотези Боголюбова одержано кінетичне рівняння для цієї броунівської частинки. Розв’язок кінетичного рівняння чисельно порівняно з точним розв’язком, одержаним Боголюбовим. Підсумки порівняння представлені в простій графічній формі.
Рассмотрен одномерный гармонический осциллятор в квазиравновесной среде, которая состоит из невзаимодействующих гармонических осцилляторов. На основе функциональной гипотезы Боголюбова получено кинетическое уравнение для этой броуновской частицы. Решение кинетического уравнения численно сравнено с точным решением, полученным Боголюбовым. Результаты сравнения представлены в простой графической форме.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111016 |
| citation_txt |
Using of numerical modeling for verification of the functional hypothesis / A.I. Sokolovsky, Z.Yu. Chelbaevsky // Вопросы атомной науки и техники. — 2007. — № 3. — С. 331-334. — Бібліогр.: 3 назв. — англ. |
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2025-11-27T06:57:10Z |
| last_indexed |
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1850802695634419712 |
| fulltext |
USING OF NUMERICAL MODELING
FOR VERIFICATION OF THE FUNCTIONAL HYPOTHESIS
A.I. Sokolovsky and Z.Yu. Chelbaevsky
Dnipropetrovsk National University, Dnipropetrovsk, Ukraine;
e-mail: alexsokolovsky@mail.ru
One-dimensional harmonic oscillator in a quasi-equilibrium medium which consists of non-interacting harmonic
oscillators has been considered. Kinetic equation for this Brownian particle has been derived on the basis of the Bo-
golyubov functional hypothesis. Solution of the kinetic equation was numerically compared with an exact solution
obtained by Bogolyubov. The results of this comparison are presented in a simple graphic form.
PACS: 02.70.-c, 05.20.Dd
1. INTRODUCTION
This work is devoted to justification of applicability
domain of the Bogolyubov functional hypothesis and
his reduced description method (RDM) on the basis of
an exact solvable model. The RDM (which is based on
the functional hypothesis) is one of basic approaches to
investigation of nonequilibrium processes, therefore the
interest in its verification has become clear.
The kinetics of a one-dimensional harmonic oscilla-
tor in a quasi-equilibrium medium which consists of
non-interacting harmonic oscillators is investigated in
this work. The interaction between the oscillator and the
environment is supposed to be weak. This model was
considered in the paper of Bogolyubov [1], where he
has found its exact solution in a very complex form.
The state of the aforesaid system is completely de-
scribed by the distribution function , where
are phase variables of the oscillator and are
phase variables of the medium. The quasi-equilibrium
environment is described by the equilibrium thermody-
namic parameters, which in general can depend on time
because of a reverse influence of the oscillator on the
medium. The energy of the medium is selected
as such parameter.
0( , , )x X tρ
( )mE t
0x X
We consider simplification in description of the sys-
tem when its state is determined by the distribution
function of energy of the oscillator and by the
energy of the medium . According to the func-
tional hypothesis the distribution function of the system
at long times has the structure .
We have derived an integral equation for the distribu-
tion , and also a system of kinetic equa-
tions
( , )w E t
0( , ,x X
( )mE t
( ), ( ))mw t E tρ
0( , , , )mx X w Eρ
( , ) ( , ( ), ( ))mw E t L E w t E t= ,
( ) ( ( ), ( ))m m mE t L w t E t= .
The mentioned values were calculated in a perturbation
theory in small interaction between the Brownian oscil-
lator and the medium.
The analytical results obtained for by the
RDM are numerically compared with the Bogolyubov
exact solution.
( , )w E t
2. THE KINETIC EQUATION
OF BOGOLUBOV’S MODEL IN THE
REDUCED DESCRIPTION METHOD
The Hamiltonian for our model is the following
b m bH H H H= + + m , (1)
where the subscript refers to the Brownian harmonic
oscillator with a frequency
b
0ω
2 2 2
0 0 0
1 ( )
2bH p qω= + , (2)
the subscript m refers to the medium consisting of a big
number of harmonic oscillators with frequencies aω
2 2 2
1
1 ( )
2
N
m a a
a
H p ω
=
= +∑ aq
q
. (3)
The Hamiltonian of interaction is given by bmH
0
1
N
bm a a
a
H qα
=
= ∑ , (4)
where αa are the small values ( , ). ~aα ε 1ε
Let the initial (when ) phase variables of the
Brownian oscillator are , ; and phase variables of
the medium { are canonical distributed random
variables:
0t =
0
0
0
0pq
, }a aq p
T
HF
b
b
ew
−
= (5)
(here and further stands for ). T kT
The Liouville equation for the distribution function
of the system is the following
),,(),,( 00 tXxtXx ρρ L= , (6)
where
0b m bm bm= + + = +L L L L L L ,
2
0 0 0
0 0
m p q
q p
ω∂ ∂
= − +
∂ ∂
L ,
2
1 1
N N
b a a a
a aa a
p q
q p
ω
= =
∂ ∂
= − +
∂ ∂∑ ∑L ,
0
01 1
N N
bm a a a
aa a
q q
p p
α α
= =
∂
= +
∂ ∂∑ ∑L ∂ (7)
and
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 331-334. 331
),( 000 qpx ≡ , , . (8) ),( aaa qpx ≡ ),,( 1 nxxX …≡
The distribution function satisfies the nor-
malization condition
0( , , )x X tρ
∫ =1),,( 00 tXxdXdx ρ . (9)
The formal solution of the Liouville equation can be
written as
),(),,( 000 XxetXx tρρ L= , (10) (ρ
where
0 0 0( , ) ( , ,0)x X x Xρ ρ= .
m
)
We will consider a kinetic stage of the evolution,
when the simplification in the description of the system
state takes place and it is determined by the distribution
function of the Brownian oscillator energy and
by the medium energy . It means that at time,
considerably larger that time τ
( , )w E t
( )mE t
, )x X t
0, according to a func-
tional hypothesis, should have the following
structure
0( ,ρ
00 0( , , ) ( , , ( ), ( ))mtx X t x X w t E tτρ ρ→ , (11)
where reduced description parameters , and
are defined by relations
( , )w E t
( )mE t
00 0( , , ) ( ) ( , )s tdx dX x X t E H w E tτρ δ − →∫ ,
00 0( , , ) ( )m tdx dX x X t H E tτρ →∫ . (12)
The Liouville equation (6) at times takes the
form
0t τ
0 0( , , ( ), ( )) ( , , ( ), ( )).m mx X w t E t x X w t E tρ ρ= L (13)
This equation and definitions Eqs.(14) lead to the
following equation for ( , )w E t
( , ) ( , ( ), ( ))mw E t L E w t E t= , , (14) 0(t τ
where
( , , )mL E w E ≡
( )0 0
1
, , , ( )
N
m s a
a
dx dX x X w E E H q p
E
ρ δ α
=
∂
≡ −
∂ ∑∫ 0a
,
.aα
.(15)
The kinetic equation for is given by the for-
mula
( )mE t
( ) ( ( ), ( ))m m mE t L w t E t= , (16) 0( )t τ
where
( , )m mL w E ≡ .
( )0 0 0
1
, , ,
N
m a
a
dx dX x X w E q pρ
=
≡ − ∑∫ (17)
The Liouville equation at the reduced description
Eq.(13) can be written due to Eqs. (14), (16) in the form
0
0
( , , , )
( , , , ) ( , , )
( )
m
m m
x X w E
x X w E dE L E w E
w E
δρ
ρ
δ
= ∫L
0( , , , )
( , ).b
b b
b
x X w E
L w E
E
ρ∂
+
∂
(18)
For this equation we use the following boundary condi-
tion of the complete correlation
0 0
00 0( , , , ) ( , , , )m qte x X w E e x X w Eτ τ
τρ ρ→L L
m ,
0( , ) ( ) ( ).
2q m b m mw E w H w E
ω
ρ
π
≡ (19)
By an usual way [2,3] Eq. (18) with condition (19)
give the following integral equation for distribution
function , )mw E
{0
0
( , ) ( , ) ( , )m q m bm mw E w E d e w Eτρ ρ τ ρ
+∞
= + ∫ L L
}) .m
( , ) ( , )
( , , ) ( ,
( )
m m
m
m
w E w E
dE L E w E L w E
w E E
δρ ρ
δ
∂
− −
∂∫ m .(20)
We solve this Eq. (20) for with expres-
sions Eqs. (15), (17) for and by
iterations in a perturbation theory in interaction constant
.
0( , , , )mx X w Eρ
( , , )mL E w E ( , )m mL w E
ε
The results of the calculations up to second order in
ε are the following
(0) ( , )q mw Eρ ρ= ; , ; (1) 0L = (1) 0mL =
0(1) 0
0
10
( )
2
N
b a a
a
w Ew d e q p
E
τω
ρ τ α
π
∞
=
∂=
∂
∑∫ L
0
1 ( )
b
a
E H
q p w E
T →
−
, (21) (2) 0.mL =
The formula for gives kinetic equation for (1)ρ ( , )w E t
0
1( , ) ( ) ( , ) ,
2
w E t TI E w E t
E E T
π ω ∂ ∂ = + ∂ ∂
(22)
where
2
2
1
( ) ( ).
N
a
a
a a
I α
ω δ ω
ω=
= −∑ ω (23)
So, we have obtained the necessary kinetic equation on
the basis of the RDM. In the considered approximation
the dependence of the medium energy on time is
absent.
( )mE t
3. NUMERICAL COMPARISON OF THE
RESULTS OF DIFFERENT APPROACHES
Bogolyubov has obtained the exact expression for
distribution function of the Brownian oscillator
in thermodynamic limit (when ) in
form
0 0( , , )b q p tρ N →∞
* *
0 0 0 0( , , ) ( ( ), ( ), )b q p t q q t p p t tρ = Φ − − , (24)
where
* 0 0
0 0( ) ( ) ( )q t q v t p v t′= + ,
* 0 0
0 0( ) ( ) ( )p t q v t p v t′′ ′= + , (25)
and
2 2
22
1 2( , , ) exp
2( )2
C B At
B ACAC B
ξ ξη ηξ η
π
− +
Φ = −−
. (26)
The functions A(t), B(t), C(t) are coefficients of the
quadratic form
332
2 2( ) 2 ( ) ( )A t B t B tλ λµ+ + µ
2
0 0
( ) { ( ) ( )} .
t
iT d I d e ωτω ω τ λν τ µν τ
∞
−′= +∫ ∫ (27)
Here function is the solution of the following
differential equation
( )tν
2
0
0
( ) ( ) ( ) ( )
t
v t v t d Q t vω τ τ′′ ′+ = −∫ τ , (28)
with initial conditions , , where (0) 0ν = (0) 1ν ′ =
0
( ) ( )(1 cos ).Q t d I tω ω ω
∞
= −∫ (29)
The comparison of the exact solution and the result
of the RDM consists in analysis of the difference be-
tween exact distribution of energy
0 0 0 0
( )
( , ) ( , , )
b
b
H E
P E t dq dp q p tρ
<
= ∫
0
0
( , )2
0 0 0
( , )2
( , , )
q E pE
b
q E pE
dp dq q p tρ
−−
= ∫ ∫ 0 ,
0
2
0
0
2
),(
ω
pE
pEq
−
= (30)
and distribution
0
0
( , ) ( , )
E
P E t dE w E t′ ′= ∫ (31)
calculated on the basis of the RDM.
Kinetic equation (22) for distribution function
is solved by us numerically in dimensionless
form
( , )w E t
( , ) 1 ( , )
2
w E t E w E
t E E
π∂ ∂ ∂ = + ∂ ∂ ∂
t ,
EE
T
= , t t , , (32) 2λ= 0( )Iλ ω≡
with the following boundary conditions
( , ) Ew E t e−→ , t ; →+∞
0( , ) ( )w E t E Eδ→ − 0→, t ;
( , ) 0w E t → , (33) .E → +∞
The differential equation (30) has a form of the second
Newton's law with a complex dissipative force and is
numerically integrated too.
The results of a graphical comparison of the
distributions and at different times are
shown in the figure. The parameters of this experiment
are
0P E( , )t )t( ,P E
1
0 00.006, 2.5, 0.001 .E sλ ω −= = =
First two graphics show essential distinctions between
the distributions at initial time, but their further evolu-
tion (when t ) is almost identical and they ap-
proach to the equilibrium distribution.
20.5λ−
Thus, we have performed the comparison of exact
and approximate solutions and have displayed their
closeness at long times.
Comparison of the distributions and
at different times. Dotted line shows the exact
solution, solid line shows the result of the reduced de-
scription method
( , )P E t
0 ( , )P E t
333
4. CONCLUSION
We have numerically compared solution of the ap-
proximate kinetic equation, which is obtained with the
help of functional hypothesis, with the exact solution. It
was shown that at long times the RDM leads to energy
distributions which are close to one another.
In the next paper we plan to compare the Bo-
golyubov exact result with consequences of the kinetic
equation of the third order approximation in interaction.
REFERENCES
1. N.N. Bogolyubov. Elementary example of transition
to equilibrium in system, connected with a bath /In:
N.N. Bogolyubov. About some statistical methods in
mathematical physics. Kiev: AN USSR, 1945,
p. 115–137 (in Russian).
2. A.I. Akhiezer, S.V. Peletminsky. Methods of statis-
tical physics. M.: “Nauka”, 1977, 368 p. (in Rus-
sian).
3. A.I. Sokolovsky, M.Yu. Tseitlin. On the theory of
the Brownian motion in the reduced description
method //Тeor. i Mat. Fizika. 1977, v. 33, N3,
p. 409-418 (in Russian).
ИСПОЛЬЗОВАНИЕ ЧИСЛЕННОГО МОДЕЛИРОВАНИЯ ДЛЯ ПРОВЕРКИ
ФУНКЦИОНАЛЬНОЙ ГИПОТЕЗЫ
А.И. Соколовский, З.Ю. Челбаевский
Рассмотрен одномерный гармонический осциллятор в квазиравновесной среде, которая состоит из не-
взаимодействующих гармонических осцилляторов. На основе функциональной гипотезы Боголюбова полу-
чено кинетическое уравнение для этой броуновской частицы. Решение кинетического уравнения численно
сравнено с точным решением, полученным Боголюбовым. Результаты сравнения представлены в простой
графической форме.
ВИКОРИСТАННЯ ЧИСЕЛЬНОГО МОДЕЛЮВАННЯ ДЛЯ ПЕРЕВІРКИ
ФУНКЦІОНАЛЬНОЇ ГІПОТЕЗИ
О.Й. Соколовський, З.Ю. Челбаєвський
Розглянуто одновимірний гармонічний осцилятор у квазірівноважному середовищі, яке складається з га-
рмонічних осциляторів, що не взаємодіють. На основі функціональної гіпотези Боголюбова одержано кіне-
тичне рівняння для цієї броунівської частинки. Розв’язок кінетичного рівняння чисельно порівняно з точним
розв’язком, одержаним Боголюбовим. Підсумки порівняння представлені в простій графічній формі.
334
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