Electromagnetic field correlations and sound waves
Statistical operator of the many component plasma has been found on the basis of the Bogolyubov reduced description method and quasi-relativistic quantum electrodynamics. Calculations were carried out in the Hamilton gauge up to the second order of a perturbation theory in interaction. Closed system...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Цитувати: | Electromagnetic field correlations and sound waves / A.I. Sokolovsky, A.A. Stupka // Вопросы атомной науки и техники. — 2007. — № 3. — С. 335-339. — Бібліогр.: 7 назв. — англ. |
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Sokolovsky, A.I. Stupka, A.A. 2017-01-07T17:32:31Z 2017-01-07T17:32:31Z 2007 Electromagnetic field correlations and sound waves / A.I. Sokolovsky, A.A. Stupka // Вопросы атомной науки и техники. — 2007. — № 3. — С. 335-339. — Бібліогр.: 7 назв. — англ. 1562-6016 PACS: 02.70.-c, 05.20.Dd https://nasplib.isofts.kiev.ua/handle/123456789/111015 Statistical operator of the many component plasma has been found on the basis of the Bogolyubov reduced description method and quasi-relativistic quantum electrodynamics. Calculations were carried out in the Hamilton gauge up to the second order of a perturbation theory in interaction. Closed system of equations for binary correlations of electromagnetic field and hydrodynamic variables of medium has been obtained and investigated near equilibrium. Classical Maxwell plasma approximation was studied. Coupled states of sound waves and waves of transversal correlation of the field were predicted. Waves of correlations of electromagnetic field can be excited by sound waves in plasma. Статистичний оператор багатокомпонентної плазми знайдено на основі метода скороченого опису Боголюбова та квазірелятивістської квантової електродинаміки. Обчислення проведено у калібровці Гамільтона з точністю до другого порядку теорії збурень за взаємодією. Одержано замкнену систему рівнянь для бінарних кореляцій поля та гідродинамічних змінних середовища і досліджено біля рівноваги. Вивчено наближення класичної максвеллівської плазми. Передбачено зв’язані стани звукових хвиль та хвиль поперечних кореляцій поля. Хвилі кореляцій електромагнітного поля можуть бути збуджені звуковими хвилями у плазмі. Статистический оператор многокомпонентной плазмы найден на основе метода сокращенного описания Боголюбова и квазирелятивистской квантовой электродинамики. Вычисления проведены в калибровке Гамильтона с точностью до второго порядка теории возмущений по взаимодействию. Получена замкнутая система уравнений для бинарных корреляций электромагнитного поля и гидродинамических переменных среды и исследована около равновесия. Изучено приближение классической максвелловской плазмы. Предсказаны связанные состояния звуковых волн и волн поперечных корреляций поля. Волны корреляций электромагнитного поля могут быть возбуждены звуковыми волнами в плазме. This work was supported by the State Foundation for Fundamental Research of Ukraine (project No. 2.7/418). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory Electromagnetic field correlations and sound waves Кореляції електромагнітного поля та звукові хвилі Корреляции электромагнитного поля и звуковые волны Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Electromagnetic field correlations and sound waves |
| spellingShingle |
Electromagnetic field correlations and sound waves Sokolovsky, A.I. Stupka, A.A. Kinetic theory |
| title_short |
Electromagnetic field correlations and sound waves |
| title_full |
Electromagnetic field correlations and sound waves |
| title_fullStr |
Electromagnetic field correlations and sound waves |
| title_full_unstemmed |
Electromagnetic field correlations and sound waves |
| title_sort |
electromagnetic field correlations and sound waves |
| author |
Sokolovsky, A.I. Stupka, A.A. |
| author_facet |
Sokolovsky, A.I. Stupka, A.A. |
| topic |
Kinetic theory |
| topic_facet |
Kinetic theory |
| publishDate |
2007 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Кореляції електромагнітного поля та звукові хвилі Корреляции электромагнитного поля и звуковые волны |
| description |
Statistical operator of the many component plasma has been found on the basis of the Bogolyubov reduced description method and quasi-relativistic quantum electrodynamics. Calculations were carried out in the Hamilton gauge up to the second order of a perturbation theory in interaction. Closed system of equations for binary correlations of electromagnetic field and hydrodynamic variables of medium has been obtained and investigated near equilibrium. Classical Maxwell plasma approximation was studied. Coupled states of sound waves and waves of transversal correlation of the field were predicted. Waves of correlations of electromagnetic field can be excited by sound waves in plasma.
Статистичний оператор багатокомпонентної плазми знайдено на основі метода скороченого опису Боголюбова та квазірелятивістської квантової електродинаміки. Обчислення проведено у калібровці Гамільтона з точністю до другого порядку теорії збурень за взаємодією. Одержано замкнену систему рівнянь для бінарних кореляцій поля та гідродинамічних змінних середовища і досліджено біля рівноваги. Вивчено наближення класичної максвеллівської плазми. Передбачено зв’язані стани звукових хвиль та хвиль поперечних кореляцій поля. Хвилі кореляцій електромагнітного поля можуть бути збуджені звуковими хвилями у плазмі.
Статистический оператор многокомпонентной плазмы найден на основе метода сокращенного описания Боголюбова и квазирелятивистской квантовой электродинамики. Вычисления проведены в калибровке Гамильтона с точностью до второго порядка теории возмущений по взаимодействию. Получена замкнутая система уравнений для бинарных корреляций электромагнитного поля и гидродинамических переменных среды и исследована около равновесия. Изучено приближение классической максвелловской плазмы. Предсказаны связанные состояния звуковых волн и волн поперечных корреляций поля. Волны корреляций электромагнитного поля могут быть возбуждены звуковыми волнами в плазме.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111015 |
| citation_txt |
Electromagnetic field correlations and sound waves / A.I. Sokolovsky, A.A. Stupka // Вопросы атомной науки и техники. — 2007. — № 3. — С. 335-339. — Бібліогр.: 7 назв. — англ. |
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| first_indexed |
2025-11-27T00:33:57Z |
| last_indexed |
2025-11-27T00:33:57Z |
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| fulltext |
ELECTROMAGNETIC FIELD CORRELATIONS
AND SOUND WAVES
A.I. Sokolovsky and A.A. Stupka
Dnipropetrovsk National University, Dnipropetrovsk, Ukraine;
e-mail: stupka_a@mail.ru
Statistical operator of the many component plasma has been found on the basis of the Bogolyubov reduced
description method and quasi-relativistic quantum electrodynamics. Calculations were carried out in the Hamilton
gauge up to the second order of a perturbation theory in interaction. Closed system of equations for binary
correlations of electromagnetic field and hydrodynamic variables of medium has been obtained and investigated
near equilibrium. Classical Maxwell plasma approximation was studied. Coupled states of sound waves and waves
of transversal correlation of the field were predicted. Waves of correlations of electromagnetic field can be excited
by sound waves in plasma.
PACS: 52.25.Dg, 52.25Gj, 52.35.Dm
1. INTRODUCTION
Recent theories of electromagnetic (ЕМ) processes,
which take into account fluctuations, are based on the
Langevin equations and are semiphenomenological [1].
Besides, usual quasi-relativistic theories consider
effective direct Coulomb interaction between charged
particles [2] and use the Coulomb gauge of the vector
potential . Our purpose is to build kinetics of
EM field in hydrodynamic medium based on the
Hamilton gauge and quasi-relativistic quantum
electrodynamics using. In the framework of this gauge
one does not need to introduce the scalar potential
and the Maxwell equations have the form of the
Hamilton equations. The medium (plasma) consists of a
few components of charged and neutral particles.
div 0A =
ϕ 0=
ϕ
Description of nonequilibrium states of the system is
based on the Bogolyubov reduced description method
(RDM) [3]. For the first time this method was applied
to the considered system in our paper [5]. In the present
paper we pay the main attention to the study of the
influence of binary correlations of the field on dynamics
of the system. Mass densities of a neutral and charged
components, mass speed and temperature of the plasma,
electric field, vector potential and their binary
correlations are chosen as variables that describe time
evolution of the system (reduced description
parameters). Therefore, in the considered model usual
plasma waves (longitudinal EM waves) are absent
because of equilibrium between particles of the
components. Besides, following to [6] we restrict
ourselves by consideration of the ideal liquid
approximation. In the framework of the RDM statistical
operator of the system is built using the Bogolyubov
condition of the complete correlation weakening.
Additional convenience in the consideration is possible
because the field variables satisfy the Peletminsky-
Yatsenko commutation condition [3].
As a small parameter of the theory ratio of the
plasma frequency Ω and Cherenkov’s frequency of
absorption of the field ( is a characteristic
equilibrium velocity) is chosen, that is equivalent to
consideration of wave vectors bounded from the bottom
by inverse Debye radius , . On the
other hand, in quasi-relativistic theory wave vectors of
the field are bounded from the above too . For
our purpose one can estimate as reverse average
interparticle distance . As a result closed
system of equations of hydrodynamics and fluctuation
electrodynamics is built within the first order of the
perturbation theory for statistical operator.
λ
Tkυ Tυ
mink k≥
maxk
1
x 0~k r−
(nA x
( ))n l tE x′
( )i t
1
min ~ Dk r−
k k≤
, )t
( ( )E x
max
(n lA
ma
( )E x
η
( ,x t)
ˆ ˆ2E−)n lE A
ˆ( )m fH+ +
{ , }n lE A=
1 2
ˆ ˆ( )V+
t
VĤ
2ˆ ( )}x Hˆ )E= + x
ˆ (n ox )nj x 3 2
2
1 ˆ
c ∫2̂ 2
= ˆ( )x A χ
ˆ
E A
2
2
ˆ ( )a
a
a
ex
m
ˆ ( )
a
χ =∑ xσ
)x
ˆ
oaπ
a
( )x
2. REDUCED DESCRIPTION
OT THE SYSTEM
We will describe our system by electric field
, vector potential , their binary
correlations , ,
(variables
( , )nE x t
( ( )n lA x
( ))x′ t
t( ))A x′ ) and densities of mass
( a is the component number), momentum
and energy of the medium (variables
). Exact definition of the correlations is given
by the relation of the type
( , )a x tσ
( , )n x tπ
( , )x tµζ
ε
ˆ ˆ
n lA2( .
In the Hamilton gauge quasi-relativistic Hamilton
operator of the system has the form
0 int
ˆ ˆ ˆH H H= + = ,
3 21ˆ { (
8fH d x
π ∫ ,
3
1
1 ˆˆ ( )V d x A
c
= − ∫ , ( )xV d ; x
ˆˆ rotH A= , 1ˆ
c
= − ; ,
ˆ ˆ( ) (a
o o
a a
ej x
m
π=∑ , (1)
where is the Hamilton operator of free medium
particles. Formulae (1) contain a non-gauge invariant
density of momentum . Gauge invariant densities
of momentum and energy can be introduced by usual
way. It is easy to check that gauge invariant mass
velocity is given by the formula
ˆ
mH
(n , )u x t
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 335-339. 335
1( , ) ( , ) ( , ) ( , )n on nu x t u x t A x t x t
c
ρ= −
( , a
a
ρ ρ=∑ a
a a
a
e
m
ρ σ= ), (2)
where is non-invariant
velocity. This allows us to use the Galilei
transformation for the medium and cast the results in a
gauge invariant form.
( , ) ( , ) / ( , )on onu x t x t x tπ σ=
Equations of motion for operators of reduced
description parameters in the terms of the gauge
invariant densities have usual form [5]
ˆˆ ˆ4t n nl l nE c A jπ∂ = ∆ −
ˆ ˆ
, ˆ ˆ
t n nA c E∂ = −
ˆ ˆˆ ˆ{ , } { , } { , }t n l nm m l lm n mE E c A E c E A′ ′ ′∂ = ∆ + ∆ ′ −
ˆ ′
′
′
ˆ ˆˆ ˆ4 ({ , } { , })n l n lj E E jπ ′ ′− +
ˆ ˆ ˆˆ ˆ ˆ ˆ{ , } { , } { , } 4 { , }t n l nm m l n l n lE A c A A c E E j Aπ′ ′ ′∂ = ∆ − − ,
ˆ ˆ ˆˆ ˆ ˆ ˆ{ , } { , } { , } 4 { , }t n l n l lm n m n lA E c E E c A A A jπ′ ′ ′ ′∂ = − + ∆ − , ˆ The last equation in (5) allows to find functions
. ( , )oY xµ ζˆ ˆ ˆ ˆˆ ˆ{ , } { , } { , }t n l n l n lA A c E A c A E′ ′∂ = − − ;
ˆ
ˆ an
t a
n
i
x
σ ∂
∂ = −
∂
,
ˆ 1 ˆˆˆ { , }
2
n
t n
n
q j E
x
ε
∂
∂ = − +
∂ n
ˆ 1 ˆˆ ˆˆˆ { , rot }
2
ln
t l l lnm n m
n
t E j
x c
π ρ ε∂
∂ = − + +
∂
A , (3)
where
nl n l nlδ∆ = ∂ ∂ − ∆ , ( ) ( ) ( ) ( )1 ˆˆ ˆ ˆn on nj x j x A x x
c
χ≡ − ,
({ , and so on; is gauge
invariant current; here ∂ is the Schrödinger time deri-
vative). Averaging equations (3) with nonequilibrium
statistical operator (SO) of the system
we obtain a closed system of equations for parameters
describing its state (it is convenient to do this using non-
gauge invariant medium variables ).
ˆ ˆ ˆ ˆ} { ( ), ( )}n l n lE E E x E x′ ≡
t
′
)
ˆ ( )nj x
ρ η
, )x t
( ) ( )( ), ot tζ
(oµζ
To construct SO for this case one can
use the RDM [3-5] starting from the Liouville equation
( ) ( )( , ot tρ η ζ
( ) ( )( ) ( ) ( )( )ˆ, [ , ,t o o
it t H t tρ η ζ ρ η ζ∂ = − ] ≡
( ) ( )( ), ot tρ η ζ≡ L . (4)
This consideration is simplified by the remark that
operators of the EM field and its correlations satisfy the
Peletminsky-Yatsenko condition . In
this work we do not introduce direct interaction among
charged particles and in the leading approximation the
medium is described by the local equilibrium
distribution of ideal gas
ˆ ˆf i ii ii
i cη η′ ′′
= − ∑L
3 ˆ( ) exp{ ( ) ( ) ( )}a o
a
w Y F Y d xY x xµζ= −∑∫ .
Using boundary condition of the complete
correlation weakening according to [3-5] we obtain the
following integral equation for statistical operator
( ), oρ η ζ
( ){0
int
0
( , ) ( ) ( ( )) ,o q o ow Y d eτρ η ζ ρ η ζ τ ρ η ζ
+∞
= + ∫ L L
( ) ( ) ( )
,
( ( )) ,o
q m o i
i i
w Y L
ρ η ζ
ρ η ζ η ζ
η
∂
+ −
∂∑L o
( )
( ) ( ),
, , }o
o
o
ie
dx M x
x µ
µ µ
τη η
δρ η ζ
η ζ
δζ −→
−∑∫ c ;
( ) ( ) int ˆ, Spi o oM η iζ ρ η ζ η≡ − , L ,
( ) ( ) (int
ˆ, , Sp , ( )o o mM x xµ µη ζ ρ η ζ ζ= − +L L )o ;
ˆSp ( , ) ( ) ( )o o ox xµ µρ η ζ ζ ζ= . (5)
Here is statistical operator of the free field,
which satisfy the Liouville equation
( )qρ η
( )
( )q
f qii i
ii i
ic
ρ η
η ρ
η ′ ′
′
∂
=
∂∑ L η . (6)
Equation (5) is solvable in perturbation theory in
small interaction. In the Baryakhtar-Peletminsky picture
in the frame of the local rest its solution has the form
[3-5]
( ) ( ) 2
0
,, , ( ( )) ( )o o
o q o nx i
i
ix w x d dx v
c
ρ η ζ ρ η ζ τ τ′
−∞
′= − ∑∫ ∫
( ) ( )( )ˆ ˆˆ{[ ( ( )), , , ( ) ]o
q o oni onw x x x x x u xjρ ζ τ ρ τη ′ ′× − + −
[ ] 2ˆˆ( ( )) ( ) ( ) Sp } ( )qo
o on q iif
i i
w x x u x O
ρ
ζ ρ ρ η λη
η ′
′ ′
∂
+ ,
∂∑ +
ˆ
a aM
τ η
nx
τ
, (7)
where
( ) ˆ( ) exp { , }o
o m a
w Hζ β β µ µ= Ω − +∑
( ). Here zero and first order contributions in
interaction are given in zero approximation in gradients
of hydrodynamic variables. The Dirac picture for
operators and was introduced in (7) by
usual formulae
1Tβ −≡
ˆ ( )nA x ˆ ( )onj x
( ) 2 ,
ˆ ˆ ˆ( , ) ( )f
n n nx i i
i
A x e A xττ ν−≡ =∑L , (8)
ˆ ˆ( , ) ( )m
on onj x e j xττ −= L , (9)
( , ) where entering first relation
Fourier components of are given by expressions
2
ˆ ˆ( )n nA x η≡ x 1
ˆ ˆ( )nE x η≡
ii ( )ν τ′
( , ) cost
nl n l nl kk k kµ τ δ ω= + ,
sin( , ) t k
nl n l nlk c k k
k
ω τν τ τ δ= − − ,
( , ; 2 1, ( ) ( , )nx lx nl x xν τ ν′ ′= − τ
2 2, ( ) ( , )nx lx nl x xν τ µ′ ′= − τ
n nk k= k , ). (10) t
nl nl n lk kδ δ= −
3. EQUATIONS OF MOTION
Let us consider averaging of relations (3) with
statistical operator (7). The first term in (7) can be
transformed with the help of formula [1,7]
336
( ) ( )
0
1
ˆˆ ˆ[ , , ] [ , , , ]o o
on on mj x w d j x H wτ β λ τ λ
−
= − ∫
0
1
ˆ ( , , ) oonj xi d wτ λβ λ
τ−
∂
= −
∂∫ (11)
( ). Integrating by parts and taking into
account, that lower limit vanishes due to the principle of
correlation weakening, we obtain instead of (7)
ˆ ˆ( ) o oA w Awλλ −≡ λ
( ) ( )
0
ˆ ˆ[ , , ] ,o o
q n q on
iw d dx A x j x
c
ρ ρ τ τ ρ τ
−∞
= + ∫ ∫ ow
( ) ( )
0 0
1
ˆ ˆ, , , o
n on qd dx d E x j x wβ τ λ τ τ λ ρ
−∞ −
+ ∫ ∫ ∫
( ) ( )
0
1
ˆ ˆ , o
n n qdx d A x j x w
c
β λ λ
−
+ ∫ ∫ ρ
( )
0
ˆˆ[ o
on q n
i u d dx w x x A x
c
ρ τ ρ ρ τ
−∞
′ ′− , − ,∫ ∫ ( , )]τ′
l
2( )O λ+ . (12) M N
Further we restrict ourselves by consideration of
classical medium. Waves of correlations will be studied
on the basis of equations of motion linearized near
equilibrium, therefore, we will drop terms, which vanish
after linearization, and only indicate them. Then
formula (12) leads to the following expression for
average gauge invariant current
0
( ) ( , ) ( , )x
n nlj x d dx I x x E xβ τ τ τ
−∞
′ ′ ′≡ − −∫ ∫
( ) 21( , 0) ( ) (x
nl l n odx I x x A x A x O u
c c
β τ χ′ ′ ′+ − = − +∫ )
ˆτ ′
)}
τ
,
ˆ ˆ( , ) Sp ( ( )) ( , ) (0)x o
nl m o on olI x w x j x jτ ζ τ′ ′≡ . (13)
According (1), (9), (10) the Dirac picture for operator
has the form ( )ˆ
nE x
0ˆ ˆ( , ) ( ) { ( , ) ( )n n nl lE x e E x dx x x E xττ µ− ′ ′= = −∫L
ˆ( , ) (nl lx x A xλ τ′ ′+ − ,
( , ) sint
nl nl kk kλ τ δ ω≡ . (14)
It the considered case correlation function in
(13) corresponds to the Maxwell plasma and their
Fourier transform can be written in the form
( , )nlI x τ
2 3( , ) 2 ( ) ( )x
nl a a n l a
a
I k e n d f kω π υυ υ υ δ ω υ= −∑ ∫ ,
( , 0) ( )x
nl nlI x xχτ δ δ
β
′ ′= = ( ∫ ) (15) 3 ( ) 1ad fυ υ =
( is the Maxwell distribution; ( )af υ
2
a a ae n m
a
χ =∑ ; here for simplicity we do not
show dependence of , , on ). This leads
to the following material equation
an β χ x
c
( ) { ( ) ( ) ( ) ( )}x x
n nl l nlj x dx M x x E x N x x A x′ ′ ′ ′= − + −∫ l ′
)′′
)
′ ′ +
)
)
)
)′ ′
)
2( )oO u+ , (16)
with material coefficients
0
( ) ( , ) ( , )x x
nl nl mlM k d I k kβ τ τ µ τ
−∞
= ∫ ,
0
( ) ( , ) ( , )x x
nl nm mlN k d I k kβ τ τ λ τ
−∞
= ∫ . (17)
Analogous to (16) calculation gives field-current
correlations
( )( ) { (x x x x x
n l lm n mA j dx M x x A E′ ′′′ ′ ′′= −∫
′ ′′ ′
( ) ( ) 2( )} (x x x x
lm n m nl oN x x A A S x x O u′ ′′ ′+ − + − + ,
( )( ) { ( )x x x x x
n l nm m lj A dx M x x E A′ ′′′ ′′= −∫
′′ ′
( ) ( ) 2( )} (x x x x
nm m l nl oN x x A A S x x O u′′ ′+ − + − + ,
( ) "( ) { (x x x x x
n l lm n mE j dx M x x E E′ ′′′ ′ ′′= −∫
( ) ( ) 2( )} (x x x x
lm n m nl oN x x E A T x x O u′ ′′ ′′ ′′ ′+ − + − + ,
( )( ) { (x x x x x
n l nm m lj E dx M x x E E′ ′′′ ′′= −∫
′′ ′
( ) ( ) 2( )} (x x x x
nm m l nl oN x x A E T x x O u′′ ′+ − + − + , (18)
where the functions , are also expressed
through ,
( )x
nlS x′
( )x
( )x
nlT x′
( )x
nl x′ x
nl
′
2
8 ( )( ) ( )x x
nl nl
T xS k N k
k
π
= − , T k (19) ( ) ( )8 ( )x x
nl nlT x M kπ= −
(in (22) and further notation of the
type ( ( are used for
binary correlations). Relations (18) are additional
material equations of the developed theory.
) ( )) ( ) ( )x x
n l t n l nE x E x E E E E′′ = = l′
tr
Material coefficients (17) determinate electromag-
netic properties of the medium. In homogenous and
isotropic media all second rank tensors can be presented
as sum of longitudinal and transversal parts
( )( ) ( ) ( )l
mn n m nm n mC k k k C k k k C kδ= + − . (20)
Using correlation function (15) and formulas (10), (14)
we obtain the expressions for nonzero components of
tensors and ( , )nlM k ω ( , )nlN k ω
, ( ) 0x lM k = ,
2
, ( ) Im ( )x tr a a
aa a
n e cM k J
m c υ+= −∑ ,
2
, ( ) Re ( )x tr a a
aa a
n e cN k J
m c υ+=∑ , (21)
where the function ( )
a
cJ υ+ is given by integral [7]
2
3 [ , ] 2( ) ( )
0a
aa
k kTd f J
m ckc k i
υυ υ υυ +=
− +∫
c ,
( 3a aT mυ ≡ ); . (22) ( ) 1J+ +∞ =
In the considered here nonrelativistic approximation
1aυ relations (21), (22) give
, ( ) 0x trM k , , 2( ) 4x trN k πΩ c (23)
( 4πχΩ = is the plasma frequency). So, introduced
material coefficients do not depend on wave vector and
temperature.
Operator equations (3) after averaging over
nonequlibrium statistical operator lead to the following
set of electrodynamic and hydrodynamic equations [5]
4t n nl l nE c A jπ∂ = ∆ − , ∂ = , t n nA c E−
( ) ( ) ( ) 4 (t n l nm m l n l n lE A c A A c E E j Aπ′ ′ ′∂ = ∆ − − )′
)′
,
( ) ( ) ( ) 4 (t n l n l lm n m n lA E c E E c A A A jπ′ ′ ′ ′∂ = − + ∆ − ,
337
( ) ( ) (t n l n l n lA A c E A c A E′ ′∂ = − − )′
)′
}
,
( ) ( ) (t n l nm m l lm n mE E c A E c E A′ ′ ′∂ = ∆ + ∆
4 {( ) ( )n l n lj E E jπ ′ ′− + ;
1( ) {( )ln
t l l l lnm n m n m
n
t E E j B j B
x c
π ρ ρ ε∂
∂ = − + + + +
∂
} ,
( )n
t n n
n
q j E j E
x
ε
∂
∂ = − + +
∂ n n , an
t a
n
i
x
σ ∂
∂ = −
∂
. (24)
Material equations (16), (18) express the right hand
sides of these equations through independent variables
of the present theory by the relations
2
, 2( ) ( )
4
x x tr
n n
xj A O
cπ
Ω
= + ou ,
2
, 2( )( ) ( ) (
4
x x x tr x
n l n l o
xj E A E O u
cπ
′ ′Ω
= + ) ,
2
, 2( )( ) ( ) (
4
x x x x tr
n l n l o
xE j E A O u
cπ
′ ′′Ω
= + ) ,
2
,( )( ) ( ) ( ) (
4
x x x tr x x
n n n l nl o
xj A A A S x x O u
cπ
′ ′Ω ′= + − , 2
The Maxwell equations from (24), (25) are absent in
(27) because in the considered approximation they are
not connected with hydrodynamic variables and
correlations of the field. They describe transverse
electromagnetic waves with dispersion law
2 2 2
f0( )kω = Ω + c k
u
. Equation for from (24),
(25) can be solved after consideration of the set (27) and
does not lead to new branches of oscillations.
( )n l kE Eδ
)+
2
,( )( ) ( ) ( ) (
4
x x x x tr x
n l n l nl o
xA j A A S x x O u
cπ
′ ′ ′′Ω ′= + − 2 )+ ;
2
2
2 ( ) ( )( )x
nl nl
T x xS k
ck
δΩ
= −
( , , ). (25) tr l
nk nk nkA A A= + tr
nk lk nlA A δ≡ nk lk n lA A k k=
4. SUBDYNAMICS
OF ELECTROMAGNETIC CORRELATIONS
OF ZERO RADIUS
In this section it will be demonstrated an influence
of correlations of the electromagnetic field on dynamics
of the system. It is convenient [6] to introduce auxiliary
field by its Fourier transform
and discuss correlations of the
electromagnetic field in the terms of , ,
. Equations (24), (25) must be linearized close to
the equilibrium. We will restrict ourselves by a zero
correlation length approximation, in which deviations of
correlations have the structure
( )nZ x
k k
lm l mZ k B=
)
trkA
)−
)−
)
k
n n nikε = −
( n lZ Z ′
( )n lE E′ ( )n lE Z ′
( ) ( ) (x x
n l n l xE E E E x xδ δ δ′ ′=
′
, ∂ =
( ) ( ) (x x
n l n l xZ E Z E x xδ δ δ ′=
′
,
( ) ( ) (x x
n l n l xZ Z Z Z x xδ δ δ ′= − (26)
(similar formulas are true for equilibrium correlations of
the field). Simple calculation leads to following closed
set of equations with full mass density as an
independent variable
t k n nik uδσ σδ∂ = − ,
22 ( )t k n nk n n kT ik w u i E Z
qc
δ δ δΩ
∂ = − + ,
( )t nk n k T ku ik Tσδ α δσ α δ∂ = − + +
2
2
2 { ( ) 4 }n l l k ki Z Zkr c
δ πΩ
+ − Tδ
)k
)k
;
( ) (t n l k n lZ Z ick E Zδ δ∂ = ,
( ) (t n l k n lE Z ick Z Zδ δ∂ = −
2
{ ( ) 2 }n l k nl ki Z Z T
ck
δ πδΩ
− − δ , (27)
where
max
min
3
1
k
Tk
d kq
kε
− = ∫ ,
max
min
3
1
4 | |
k
n
n
k
d k kk r
k kπ σ
− ′ ′
≡
′ −∫ ;
T
mσα σ
= , 1
T m
α = , 3
2T m
σε = , 2
3
Tw = . (28)
According to the mentioned above here ,
( r is the Debye radius, r is the average
interparicle distance). Equations (27) are based on non-
dissipative hydrodynamic fluxes and expressions for
energy density and pressure
1
min ~ Dk r−
1
max 0~k r−
D 0
3 2T mε σ= , p T mσ= in
ideal gas approximation.
Hydrodynamic equations (27) without correlations
of the field describe sound waves with dispersion law
(sound velocity s0 ( )k kω = 5 3u T= m
k
). Equations
for correlations (27) contain only transversal ,
parts of the field correlations. In the case of
equilibrium medium these equations describe waves
with dispersion law .
( )tr
kZZδ
( )tr
kEZδ
f0 ( )kω
5. CONNECTED SOUND AND
CORRELATION OF THE FIELD WAVES
Equations (27) in the terms of variables ,
, , , give
kδσ
l
k n nu k uδ δ≡ k kTδ ( )tr
kZZδ ( )tr
kEZδ
l
t k ik uδσ σδ∂ = − ,
22 ( )l t
t k k kT ikw u i EZ
qc
δ δ δΩ
∂ = − + r ,
2
2
2( ) { ( )l t
t k k T k k ku ik T i ZZ T
r cσδ α δσ α δ δ πδΩ
− + + − , 2 }r
2
2 2 2 2( ) { } ( )tr tr
t k k
iEZ c k ZZ i T
ck ck
πδ δ Ω
∂ = − +Ω + kδ
k
k
,
( ) ( )tr tr
t kZZ ick EZδ δ∂ = . (29)
The matrix of coefficients of this set of linear equations
for , , , 2 , 2 has
the form
kδσ l
k n nu k uδ δ≡ kTδ ( )tr
kEZδ ( )tr
kZZδ
2 2
2 2
2
2 2
0 0 0
0 4 0
0 0
40 0 0
0 0 0 0
T
ik
ik ik i i
r c r c
ikw i
q c
i i
k c k c
i kc
σ
σ
α α π
π
−
Ω Ω − − −
Ω −
Ω Ω − −
−
0
0
ikc
(30)
338
Nonzero eigenvalues of this matrix, which correspond
to frequencies of own oscillations in the system, can be
written as it follows
4 3 2 2 2 2
2
1{ ( (
2
i c k qr c kqr k u
c kqr
λ = ± + +Ω )
)
4 2)
Ω
}
2 2 2 2 3 2 24 ( ) {( (k qw r c k qr c uπ+ Ω + Ω ± +
2 2( 4 ) 4kq c r k w rπ π+ + Ω + Ω
2 3 4 3 2 2 2 24 ( ( 4 )c k qr c k qru c kq ru k wπ− + +
4 1 2 1 24 )} )T rπα / /+ Ω . (31)
In general solution of equations (30) all combinations of
the signs must be used. In the leading order in
dispersion laws is given by formula
1c−
1 2 2 2 2 2( ) 2 { ( )k k c uω −= + +Ω ±
2 2 2 2 2 2 3 1 2 1 2(( ( ) ) 16 ) }k c u w k rπ± − +Ω − Ω . (32)
In the small interaction limit field correlation and sound
frequencies (32) take the form
2
f ( )
2
k ck
ck
ω Ω
= +
2 2
4
3 2 2 3
(16 ) 16 16
8 ( )
Tc k q r qru qwk r k
c qr c u k
π π πα− + + −
+
−
σ 5( )O ΩΩ + ,
2
s 2
2( ) wk uk
c ru
πω Ω
= +
2 2 2 2 2 2 4 2
4
4 2 3 2 2 2
{ (2
( )
Tqu w k c qru w qw k r u u
c qr u c u k
π π απ − + + −
+
−
)}σ
Ω
5( )O+ Ω . (33)
In the leading approximation in c these expressions
can be written as
1−
2
4 5
f 3 3
(16 )( ) ( )
2 8
k qk ck O
ck c qk
πω Ω −
+ + Ω + Ω ,
2
s 2
2( ) wk uk
c ru
πω Ω
+ (34)
2 2 2 4 2 0
4
4 2 3 2
( )2 Tqru w qw k r u u
c qr u k
π α σπ + + −
− Ω + 5( )O Ω .
6. CONCLUSION
Obtained on the base of the Bogolyubov reduced
description method results [5], were applied to the
system of the electromagnetic field in hydrodynamic
plasma considered as an ideal gas [6,7]. A closed
system of equations (24), (25) for density, mass velocity
and temperature of plasma and field correlations was
built, which linearized near the equilibrium. The
subdynamics of correlations with zero radius was
considered. New modes of oscillations in this system
were studied (31), which correspond to coupled due to
electromagnetic interaction sound and transversal
correlation modes. The case of small interaction (33)
was considered.
This work was supported by the State Foundation
for Fundamental Research of Ukraine (project No.
2.7/418).
REFERENCES
1. Electrodynamics of Plasma. Editor A.I. Akhiezer.
M.: “Nauka”, 1974, 720 p. (in Russian).
2. D. Bohm. General Theory of Collective
Coordinates. London: “Methuen”, 1959.
3. A.I. Akhiezer, S.V. Peletminskii. Methods of
statistical physics. M.: “Nauka”, 1977, 368 p. (in
Russian).
4. M.Yu. Kovalevsky, S.V. Peletminskii. Statistical
mechanics of quantum liquids and crystals. M.:
“FizMatLit”, 2006, 368 p. (in Russian).
5. A. Sokolovsky, A. Stupka. Equations of electro-
dynamics in hydrodynamic medium with regard for
nonequilibrium fluctuations //Ukrainian Mathe-
matical Journal. 2005, v. 57, N.6, p. 1004-1019.
6. A. Sokolovsky, A. Stupka. Waves of electro-
magnetic field correlations in hydrodynamic plasma
//Proc. Of the 11th MMET Conf. 2006, p. 434-436.
7. A. Sokolovsky, A. Stupka. Linear fluctuation
electrodynamics //Journal of Physical Studies. 2006,
N1, p. 12-23 (in Ukrainian).
КОРРЕЛЯЦИИ ЭЛЕКТРОМАГНИТНОГО ПОЛЯ И ЗВУКОВЫЕ ВОЛНЫ
А.И. Соколовский, А.А. Ступка
Статистический оператор многокомпонентной плазмы найден на основе метода сокращенного описания
Боголюбова и квазирелятивистской квантовой электродинамики. Вычисления проведены в калибровке
Гамильтона с точностью до второго порядка теории возмущений по взаимодействию. Получена замкнутая
система уравнений для бинарных корреляций электромагнитного поля и гидродинамических переменных
среды и исследована около равновесия. Изучено приближение классической максвелловской плазмы.
Предсказаны связанные состояния звуковых волн и волн поперечных корреляций поля. Волны корреляций
электромагнитного поля могут быть возбуждены звуковыми волнами в плазме.
КОРЕЛЯЦІЇ ЕЛЕКТРОМАГНІТНОГО ПОЛЯ ТА ЗВУКОВІ ХВИЛІ
О.Й. Соколовський, А.А. Ступка
Статистичний оператор багатокомпонентної плазми знайдено на основі метода скороченого опису
Боголюбова та квазірелятивістської квантової електродинаміки. Обчислення проведено у калібровці
Гамільтона з точністю до другого порядку теорії збурень за взаємодією. Одержано замкнену систему
рівнянь для бінарних кореляцій поля та гідродинамічних змінних середовища і досліджено біля рівноваги.
Вивчено наближення класичної максвеллівської плазми. Передбачено зв’язані стани звукових хвиль та
хвиль поперечних кореляцій поля. Хвилі кореляцій електромагнітного поля можуть бути збуджені
звуковими хвилями у плазмі.
339
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