Classical fluctuation electrodynamics

A system consisting of an equilibrium medium formed by charged particles and electromagnetic field is considered in the classical case at weak interaction between subsystems. The field is described with all the statistical moments of electric and magnetic fields. The moments are reduced descript...

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Datum:	2005
Hauptverfasser:	Sokolovsky, A.I., Stupka, A.A.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2005
Schriftenreihe:	Condensed Matter Physics
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Zitieren:	Classical fluctuation electrodynamics / A.I. Sokolovsky, A.A. Stupka // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 685–700. — Бібліогр.: 10 назв. — англ.

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spelling	irk-123456789-1205142017-06-13T03:04:55Z Classical fluctuation electrodynamics Sokolovsky, A.I. Stupka, A.A. A system consisting of an equilibrium medium formed by charged particles and electromagnetic field is considered in the classical case at weak interaction between subsystems. The field is described with all the statistical moments of electric and magnetic fields. The moments are reduced description parameters of the herein developed theory based on the Bogolyubov reduced description method of nonequilibrium states. The generalized Bogolyubov condition of the complete correlation weakening between the subsystems is used as a boundary condition to the Liouville equation. Distribution function of the system is calculated up to the third order in electromagnetic interaction. Time equations for the reduced description parameters are written in a compact form using a generating functional for the field moments and a generating functional for field correlations (centered moments, fluctuations). The obtained equations generalize the nonlinear electrodynamics in equilibrium media for the case of fluctuations of electromagnetic field being taken into account. Cистема, яка складається з рівноважного середовища із заряджених частинок і електромагнітного поля, розглянута в класичному випадку та при малій взаємодії між підсистемами. Функцію розподілу системи розраховано з точністю до третього порядку за електромагнітною взаємодією. Поле описується усіма моментами електричного та магнітного поля, які обрані параметрами скороченого опису в рамках метода скороченого опису нерівноважних станів Боголюбова. Як гранична умова до рівняння Ліувілля застосована узагальнена умова Боголюбова повного ослаблення кореляцій між підсистемами. Одержані часові рівняння для параметрів скороченого опису записані в компактній формі за допомогою породжуючого функціоналу для моментів поля та породжуючого функціоналу для кореляцій поля (центрованих моментів, флуктуацій). Отримані рівняння узагальнюють нелінійну електродинаміку в рівноважному середовищі на випадок врахування флуктуацій електроманітного поля. 2005 Article Classical fluctuation electrodynamics / A.I. Sokolovsky, A.A. Stupka // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 685–700. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 05.20.-y, 05.40.-a, 11., 52.40.Db DOI:10.5488/CMP.8.4.685 http://dspace.nbuv.gov.ua/handle/123456789/120514 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	A system consisting of an equilibrium medium formed by charged particles and electromagnetic field is considered in the classical case at weak interaction between subsystems. The field is described with all the statistical moments of electric and magnetic fields. The moments are reduced description parameters of the herein developed theory based on the Bogolyubov reduced description method of nonequilibrium states. The generalized Bogolyubov condition of the complete correlation weakening between the subsystems is used as a boundary condition to the Liouville equation. Distribution function of the system is calculated up to the third order in electromagnetic interaction. Time equations for the reduced description parameters are written in a compact form using a generating functional for the field moments and a generating functional for field correlations (centered moments, fluctuations). The obtained equations generalize the nonlinear electrodynamics in equilibrium media for the case of fluctuations of electromagnetic field being taken into account.
format	Article
author	Sokolovsky, A.I. Stupka, A.A.
spellingShingle	Sokolovsky, A.I. Stupka, A.A. Classical fluctuation electrodynamics Condensed Matter Physics
author_facet	Sokolovsky, A.I. Stupka, A.A.
author_sort	Sokolovsky, A.I.
title	Classical fluctuation electrodynamics
title_short	Classical fluctuation electrodynamics
title_full	Classical fluctuation electrodynamics
title_fullStr	Classical fluctuation electrodynamics
title_full_unstemmed	Classical fluctuation electrodynamics
title_sort	classical fluctuation electrodynamics
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2005
url	http://dspace.nbuv.gov.ua/handle/123456789/120514
citation_txt	Classical fluctuation electrodynamics / A.I. Sokolovsky, A.A. Stupka // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 685–700. — Бібліогр.: 10 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT sokolovskyai classicalfluctuationelectrodynamics AT stupkaaa classicalfluctuationelectrodynamics
first_indexed	2025-07-08T18:00:33Z
last_indexed	2025-07-08T18:00:33Z
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fulltext	Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 685–700 Classical fluctuation electrodynamics A.I.Sokolovsky, A.A.Stupka Dnipropetrovs’k National University, 13 Naukova Str., 49050 Dnipropetrovs’k, Ukraine Received August 3, 2005, in final form November 8, 2005 A system consisting of an equilibrium medium formed by charged parti- cles and electromagnetic field is considered in the classical case at weak interaction between subsystems. The field is described with all the statis- tical moments of electric and magnetic fields. The moments are reduced description parameters of the herein developed theory based on the Bo- golyubov reduced description method of nonequilibrium states. The gen- eralized Bogolyubov condition of the complete correlation weakening be- tween the subsystems is used as a boundary condition to the Liouville equation. Distribution function of the system is calculated up to the third or- der in electromagnetic interaction. Time equations for the reduced descrip- tion parameters are written in a compact form using a generating function- al for the field moments and a generating functional for field correlations (centered moments, fluctuations). The obtained equations generalize the nonlinear electrodynamics in equilibrium media for the case of fluctuations of electromagnetic field being taken into account. Key words: Bogolyubov reduced description method, complete correlation weakening, equilibrium medium, fluctuation electrodynamics, generating functional PACS: 05.20.-y, 05.40.-a, 11., 52.40.Db 1. Introduction Almost all the observed phenomena are connected with electromagnetic interac- tions. Therefore, deep understanding of electromagnetic phenomena and their ade- quate description will always attract the theoretical thought. The rapid development of quantum optics with special attention to new states of electromagnetic field and the advances of plasma physics have urged the researchers to take into account hi- gher correlation functions of the field (see, for example, [1]). The idea to consolidate the modern directions of nonlinear, stochastic, and fluctuation electrodynamics in the media in a more general theory taking into account all correlations and having a microscopic basis seems to be an important goal. Substantial steps have been made in this way by Klimontovich [2]. However, he studied only binary correlations of the field. Balescu and other representatives of the Brussels school have applied a power- c© A.I.Sokolovsky, A.A.Stupka 685 A.I.Sokolovsky, A.A.Stupka ful diagram technique to generalize the equations of nonlinear electrodynamics, but they did not study the own degrees of freedom of the field in detail [3]. Phenomeno- logical nonlinear electrodynamics is based on the idea of expanding the average electric current in powers of small electric and magnetic fields. The coefficients of this expansion (generalized conductivities) remain indefinite in this approach. It is obvious that such material equations do not take field correlations into account. In the present paper an electromagnetic field in equilibrium medium is described with all the moments of electric and magnetic fields based on the Bogolyubov method of reduced description of nonequilibrium processes [4,5]. The analogous problem for hydrodynamics was considered in [6]. This way the correlation contributions to mate- rial equations are studied and microscopic expressions for generalized conductivities are obtained (about these problems see [7]). 2. Construction of the system distribution function We consider an electromagnetic field (f-subsystem) interacting with charged and neutral particles (m-subsystem, medium). The Hamilton function of the system f + m in a quasi-relativistic approximation takes the form Ĥ = ∑ i (pi − ei c Â(xi)) 2 2mi + Ĥf + Ĥint , Ĥf = 1 8π ∫ d3x(Ê2 + B̂2); B̂(x) ≡ rotÂ(x), Ê(x) = −4πcP̂ (x). (1) Here Ĥf is free electromagnetic field contribution, Ĥint corresponds to a direct inter- action between particles which is additional to the interaction via the field. There- fore, we have Ĥ = Ĥf + Ĥm + Ĥmf , Ĥm ≡ ∑ i p2 i 2mi + Ĥint, Ĥmf = Ĥ1 + Ĥ2, Ĥ1 ≡ − 1 c ∫ dxÂn(x)ĵn(x), Ĥ2 ≡ 1 2c2 ∫ dxχ̂(x)Â(x)2, ĵn(x) ≡ ∑ i pin mi eiδ(x − xi), χ̂(x) ≡ ∑ i e2 i mi δ(x − xi). (2) Here Ĥmf is the interaction between electromagnetic field and particles, Ĥm is the Hamilton function of particles. We use the Hamilton gauge of electromagnetic field with ϕ = 0 and consider vector potential Ân(x) as a generalized coordinate of the field. P̂n(x) is a corresponding generalized momentum. Hereinafter we mark the functions of the system phase variables with a cap (except the coordinates and momenta of particles). We also assume that some values are vectors but we reflect this in notation only someplace. In this paper the Hamilton technique with the 686 Classical fluctuation electrodynamics following definition of the Poisson brackets is used {f̂ , ĝ} = ∑ i ( ∂f̂ ∂xin ∂ĝ ∂pin − ∂f̂ ∂pin ∂ĝ ∂xin ) + ∫ d3x ( δf̂ δÂn(x) δĝ δP̂n(x) − δf̂ δP̂n(x) δĝ δÂn(x) ) , (3) (f̂ , ĝ denote arbitrary functions of phase variables). Therefore, the following formulae for the Poisson brackets of the vector potential and electric field {Ân(x), Âl(x ′)}=0, {Ên(x), Âl(x ′)} = 4πc δnlδ(x − x′), {Ên(x), Êl(x ′)}=0 (4) are true. The relation of electrodynamics Ên(x) = − 1 c ˆ̇An(x), (5) ( ˆ̇f ≡ {f̂ , Ĥ}) is obviously true, too. Distribution function ρ(t) of the phase variables of the system satisfies the Liouville equation ∂tρ (t) = Lρ(t); Lρ(t) ≡ {Ĥ, ρ(t)}, L = L0 + Lmf , L0 ≡ Lm + Lf , Lmf = L1 + L2 , (6) where contributions to the Liouville operator correspond to contributions to the Hamilton function. We will study the states of the system for which relations Ĥf ∼ Ĥm, Ĥf � Ĥ1, Ĥ2; B(x) ∼ E(x) (7) are valid. Formulae (1), (2) give the following estimates Ĥm ∼ nV T, Ĥf ∼ V E2; E ∼ kA, ω ∼ kc; ĵ ∼ enυT , χ̂ ∼ e2n m ; Ĥ1 ∼ nV T Ω ω , Ĥ2 ∼ nV T Ω2 ω2 . (8) Here n, T are density and temperature of the medium, ω, k are frequency and wave number of the field, V is volume of the system f + m. Characteristic velocity υT and plasma frequency Ω are given by the expression υT = √ 3T m , Ω = √ 4πe2n m , (9) (m is the electron mass and e is the module of its charge). Therefore, such relations Lm ∼ ω, ∼ Lf ∼ ω, L1 ∼ Ω, L2 ∼ Ω2 ω (10) 687 A.I.Sokolovsky, A.A.Stupka are valid (ωm is a characteristic frequency of medium processes). This allows us to study the system in a perturbation theory in g with estimates Lm ∼ g0, Lf ∼ ω ωm g0, L1 ∼ g1, L2 ∼ ωm ω g2; g ≡ Ω/ωm. (11) Let us describe the electromagnetic field in the equilibrium medium by all the moments of electric Ên(x) and magnetic B̂n (x) fields. In order to simplify the con- sideration, we will use cumulative indices a (b, c), µ and cumulative notation for electromagnetic field a ≡ (µ1, . . . , µs), µ ≡ (i, n, x); ξ̂µ ≡ ξ̂in(x) : ξ̂1n(x) = B̂n(x), ξ̂2n(x) = Ên(x); η̂a ≡ η̂µ1...µs = ξ̂µ1 · · · · · ξ̂µs (1 6 s 6 ∞); ∑ a ≡ ∑ µ1...µs , ∑ µ ≡ ∑ i,n ∫ dx; ∂ ∂ξµ ≡ δ δξin(x) . (12) In a general case the mentioned description of the field is possible at long times t � τ0 where a characteristic time τ0 depends on the initial state ρ0 of the system. Our consideration is based on the Bogolyubov idea of the functional hypothesis [4,5] ρ(t)−−−→ t�τ0 ρ(η(t, ρ0)), (13) where the reduced description parameters (RDP) ηa(t, ρ0) are defined by the formula Spρ(t)η̂a −−−→ t�τ0 ηa(t, ρ0), ρ0 ≡ ρ(t = 0), (14) (Sp denotes the integration over phase variables). The distribution function of the system ρ(η) in (13) does not depend on its initial state ρ0 and time t. For the system under consideration the following relations L0ξ̂µ = −i ∑ µ′ cµµ′ ξ̂µ′ , L0η̂a = −i ∑ b cabη̂b (15) are true because they are a matrix form of the microscopic Maxwell equations for a free electromagnetic field L0Ê(x) = −c rotB̂(x), L0B̂(x) = c rotÊ(x). (16) Non-zero elements of matrix cµµ′ are cµµ′ ≡ cin,i′n′(x, x′); c1n,2l(x, x′) = ic εnml ∂δ(x − x′) xm , c1n,2l(x, x′) = −ic εnml ∂δ(x − x′) xm . (17) 688 Classical fluctuation electrodynamics Matrix cab has only diagonal elements and, for example, cµ1µ2,µ′ 1 µ′ 2 = cµ1µ′ 1 δµ2µ′ 2 + δµ1µ′ 1 cµ2µ′ 2 . (18) We call a nonequilibrium theory with the property (15) of η̂a the Peletminskii- Yatsenko model [5]. The RDP’s ηa(t, ρ0) satisfy the following time equation ∂tηa(t, ρ0) = La(η(t, ρ0)), La(η) = i ∑ b cabηb + L̃a(η), L̃a(η) ≡ Spη̂aLmfρ(η). (19) For the system distribution function ρ(η) we have the following equations Lρ(η) = ∑ a ∂ρ(η) ∂ηa La(η), Spρ(η)η̂a = ηa. (20) According to Bogolyubov [4], to find their unique solution one has to add to equati- ons (20) a boundary condition. We shall use the boundary condition of the complete correlation weakening eτ L0ρ(η)−−−→ t�τ0 eτ L0wρq(η), ( w ≡ e F−Ĥm T ) , (21) where w is the Gibbs distribution for the equilibrium medium with temperature T , ρq(η) is a quasi-equilibrium distribution function of the field ρq(η) = exp{Ω(η) − ∑ a Za(η)η̂a} , Spfρq(η)η̂a = ηa, Spfρ(η) = 1, (22) (Spf denotes the integration over phase space of the f-subsystem). The second and the third formulae here yield the function Za(η) and Ω(η). The distribution function ρq(η) satisfies the Liouville equation for free subsystem f Lfρq(η) = ∑ a,b ∂ρq(η) ∂ηa icabηb , (23) (see [5]) and has the following property eτL0ρq(η) = ρq(e icτη). (24) The boundary condition means that the free evolution of the system breaks all the spatial correlations between particles and field because the distance between these subsystems increases. The idea of spatial correlation weakening by means of the free evolution operator eτL0 belongs to Bogolyubov [4]. Using the free evolution operator in the boundary condition does not mean that it is correct only approximately. In fact here we do not need a concrete expression for distribution function ρq(η) due to the of relation η̂aη̂b = η̂a∪b and the second formula in (22). 689 A.I.Sokolovsky, A.A.Stupka It is convenient to rewrite the boundary condition (14) as follows: lim τ→+∞ eτ L0ρ(e−i cτη) = wρq(η) (25) proceeding from the formula (24). Using the standard procedure [5] we get the following integral equation for the system distribution function ρ(η) ρ(η) = ρq(η)w + +∞ ∫ 0 dτeτ L0 ( Lmfρ(η) − ∑ a ∂ρ(η) ∂ηa L̃a(η) ) η→e−i cτ η . (26) For the case of a closed system this equation was first obtained by Peletminskii and Yatsenko [5]. The obtained equation is solvable in a perturbation theory in small parameter g but it is not very convenient to study high order approximations with it. Therefore, let us introduce an auxiliary distribution function ρ(η, τ) = e−τ L0ρ(ei cτη). (27) Simple considerations show that it satisfies the following integral equation ρ(η, τ) = ρq(η)w + τ ∫ −∞ dτ ′ ( Lmf(τ ′)ρ(η, τ ′) − ∑ a ∂ρ(η, τ ′) ∂ηa L̃a(η, τ ′) ) , (28) where L̃a(η, τ) = Spη̂aLmf(τ)ρ(η, τ); Lmf(τ)f̂ = {Ĥmf(τ), f̂}, f̂(τ) = e−τ L0 f̂ . (29) f̂ denotes an arbitrary function of the phase variables; therefore, f̂(τ) gives the corresponding value at the time moment τ after free evolution. This equation is also solvable in the perturbation theory at a small parameter g introduced in (11) ρ(η, τ) = ∞ ∑ s=0 ρ(s)(η, τ), ρ(s)(η, τ) ≡ wRs(η, τ), Rs(η, τ) ∼ gs, R0(η, τ) = %q(η)w. (30) Moreover, in this paper we restrict ourselves to the consideration of the nonrelati- vistic approximation taking into account that the value υT /c is an additional small parameter. We shall also assume that the average equilibrium current is equal to zero Spmwĵn(x) = 0, (31) (Spm denotes the integration over phase space of m-subsystem; SpÂ = SpmSpfÂ). Let us note that contributions Ĥ1 and Ĥ2 to Ĥmf are of the order 1/c, 1/c2 respectively. There are two origins of c powers in numerators of the considered expressions for ρ(η, τ). The Hamilton functions Ĥ1 , Ĥ2 contribute to ρ(η, τ) through 690 Classical fluctuation electrodynamics the Poisson brackets. One Poisson bracket can give only c1 in a numerator because formulae {Ân(x), Êl(x ′)} = −4πcδnlδ(x − x′), ∂tÂn(x, t) = −cÊn(x, t). (32) Therefore, we have to omit a Ĥ2 contribution to the Hamilton function Ĥmf in the considered approximation. Taking into account (32), we can rewrite our typical expression in the perturbation theory as τ ∫ −∞ dτ{Ĥ1(τ ′), ρ(s)(η, τ ′)}f(τ ′) = 1 c τ ∫∫ −∞ dx{wRs(η, τ ′), ĵn(x, τ ′)Ân(x, τ ′)}f(τ ′) = 1 c τ ∫ −∞ dτ ′ ∫ dxw ( − 1 T {Ĥm, ĵn}RsÂn + {Rs, jn}Ân + {Rs, Ân}ĵn ) f(τ ′) = τ ∫ −∞ dτ ′ ∫ dxw M(x, τ ′)Rs(η, τ ′)f(τ ′) + O (υT /c). (33) Here f(τ) is an arbitrary function. A useful operator in the phase space M(x, t)f̂ ≡ ĵn(x, t) ( 1 c {f̂ , Ân(x, t)} + 1 T Ên(x, t)f̂ ) (34) was introduced (we denote operators in the phase space with bold letters). Applying the mentioned ideas, we obtain the following expressions for functions Rs(η, τ) defined by (30) R1(η, τ) = τ ∫ −∞ dτ ′ ∫ dxM(x, τ ′)ρq(η), R2(η, τ) = τ ∫ −∞ dτ ′ ∫ dx ( M(x, τ ′)R1(η, τ ′) − ∑ a ∂ρq ∂ηa Spη̂awM(x, τ ′)R1(η, τ ′) ) , R3(η, τ) = τ ∫ −∞ dτ ′ ∫ dx ( M(x, τ ′)R2(η, τ ′) − ∑ a ∂ρq ∂ηa Spη̂awM(x, τ ′)R2(η, τ ′) − ∑ a ∂R1(η, τ ′) ∂ηa Spη̂awM(x, τ ′)R1(η, τ ′) ) , (35) (for simplicity we omit here and further the estimations O(υT /c)). 691 A.I.Sokolovsky, A.A.Stupka 3. Equations for the generating functional In order to make the obtained result more visible, let us introduce the generating functional (GF) F (η, u) for all the moments of electromagnetic field [6] F (η, u) = 1 + ∞ ∑ s=1 1 s! ∑ µ1...µs uµ1 . . . uµs ηµ1...µs , (36) where uµ ≡ uin(x) are auxiliary functions. Formulae (12), (20), (22) show that the following relations F̂ (u) ≡ F (η̂, u) = ∑ a ∂F (η, u) ∂ηa η̂a = exp ∑ µ uµξ̂µ ; F (η, u) = Spρ(η)F̂ (u) = Spρq(η)F̂ (u) (37) are true. It is easy to derive a closed time equation for F (η(t), u) (we omit ρ0 from ηa(t, ρ0) for simplicity). Statistical operator ρ(η(t)) satisfies the Liouville equation (6) and therefore we have ∂tF (η (t) , u) = SpF̂ (u)Lρ(η(t)) = ∑ µ uµSpρ(η(t))F̂ (u) ˆ̇ξµ . (38) Value ˆ̇ξµ = −Lξ̂µ is given by the microscopic Maxwell equations ˆ̇ξµ = i ∑ µ′ cµµ′ ξ̂µ′ − 4πĵµ − L2ξ̂µ; ĵµ ≡ ĵin(x) = δi2ĵn(x), (39) (see (6), (15), (16)). Here the term L2ξ̂µ ∼ 1/c must be omitted in the considered approximation. Note that the calculation of the trace Spm over the phase space of the medium gives a formula Spρ(η)F̂ (u)ĵn(x) ≡ Spfρq(η)F̂ (u)In(x, u, ξ̂) (40) with a certain function of a microscopic electromagnetic field In(x, ξ̂). Therefore, using (38)–(40), we have ∂tF (η (t) , u) = ∑ µ uµSpρq(η(t))F̂ (u)Lµ(u, ξ̂), (41) where the notation Lµ(u, ξ) = i ∑ µ′ cµµ′ξµ′ − 4πIµ(u, ξ), Iµ(u, ξ) ≡ Iin(x, u, ξ) = δi2In(x, u, ξ) (42) 692 Classical fluctuation electrodynamics is introduced. The RHS of equation (41) for F (η(t), u) can be expressed through F (η(t), u) using the formula F̂ (u)f(ξ̂) = F̂ (u + ∂ ∂ξ )f(ξ) ∣ ∣ ∣ ∣ ξ→0 (43) which follows from (37). Really, (37), (43) give the necessary closed equation for the generating functional F (η(t), u) ∂tF (η(t), u) = F (η(t), u + ∂ ∂ξ ) ∑ µ uµLµ(u, ξ) ∣ ∣ ∣ ∣ ∣ ξ→0 . (44) This equation is the equation of fluctuation electrodynamics (FED) in the equilib- rium medium, i.e. the equation for all the moments of electromagnetic field. Instead of the GF for moments F (η, u) a GF G(g, u) for correlations (fluctuations, centered moments) ga ≡ gµ1...µs of the field can be introduced [6] so that G(g, u) = ∞ ∑ s=2 1 s! ∑ µ1...µs uµ1 . . . uµs gµ1...µs , F (η, u) = exp ( ∑ µ uµξµ + G(g, u) ) .(45) Simple calculation based on the formula F (η(t), u + ∂ ∂ξ )f(ξ) ∣ ∣ ∣ ∣ ξ→0 = e ∑ µ ξµ(t)uµ eG(g(t),u+ ∂ ∂ξ )f(ξ) ∣ ∣ ∣ ∣ ξ→ξ(t) (46) gives the following equations for an average value of electromagnetic field and its correlations ∂tξµ(t) = eG(g, ∂ ∂ξ )Lµ(u = 0, ξ) ∣ ∣ ∣ ξ→ξ(t), g→g(t) , ∂tG(g(t), u) = ∑ µ uµ { eG(g,u+ ∂ ∂ξ )−G(g,u)Lµ(u, ξ) − eG(g, ∂ ∂ξ )Lµ(u = 0, ξ) } ξ→ξ(t), g→g(t) (47) (compare with [6]). Contribution to (47) from linear in ξ̂µ terms from (42) (con- tribution of free electromagnetic field) can be easily calculated using a method of differential equation. Really, function fµ(λ) ≡ eλG(g,u+ ∂ ∂ξ )ξµ (48) has a derivative ḟµ(λ) ≡ eλG(g,u+ ∂ ∂ξ ) ( G(g, u) + ∞ ∑ s=1 1 s! ∑ µ1...µs Gµ1...µs (g, u) ∂s ∂ξµ1 . . . ∂ξµs ) ξµ , where Gµ1...µs (g, u) = ∂sG(g, u) ∂uµ1 . . . ∂uµs . (49) 693 A.I.Sokolovsky, A.A.Stupka Therefore, it satisfies the following differential equation ḟµ(λ) = G(g, u)fµ(λ) + Gµ(g, u) eλG(g,u), fµ(0) = ξµ . (50) Solution of this equation leads to identity eG(g,u+ ∂ ∂ξ )−G(g,u)ξµ = ξµ + Gµ(g, u). (51) So, equation (47) can be rewritten in the form ∂tξµ(t) = i ∑ µ′ cµµ′ξµ′(t) − 4πjµ(ξ(t), g(t)), ∂tG(g(t), u) = i ∑ µµ′ cµµ′Gµ′(g(t), u) + ∑ µ uµ { eG(g,u+ ∂ ∂ξ )−G(g,u)Iµ(u, ξ) − eG(g, ∂ ∂ξ )Iµ(u = 0, ξ) } ξ→ξ(t), g→g(t) , (52) where Iµ(u, ξ) is expressed through In(x, u, ξ) by the formula (42) and jµ(ξ, g) ≡ jin(x, ξ, g) = δi2jn(x, ξ, g), jn(x, ξ, g) = eG(g, ∂ ∂ξ )In(x, u = 0, ξ). (53) According to (12), (15), (16) the first formula (52) gives the Maxwell equations; therefore, jn(x, ξ, g) is the average electric current in the system. Equations (52) are equations of the FED in the terms of average electromagnetic field ξµ(t) and its correlations (fluctuations, centered moments) gµ1...µs (t). These equations are completely defined by function In(x, u, ξ) which is introduced by the formula (40). Therefore, the last step of our investigation is to calculate this function. A typical value to be calculated is SpwF̂ ĵn(x)M(x′, τ)Rs = Spwĵn(x)ĵl(x ′, τ) × Rs ( 1 c {Âl(x ′, τ), F̂} + 1 T Êl(x ′, τ)F̂ ) = 1 T Spwĵn(x)ĵl(x ′, τ)Rsαl(x ′, τ, ξ̂ + 4πTu)F̂ , (54) where αn(x, τ, u) ≡ ∑ i ∫ dx′θinl(x − x′, τ)uil(x ′), θ1nl(x, t) ≡ λnl(x, t), θ2nl(x, t) ≡ µnl(x, t); λnl(k, t) = iεnlmk̃m sin ωkt, µnl(k, t) = k̃nk̃l + δ̃nl cos ωkt, k̃n ≡ kn/k, δ̃nl ≡ δnl − k̃nk̃l . (55) 694 Classical fluctuation electrodynamics Relations (54) also contain the solutions of equations for free electromagnetic field Ân(x, t) = e−tL0Ân(x) = ∫ dx′ ( µnl(x − x′, t)Âl(x ′) + νnl(x − x′, t)Êl(x ′) ) , Ên(x, t) = e−tL0Ên(x) = ∫ dx′ ( λnl(x − x′, t)B̂l(x ′) + µnl(x − x′, t)Êl(x ′) ) , νnl(k, t) = −k̃nk̃lct − δ̃nl sin ωkt k (56) which have the following properties Ên(x, t) = αn(x, t, ξ̂), {Ân(x, t), F̂ (u)} = 4πc αn(x, t, u)F̂ (u), {Ân(x, t), ξ̂il(x ′, t′)} = 4πc θinl(x − x′, t − t′). (57) From the obtained results one can see that In(x, u, ξ) = In(x, u = 0, ξ + 4πTu), (58) where function In(x, u = 0, ξ) is given by formulae: In(x, u = 0, ξ) = ∞ ∑ s=2 I(s) n (x, u = 0, ξ), I(2) n (x, u = 0, ξ) = 1 T 0 ∫ −∞ dτ1 ∫ dx1Inl(x1 − x, τ1) αl(x1, τ1, ξ), I(3) n (x, u = 0, ξ) = 1 T 2 0 ∫ −∞ dτ1 ∫ dx1 τ1 ∫ −∞ dτ2 ∫ dx2Inlm(x1 − x, τ1; x2 − x, τ2) × [αl(x1, τ1, ξ)αm(x2, τ2, ξ) + 4πTµlm(x1 − x2, τ1 − τ2)], I(4) n (x, u = 0, ξ) = 1 T 3 0 ∫ −∞ dτ1 ∫ dx1 τ1 ∫ −∞ dτ2 ∫ dx2 τ2 ∫ −∞ dτ3 ∫ dx3 × Inlms(x1 − x, τ1; x2 − x, τ2; x3 − x, τ3)[αl(x1, τ1, ξ)αm(x2, τ2, ξ)αs(x3, τ3, ξ) + 4πTαm(x2, τ2, ξ)µls(x1 − x3, τ1 − τ3) + 4πTαl(x1, τ1, ξ)µms(x2 − x3, τ2 − τ3) +4πTαs(x3, τ3, ξ)µlm(x1 − x2, τ1 − τ2)]. (59) Here correlation functions of currents ĵn(x, t) = e−tL0 ĵn(x) Inl(x1, τ1) = Spmwĵn(0)ĵl(x1, τ1), Inlm(x1, τ1; x2, τ2) = Spmwĵn(0)ĵl(x1, τ1)ĵm(x2, τ2), Inlsm(x1, τ1; x2, τ2; x3, τ3) = Spmwĵn(0)ĵl(x1, τ1)ĵm(x2, τ2)ĵs(x3, τ3) − Spmwĵn(0)ĵl(x1, τ1)Spmwĵm(x2, τ2)ĵs(x3, τ3) − Spmwĵn(0)ĵm(x2, τ2)Spmwĵl(x1, τ1)ĵs(x3, τ3) − Spmwĵn(0)ĵs(x3, τ3)Spmwĵl(x1, τ1)ĵm(x2, τ2) (60) 695 A.I.Sokolovsky, A.A.Stupka are introduced. In the considered approximation (59) the function Iµ(u = 0, ξ) is given by the formula Iµ(u = 0, ξ) = Sµ + ∑ µ1 σµ,µ1 ξµ1 + ∑ µ1µ2 σµ,µ1µ2 ξµ1 ξµ2 + ∑ µ1µ2µ3 σµ,µ1µ2µ3 ξµ1 ξµ2 ξµ3 + O(g5), (61) where according to (12) and (42) the values Sµ, σµ,µ1 , σµ,µ1µ2 , σµ,µ1µ2µ3 have the struc- ture Sµ ≡ Sin(x) = δi2Sn, σµµ1 ≡ σin,i1n1 (x, x1) = δi2σn,i1n1 (x − x1), σµ,µ1µ2 ≡ σin,i1n1i2n2 (x, x1, x2) = δi2 σn,i1n1i2n2 (x − x1, x − x2), . . . . (62) For all of them we obtain concrete expressions Sn = 4π T 0 ∫ −∞ dτ ∫ dx τ ∫ −∞ dτ ′ ∫ dx′Inlm(x, τ ; x′, τ ′)µlm(x − x′, τ − τ ′) + O(g5), σn,il(x − x′) = 1 T 0 ∫ −∞ dτ ∫ dx′′Inm(x − x′′, τ) λiml(x ′′ − x′, τ) + O(g4), . . . (63) Formulae (58), (61) make it possible to calculate the values entering the equations of FED (52). This can be done using the differential equation method (see (50)) for functions fµ1...µs (λ) = eλG(g,u+ ∂ ∂ξ )ξµ1 . . . ξµs (64) which leads to a chain of equations of the first order. This chain can be solved successively starting from equation (50) for fµ1 (λ). This way we obtain eG(g,u+ ∂ ∂ξ )−G(g,u)ξ1ξ2 = ξ1ξ2 + G12 + ξ1G2 + ξ2G1 + G1G2 eG(g,u+ ∂ ∂ξ )−G(g,u)ξ1ξ2ξ3 = ξ1ξ2ξ3 + G123 + [ G12( 1 2 G3 + ξ3) + c.p. ] + [ G3(ξ1ξ2 + 1 2 G12 + 1 2 ξ1G2 + 1 2 ξ2G1 + 1 3 G1G2) + c.p. ] ,(65) where for simplicity a highly reduced notation ξi ≡ ξµi , G1...s ≡ Gµ1...µs (66) is used. We will not substitute expressions (65) into equations (52) completely, and re- strict ourselves to a final formula for average current in the system jµ(ξ, g) jµ(ξ, g) = Sµ + ∑ µ1 σµ,µ1 ξµ1 + ∑ µ1µ2 σµ,µ1µ2 (gµ1µ2 + ξµ1 ξµ2 ) + ∑ µ1µ2µ3 σµ,µ1µ2µ3 (3gµ1µ2 ξµ3 + gµ1µ2µ3 + ξµ1 ξµ2 ξµ3 ) + O(g5). (67) 696 Classical fluctuation electrodynamics This formula is a material equation for the FED. Values σµ,µ1 , σµ,µ1µ2 , σµ,µ1µ2µ3 can be called generalized conductivities. According to (62) they take into account a spatial dispersion. In equilibrium, the electric current in the considered system must vanish jµ(ξ = 0, geq) = 0, (ξeq = 0). (68) Therefore, the material equation (67) can be rewritten in the form jµ(ξ, g) = ∑ µ1 σµ,µ1 ξµ1 + ∑ µ1µ2 σµ,µ1µ2 (δgµ1µ2 + ξµ1 ξµ2 ) + ∑ µ1µ2µ3 σµ,µ1µ2µ3 (3gµ1µ2 ξµ3 + δgµ1µ2µ3 + ξµ1 ξµ2 ξµ3 ) + O(g5), (69) where δgµ1...µs = gµ1...µs − geq µ1...µs . (70) Let us compare the developed theory with the theories having a truncated set of nonequilibrium correlations as independent variables. In usual nonlinear electro- dynamics (NED) correlations gµ1...µs should be functions of electromagnetic field gµ1...µs (ξ) and a function j0µ(ξ) ≡ jµ(ξ, g2(ξ), g3(ξ) . . . ) is material equation of the NED (we use a more detailed notation jµ(ξ, g2, g3, . . . ) ≡ jµ(ξ, g)). Time equations for all correlations (52) should be satisfied with functions gµ1...µs (ξ(t)) (2 6 s < ∞). In FED taking into account only binary correlations gµ1µ2 (FED2) triple and more complicated correlations should be functions of electromagnetic field and binary correlations gµ1...µs (ξ, g2) (3 6 s < ∞ ). The material equation of the FED2 in the terms of function (67) is given by formula j1µ(ξ, g2) ≡ jµ(ξ, g2, g3(ξ, g2), . . . ). Time equations for triple and more complicated correlations (52) should be satisfied with functions gµ1...µs (ξ, g2) (3 6 s < ∞). Let us truncate the equations (47) of the FED taking into account only binary correlations and construct the corresponding equations of the FED2. Note that the FED2 which describes the electromagnetic field with ξµ and gµ1µ2 can be built based on our equations (19), (22), (26) with operator of RDP’s η̂a= ξ̂µ, ξ̂µ1 ξ̂µ2 (see [8,9]). In this case ρq is a Gauss distribution ρq = e Ω− ∑ µ Zµξ̂µ− ∑ µµ′ Zµµ′ ξ̂µξ̂µ′ (71) and averages Spρq ξ̂µ1 . . . ξ̂µs can be calculated in the terms of ξµ and gµµ′ using the generating functional Fq(u) = SpfρqF̂ (u) = e ∑ µ uµξµ+ 1 2 ∑ µµ′ uµuµ′gµµ′ . (72) In order to compare FED and FED2 we need gµ1...µs (ξ, g2) which can be obtained with a generating functional method. According to (30), (31), (37), (45) and using 697 A.I.Sokolovsky, A.A.Stupka the reduced notation (66) we have G(u) = ln F (u) = ln Spρ(η)F̂ (u) = ln Sp ( ρqw + ρ(1) + ρ(2) + ρ(3) + O(g4) ) F̂ (u) = ln Fq(u) + Fq(u)−1Sp ( ρ(2) + ρ(3) ) F̂ (u) + O(g4) = ln Fq(u) + [ 1 3! ∑ 123 u1u2u3Sp ( ρ(2) + ρ(3) ) ξ̂1ξ̂2ξ̂3 + 1 4! ∑ 1234 u1u2u3u4Sp ( ρ(2) + ρ(3) ) × ξ̂1ξ̂2ξ̂3ξ̂4 + O(g2u5, g3u5, g4) ][ 1 + ∑ 5 u5ξ5 + O(u2) ]−1 , ( ∑ i ≡ ∑ µi ) because Spρ(s)ξ̂1 = 0, Spρ(s)ξ̂1ξ̂2 = 0 for s > 1 (see (20), (22)). This relation leads to the following expressions for correlations g123(ξ, g2) = Sp(ρ(2) + ρ(3))ξ̂1ξ̂2ξ̂3 + O(g4) ∼ g2, g1234(ξ, g2) = Sp(ρ(2) + ρ(3)) [ ξ̂1ξ̂2ξ̂3ξ̂4 − ( ξ̂1ξ̂2ξ̂3ξ4 + c.p. )] + O(g4) ∼ g2. (73) Taking into account these estimates and formula (69) we obtain an average current for FED2 j1µ(ξ, g2) = ∑ µ1 σµ,µ1 ξµ1 + ∑ µ1,µ2 σµ,µ1µ2 (ξµ1 ξµ2 + δgµ1µ2 ) + ∑ µ1,µ2,µ3 σµ,µ1µ2µ3 (ξµ1 ξµ2 + 3gµ1µ2 ) ξµ3 + O(g5) (74) and the averaged electromagnetic field ξµ(t) satisfies the Maxwell equation ∂tξµ = i ∑ µ′ cµµ′ξµ′ − 4πj1µ(ξ, g2), (75) (see (52)). According to (49), (52), (58) time equation of the FED for the binary correlations has the form ∂tgµµ′ = i ∑ µ′′ cµµ′′gµ′′µ′ − 4πeG(g, ∂ ∂ξ ) ( Gµ(g, ∂ ∂ξ ) + 4πT ∂ ∂ξµ ) × Iµ′(u = 0, ξ) + (µ ↔ µ′)). (76) Now using formulae (45), (49), (61) and estimates (73) we obtain the following equation of the FED2 for binary correlations ∂tgµµ′ = i ∑ µ′′ cµµ′′gµ′′µ′ − 4π ∑ µ′′ (gµ′′µ′ + 4πTδµ′′µ′) × ( σµ,µ′′ + 2 ∑ µ1 σµ,µ′′µ1 ξµ1 + 3 ∑ µ1,µ2 σµ,µ′′µ1µ2 (ξµ1 ξµ 2 + gµ 1µ 2 ) ) + (µ ↔ µ′) + O(g5). (77) 698 Classical fluctuation electrodynamics Expressions (74), (75), (77) give equations of the FED2, i.e. the equations of the FED taking into account only binary correlations. Note finally that we cannot build the NED similar to our consideration of FED2 because quasi-equilibrium distribution ρq (22) does not exist. This probably indicates that a reduced description of electromagnetic field only by average electric and magnetic fields in a general case is impossible (see also [10]). In the present work the classical FED is considered. This theory is a limit of the corresponding quantum theory. In a quantum case the binary correlation func- tion gµ1µ2 is closely related to the Wigner photon distribution functions. Therefore, equation (77) is similar to the kinetic equation for photons in which square in gµ1µ2 terms describe the photon-photon collisions in the medium. 4. Conclusions Thus, there has been built a fluctuation electrodynamics of classical electromag- netic field in equilibrium medium as a theory which describes its nonequilibrium states by average field and all field correlations. Equations of the fluctuation elec- trodynamics are written in terms of a generating functional for moments of the field and in terms of an average field and a generating functional for correlations (fluctuations, centered moments). It was established that the right hand sides of the corresponding equations are completely defined by an average electric current calculated in the present paper up to the fourth order in electromagnetic interacti- on. The dependence of the current on electromagnetic field and its correlations (the material equation of the theory) is nonlinear, local in time and nonlocal in space (spatial dispersion). The paper discusses the relation of the developed theory to the description of the electromagnetic field with average field and its binary correlations. This work was supported by the State Foundation for Fundamental Research of Ukraine (project No. 2.7/418) and INTAS (project No. 00–577). References 1. Kilin S.Ya. Quantum Optics. Fields and their Detecting. Navuka i Tekhnika, Minsk, 1990 (in Russian). 2. Klimontovich Yu.L. Kinetic Theory of Nonideal Gases and Nonideal Plasmas. Nauka, Moscow, 1975 (in Russian); Pergamon Press, Oxford, 1982. 3. Balescu R. Statistical Mechanics of Charged Particles. John Wiley and Sons, London, 1963. 4. Bogolyubov N.N. Problems of Dynamical Theory in Statistical Physics. Gostekhizdat, Moscow, 1946 (in Russian). 5. Akhiezer A.I., Peletmisky S.V. Methods of Statistical Physics. Nauka, Moscow, 1977 (in Russian); Pergamon Press, Oxford, 1981. 6. Peletmisky S.V., Sokolovsky A.I., General equations of fluctuation hydrodynamics. Ukrainian Journal of Physics, 1992, 37, 1521–1528 (in Russian). 7. Alexandrov A.F., Bogdankevich L.S., Rukhadze A.A. Foundations of Plasma Electro- dynamics. Vysshaya Shkola, Moscow, 1988 (in Russian). 699 A.I.Sokolovsky, A.A.Stupka 8. Sokolovsky A.I., Stupka A.A., Kinetic theory of electromagnetic processes in equilibri- um medium. Visnyk Dnipropetrovs’kogo Natsional’nogo Universytetu. Physics, Radio Electronics, 2003, 10, 63–70 (in Ukrainian). 9. Sokolovsky A.I., Stupka A.A., Modes of electromagnetic field in equilibrium plasma. The Journal of Kharkiv National University. “Nuclei, Paryicles, Fields”, 2004, No. 628, 2(24), 87–92 (in Ukrainian). 10. Peletminskii S.V., Prikhod’ko V.I., Schelokov V.S., Low-frequency asymptotics of elecrodynamic Green functions. Teoreticheskaya i matematicheskaya fizika, 1975, 25, No. 1, 70–79 (in Russian). Класична флуктуаційна електродинаміка О.Й.Соколовський, А.А. Ступка Дніпропетровський національний університет, вул. Наукова 13, Дніпропетровськ, Україна, 49050 Отримано 3 серпня 2005 р., в остаточному вигляді – 8 листопада 2005 р. Cистема, яка складається з рівноважного середовища із зарядже- них частинок і електромагнітного поля, розглянута в класичному випадку та при малій взаємодії між підсистемами. Функцію розпо- ділу системи розраховано з точністю до третього порядку за елек- тромагнітною взаємодією. Поле описується усіма моментами елек- тричного та магнітного поля, які обрані параметрами скороченого опису в рамках метода скороченого опису нерівноважних станів Боголюбова. Як гранична умова до рівняння Ліувілля застосована узагальнена умова Боголюбова повного ослаблення кореляцій між підсистемами. Одержані часові рівняння для параметрів скороче- ного опису записані в компактній формі за допомогою породжую- чого функціоналу для моментів поля та породжуючого функціоналу для кореляцій поля (центрованих моментів, флуктуацій). Отримані рівняння узагальнюють нелінійну електродинаміку в рівноважному середовищі на випадок врахування флуктуацій електроманітного поля. Ключові слова: метод скороченого опису нерівноважних станів Боголюбова, узагальнена гранична умова повного ослаблення кореляцій, флуктуаційна електродинаміка у рівноважному середовищі, породжуючий функціонал PACS: 05.20.-y, 05.40.-a, 11., 52.40.Db 700

Classical fluctuation electrodynamics

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