Energy flux of electromagnetic field in stochastic model of radiative heat transfer in dielectric solid medium
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r MSC 85A25, 81P20
Energy flux of electromagnetic field in stochastic modelof radiative heat transfer in dielectric solid medium
Y u.P. V irchenko and Lam T an P hat
Belgorod State University, 308015, Belgorod, Russia
The stochastic model that describes radiative heat transfer in dielectric medium is built. Themodel is based on the representation that heat transfer is realized both by heat conductivity mech-anism in it and due to the electromagnetic radiation that is generated by thermal fluctuations ofatoms in the medium. Using the fluctuation-dissipative theorem, on the basis of such physicalsuppositions, the stochastic model is formulated in the form of the infinite dimensional Ornstein-Uhlenbeck process that describes medium fluctuations. In the model frameworks, the energy fluxdensity of fluctuating electromagnetic field is calculated in the form of the functional of temperaturedistribution in three-dimensional medium sample.
1. Introduction.
The heat transfer in solids is realized by two mechanisms. Theyare the proper thermal conductivity and the heat transfer by electromagnetic radiation.The last is generated by thermal fluctuations of the medium local thermodynamic state.In accordance with such physical representation, the evolution equation of the temperaturedistribution T ( x , t ) may be written phenomenologically in the form at each fixed time t (see,for example, [1]-[3]) κ ( T ) ˙ T ( x , t ) = ( ∇ , κ ( T ) ∇ ) T ( x , t ) − ( ∇ , S )( x , t ) (1) where κ ( T ) > is the medium thermal conductivity coefficient which depends on temper-ature, κ ( T ) is the medium volume heat capacity. The vector field S ( x , t ) is the energy fluxdensity of electromagnetic radiation associated with those absorption and radiation actionsof electromagnetic field by means of which the heat is transferred.The ( ∇ , S ( x , t )) value multiplied on the small volume of the spatial medium region cen-tered near the point x is the part of flux density which is spent in the medium heating itssmall volume centered in the point x at the time moment t . This term in Eq. (1) is essentialwhen heat transfer problems are solved in optically semitransparent medium that possessesa low electrical conductivity and also some sufficiently large temperature drops are presentthrough a characteristic distance ¯ L . To solve the problems of heat transfer in these cases, wemust obtain a complete evolution equation controlling the temperature distribution T ( x , t ) .So, it is necessary to find the explicit form of the functional S ( x , t ) = S [ T ( x , t )] whichtransforms Eq. (1) into the self-consistent one.Usually, the energy flux density S ( x , t ) is constructed phenomenologically in frameworksof so-called theory of radiation transfer. It is done using: the geometric optics laws whichare applied to «thermal» rays inside the medium, the phenomenological Kirchhoff law thatconcerns the radiation and absorption intensities of the optic radiation, as well as the Beer-Bouguer-Lambert law (see, for example, [1]-[4]). The thermal electromagnetic field in suchtheoretical constructions does not exist in such a theory. It seems that such a situation is1nsatisfactory from the theoretical viewpoint. It is connected with the absence of successivemicroscopic theory of radiation heat transfer which should be based on the quantum theoryof radiation and absorption of thermal photons in solid medium.Here, we shall not concentrate on detailed analysis of those problems which are relatedwith the construction of the microscopic theory of heat radiation transfer based on statisti-cal physics formalism. We point out only that the statistical approach in the theory of heatradiation transfer has been proposed in the Rytov works which are summarized in the mono-graphs [8], [9]. In connection with the complexity of microscopic theory construction, thisapproach is semi-phenomenological. It is based on the presentation that the electromagneticradiation and absorption actions in the medium are connected with thermal fluctuations ofits local thermodynamic state. These thermal fluctuations determine the microscopic fluctu-ations of charges in medium and currents induced by them. Such a fluctuation approach isthe statistical one due to its nature. Therefore, the electromagnetic field which is responsiblefor the radiation heat transfer in the medium, is the stochastic one. Besides, the microscopicmechanism of the energy field transformation into heat is not concretized in the frameworkof such a theory. It permits to avoid the quantum description of the radiation and absorptionactions. It turn, it is reasonable from theoretical viewpoint, since the heat radiation transferis not a quantum effect.We note that thermal fluctuations of electric charges which generate the stochastic elec-tromagnetic field, may be occurred in electro-neutral mediums having very low electricalconductivity, i.e. in dielectrics and high-resistance semiconductors. Thus, thermal fluctua-tions of charges lead with inevitability to induction of electric currents in such media. Butthese currents exist for very small distances. Since the amplitude of thermal fluctuations in-creases with the temperature growth, then, for sufficiently large its value, thermal vibrationsof medium atoms lead to electric charges fluctuations even in dielectric media. They areoccurred for distances having the order of interatomic ones. Their value may be appearedessential when the problem of heat transfer caused by the electromagnetic radiation is solved.The mathematical realization of the described physical considerations is performed bythe use of stochastic electromagnetic fields which are obeyed Maxwell’s equations. At presentwork, we construct the specific mathematical stochastic model of thermal radiation transferin frameworks of the above-described fluctuation approach. To avoid the account of theboundary conditions in the suggested model, we study only the case when the inhomogeneityof temperature distribution is concentrated in a limited region of boundless medium envi-ronment.
2. The mathematical model construction.
For the problem formulation, we considerthat the thermal electromagnetic field is defined by the field pair h ˜ E ( x , t ) , ˜ H ( x , t ) i , x ∈ R , t ∈ R , which are stochastic ones. Here and after we mark all random functions by the sign«tilde». On the basis of this pair, the energy flux density of thermal electromagnetic field isdetermined as ˜ S ( x , t ) = c π [ ˜ E , ˜ H ]( x , t ) (2) where c is the vacuum light velocity. So, it is a random function.The thermal electromagnetic field changes rapidly through distances having the charac-2eristic wavelength ( ∼ − cm) order that corresponds to thermal (red and infra-red) ra-diation. Therefore, the characteristic time has the order of ∼ − sec. At the same time,the characteristic length of thermal conductivity process in crystalline dielectrics has theorder − cm, and the correspondent characteristic time is − sec. Therefore, the energyflux density (2) should be averaged over spatial regions having a size which is much greaterthan the characteristic wavelength of stochastic electromagnetic field when adequate math-ematical theory is constructed. But it is much smaller than the characteristic length of theheat transfer process. In addition, the density (2) should be averaged over temporal inter-vals which are much greater than the characteristic period of thermal radiation oscillations,but it is much smaller than the characteristic time of thermal conductivity process. Suchan averaging permits to ignore the small rapid oscillations of the divergence ( ∇ , ˜ S ( x , t )) ofthe radiation flux density over space and time because they have no a relation to the heattransfer process. Due to basic statistical physics representations, the pointed out space-timeaveraging is equivalent to the averaging on the basis of the probability distribution of ran-dom electromagnetic field, when the pair of random fields ˜ E ( x , t ) and ˜ H ( x , t ) possesses the ergodicity property. Thus, the energy flux density of the field used in (1) is determined bythe mathematical expectation S ( x , t ) = hh ˜ S ( x , t ) ii on the probability distribution of the ran-dom vector field ˜ S ( x , t ) (here and after angular brackets denote such an averaging). Then,for the complete mathematical formulation of the model that describes the radiation heattransfer, it is necessary to build the adequate stochastic model of the thermal electromag-netic field and to calculate the mathematical expectation hh S ( x , t ) ii on basis of its probabilitydistribution.Thus, the stochastic electromagnetic field is represented by random realizations h ˜ E ( x , t ) , ˜ H ( x , t ) i which satisfy the system of stochastic Maxwell equations in the continuous dielectricmedium neglecting its dispersion εc ∂ ˜ E ∂t + 4 πc ˜ j = [ ∇ , ˜ H ] , ( ∇ , ˜ E ) = 4 πε ˜ ρ ,µc ∂ ˜ H ∂t = − [ ∇ , ˜E ] , ( ∇ , ˜ H ) = 0 (3) where ˜ E and ˜ H are intensities of electric and magnetic fields of thermal radiation which aregenerated by heated medium. At the same time, ε is the electric permeability of uniformdielectric medium and µ is the magnetic one. We consider them to be independent on x and t .Generally, the ε and µ values depend on the temperature. These dependencies may besubstantial at large temperature drops through distances having the characteristic size orderthat is connected with temperature non-uniformity in the medium. Temperature valuesin dependencies of ε and µ on T should be equal to the local temperature T ( x , t ) in themedium. In general case, spatial and temporal derivatives of ε ( T ( x , t )) and µ ( T ( x , t )) shouldbe appeared in the Maxwell equations when these dependencies are taken into account.However, these derivatives are extremely small in comparison with those length and temporalscales which are characteristic of thermal radiation due to slowness of dependencies pointedout. Therefore, these derivatives are not taken into account in Eqs.(3).3andom realizations of ˜ E ( x , t ) and ˜ H ( x , t ) are determined by stochastic sources ˜ j , ˜ ρ sincethey are some solutions of the system (1). These sources are some fluctuations of the electriccurrent density and the charge density. They are occurred at micro-regions having the orderof the characteristic wavelength due to thermal fluctuations.Besides, for complete determination of solutions, it is important to propose definite initialand boundary conditions corresponding to described physical situation. As for boundaryconditions, we shall study the simplest physical situation when the thermal localized non-uniformity takes place in unbounded medium. This nonuniformity is concentrated in abounded region of space with the linear size L having the order of 1cm ÷ cm. Then thelocal medium temperature T ( x , t ) tends to a constant when | x | tends to infinity. As for thespatially distributed stochastic sources which are performed by the densities ˜ j , ˜ ρ in Eqs. (3),their specific form determines completely the constructed model. The consistency conditionof the system Eqs. (3) leads to the fact that these densities satisfy the continuity equation ˙˜ ρ + ( ∇ , ˜ j ) = 0 . (4) Due to such a relation, it is sufficient to determine only the random field ˜ j ( x , t ) for completemathematical building of the model.In our model, the current density ˜ j is composed of two parts. The first is the properstochastic source of electromagnetic field. It is plays the role of an internal «electromotiveforce» in the medium. It arises as a result of the thermal fluctuations. The second isdetermined by Ohm’s law σ ˜ E . We note that the coefficient σ > . It plays the role of theelectrical conductivity. But it is not the genuine macroscopic electrical conductivity of themedium that may be very small in physical situation under consideration. It performs an«effective electrical conductivity» which should be different from zero due to the so-called fluctuation-dissipative theorem (see, for example, [9]). It is necessary to take into accountfrom the mathematical viewpoint in order that a regular dissipative constituent should be inthe system of stochastic evolution equations (1) with additive noise. In turn, it is connectedwith presence of stationary evolution regime.With probability one, the part of the fluctuation current density a ( x , t ; T ) ˜ ϕ that servesthe stochastic source of electromagnetic field, should be certainly contained the vortical term(a fluctuation «Foucault current») in spite of the radiation transfer occurs in dielectrics (orhigh-resistance semiconductors). Here, the source intensity a ( x , t ; T ) depends functionallyon the local temperature T = T ( x , t ) . Therefore, it may be varied spatially and temporally.This varying is much slower in comparison with the change of the thermal electromagneticfield. The irradiation of electromagnetic waves which transfer the heat is associated with theavailability of the vortical part. In connection with dielectric character of the medium, thefluctuation current (its correlation function) is concentrated at small space scale that hasthe order of ÷ interatomic distance. Thus, the current density ˜ j should be replaced inEqs. (3) and (4) by ˜ j ( x , t ) = ˜ ϕ ( x , t ) a ( x , t ; T ) + σ ˜ E ( x , t ) where the intensity a ( x , t ; T ) shouldbe defined on the basis of statistical physical consideration for completion of the modelconstruction. We suppose that the squared intensity is determined by thermal photonsirradiation in a small spatial region which concentrates near the point x at the time t .4herefore, a ( x , t ; T ) = ~ ∞ Z −∞ ω f (cid:16) ~ ω k T ( x , t ) (cid:17) dω (5) where f is the energy distribution function of irradiated photons. It depends on the tempera-ture T ( x , t ) distribution. Then we obtain that a ( x , t ; T ) ∼ T ( x , t ) , when f is the Planckfunction.Substitution of the explicit form of ˜ j ( x , t ) into the Eqs. (3) leads to the stochastic equa-tions system with the additive noise ˜ ϕ where the field ˜ E ( x , t ) is determined by the equation ∂ ˜ E ∂t + γ ˜ E + 4 πε a ˜ ϕ = cε [ ∇ , ˜ H ] , γ = 4 πσε . (6) Besides, the evolution equation of the charge density is valid ˙˜ ρ + γ ˜ ρ + ( ∇ , a ˜ ϕ ) = 0 (7) where, as above, we have neglected spatial derivatives of the temperature distribution. Ingeneral case, the coefficient σ depends on the local temperature which changes slowly on x and t . But we neglect this dependence for reasons above pointed out.The random field ˜ ϕ in Eqs.(6),(7) is Gaussian with the zero average value hh ˜ ϕ ( x , t ) ii = 0 due to supposed physical smallness of thermal fluctuations. At the same time, we supposethat hh ˜ ρ ( x , t ) ii = 0 . Then the Gaussian field ˜ ϕ ( x , t ) is completely determined by the paircorrelation function K j j ( x , t ; x , t ) = hh ˜ ϕ j ( x , t ) ˜ ϕ j ( x , t ) ii . Due to physical reasons,the random field ˜ ϕ ( x , t ) is translationally invariant on x in the stochastical sense and itis stationary on t in the sense of the theory random processes. Besides, we assume thatthis field is stochastically isotropic and temporally reversible. So, its correlation function isrepresented in the form K j j ( x , t ; x , t ) = K ( | x − x | , | t − t | ) δ j ,j . (8) In this case, the source a ( x , t ; T ) ˜ ϕ ( x , t ) of thermal radiation is uniform on x in Eqs. (6),(7)if we neglect the pointed out slow dependence on the local temperature T ( x , t ) . Moreover,at such conditions, it is stationary on t and it is stochastically isotropic.Further, we use some supplement assumptions about properties of the function K ( r, s ) , r, s > . These properties are associated with the locality of correlation function K j j ( x , t ; x , t ) . For physical consideration, the random field ˜ ϕ ( x , t ) should have theextreme small correlation time. Such correlations should be disappear during the temporalinterval equal to several periods of stochastic electromagnetic field oscillations. Then, wesuppose that K ( r, s ) ∼ δ ( s ) . In this case, the field ˜ ϕ ( x , t ) is transformed to a generalizedrandom Gaussian field of the «white noise» type on the temporal variable. Spatial cor-relations of the field ˜ ϕ ( x , t ) values are also short-ranged. They disappear at the distanceequal to some interatomic lengthes. So, the correlation length is the smallest parameter be-tween those which have the linear size in the problem under study. However, for the reasons5hat will become clear from the subsequent analysis, we may not assume that the function K ( r, s ) is proportional to δ ( r ) by the analogy with the temporal variable. So, we use thenext representation K ( | x − x | , | t − t | ) = K ( | x − x | ) δ ( t − t ) (9) where the function K ( r ) is absolutely integrable R R | K ( x ) | d x < ∞ and it is localized in thezero neighborhood having the r > size order that is K ( r ) = r − Q ( r / r ) where r is asmall parameter and K = R ∞ Q ( ξ / dξ < ∞ . Here the function Q ( r ) is concentrated inthe region with the linear size of order 1.After determination of the random process ˜ j ( x , t ) in the stochastic differential equationssystem (3), the fluctuation electromagnetic field is completely defined by the requirement ofits temporal stationarity. At the same time, the random function ˜ S ( x , t ) is a functional on T ( x , t ) , and its mathematical expectation hh ˜ S ( x , t ) ii = c π hh [ ˜ E , ˜ H ]( x , t ) ii (10) is determined by the probability distribution of the fluctuation field ˜ ϕ .
3. Small parameters of mathematical model.
Consistent mathematical analysis ofthe random process which is determined by the constructed mathematical model of radiationheat transfer is very complicated in the physical situation under consideration. In particular,the averaging in the resulting formula for the energy flux density S j ( x , t ) of the fluctuationelectromagnetic field leads to the complicated expressions which are uncomfortable for itspractical application when we solve heat transfer problems of electromagnetic radiation insemi-transparent medium. The significant simplification of these expressions is reached, whenthe specific physical conditions are taken into account where these transport processes occur.It leads to the detection of sequence small parameters in the problem under study. Thenthe natural setting of the mathematical problem consist of the calculation of the S j ( x , t ) expression in the form of main asymptotic term when these small parameters tend to zero.Let ¯ L be the size of temperature non-uniformity that equals to the linear size of theregion where the non-uniform distribution temperature T ( x , varies at one degree. We notethat the characteristic time during which the temperature distribution changing is occurreddue to the heat conductivity process, is significantly more than the time ¯ L/ ¯ c during whichthe thermal electromagnetic radiation overcomes the distance ¯ L and goes out of the non-uniformity region (it occurs during ∼ · − sec when ¯ L ∼ − cm) where the heat transferprocesses occurs. Therefore, this part of radiation does not effect on the heat transfer processwhen it comes out of the system. The natural time for the heat transfer process is determinedby the value ¯ L κ/ κ where the ratio κ /κ has the order of − cm / sec in the typical physicalsituation in solid high-resistance semiconductor crystal. Consequently, the typical time ofthe distribution temperature varying in problems under consideration is equal to − s. Asa result, we obtain the small parameter κ / ¯ L ¯ cκ ≪ having the order of · − where ¯ c = c /εµ and ¯ c is the light velocity in the medium, ¯ c = c /εµ .Further, we assume that the medium is very semi-transparent. The characteristic distanceof the radiation damping is much larger in it than the introduced size ¯ L . In this case, if we6se typical values of specific electrical conductivity, the parameter γ ¯ L/ ¯ c has the values inthe range · (10 − ÷ − ) ≪ in dielectrics where γ = 4 πσ/ε has the order of ÷ − sec − . For some semiconductors, the parameter γ ¯ L/ ¯ c varies in the range · (10 − ÷ ) .As mentioned above, there is another natural small parameter which is the ratio r / ¯ L .This ratio is small in view of the fact that r ∼ − cm and ¯ L ∼ − cm, so that r / ¯ L ∼ − . Thus, we conclude that the following relations κ / ¯ L ¯ cκ ≪ r /L , γL/ ¯ c ≪ r /L between the introduced small parameters are fulfilled in dielectrics. As we can see fromthe above estimates, the parameter γ ¯ L/ ¯ c is not small for semiconductors in general case.Thus, the calculation of the energy flux density of the fluctuation electromagnetic field willbe performed in the form of the main asymptotic term when these parameters tend to zero.In view of the fact that the transition to the limit is realized by several parameters whenthe asymptotic calculation is done, it is necessary to specify the transition character. Weassume that these transition are understood as repeated ones. In accordance with the theirmentioned typical physical values, the limit transition order will be realized in the order oftheir value, i.e. from small ones to large ones. Thus, the transition to the limit of r / ¯ L → will be produced at the final step of calculations. At the same time, for construction of suchcalculations, it is necessary to explicitly introduce the parameter ¯ L into the appropriateformulas. Respectively, all values of length and time dimensionalities in our model aremeasured by units of the largest spatial size ¯ L and the biggest temporal duration ¯ L κ/ κ .
4. Construction of the random stationary process.
Since the typical time ofthermal conductivity process is the largest parameter of the temporal dimensionality in ourmodel, the first step of the mentioned transition to the limit at the asymptotic value S j ( x , t ) calculation is the construction of random stationary process on the basis of random processdetermined by the stochastic Eqs. (3) with a fixed initial temperature distribution. With thisaim, we introduce the generalized Fourier expansions of the random stochastic realizationsof fields ˜ E ( x , t ) and ˜ H ( x , t ) , ˜ E ( x , t ) = Z R ˜¯ E ( k , t ) exp[ i ( k , x )] d k , ˜ H ( x , t ) = Z R ˜¯ H ( k , t ) exp[ i ( k , x )] d k . (11) Here, ˜¯ E ( k , t ) and ˜¯ H ( k , t ) are generalized random fields on k ∈ R . We substitute theexpansion (11) in Eqs. (3), (5), (6). Then, because of their uniqueness determination onthe basis of Fourier’s expansions, we obtain the finite equations system of the generalizedFourier-images for each k ∈ R , ∂∂t ˜¯ E ( k , t ) + γ ˜¯ E ( k , t ) + 4 πε ˜¯ j ( k , t ) = icε [ k , ˜¯ H ( k , t )] , (12) ∂∂t ˜¯ H ( k , t ) = − icµ [ k , ¯˜E ( k , t )] , ( k , ˜¯ E ( k , t )) = − πiε ˜¯ ρ ( k , t ) , ( k , ˜¯ H ( k , t )) = 0 , (13)˙˜ ρ ( k , t ) + γ ˜ ρ ( k , t ) + i ( k , ˜¯ j ( k , t )) = 0 . (14) ˜ ρ ( x , t ) = Z R ˜¯ ρ ( k , t ) exp[ i ( k , x )] d k . (15) As well, we introduce generalized Fourier images ˜¯ j ( k , t ) of the random field a ( x , t ) ˜ ϕ ( x , t ) realizations, a ( x , t ; T ) ˜ ϕ ( x , t ) = Z R ˜¯ j ( k , t ) exp[ i ( k , x )] d k . (16) The fields ˜¯ j ( k , t ) , ˜¯ ρ ( k , t ) are complex-valued Gaussian random ones due to the Gaussianproperty of the field ˜ ϕ ( x , t ) . They have zero average values hh ˜¯ j ( k , t ) ii = 0 , hh ˜¯ ρ ( k , t ) ii = 0 .In view of the reality of the value a ( x , t ; T ) ˜ ϕ ( x , t ) , the field ˜¯ j ( k , t ) realizations has thefollowing property ˜¯ j ∗ ( k , t ) = ˜¯ j ( − k , t ) with the probability one. Namely, it is completelycharacterized by the correlation function ¯ K j j ( k , t ; k , t ) = hh ˜¯ ϕ j ( k , t ) ˜¯ ϕ ∗ j ( k , t ) ii . Thisfunction is positively definite matrix-function on k ∈ R and t . Then, it is associated withthe correlation function ¯ K ll ′ ( k , ω, k ′ , ω ′ ) = 1(2 π ) Z R exp h i ( ω ′ t ′ − ωt ) + i (cid:16) ( k ′ , x ′ ) − ( k , x ) (cid:17)i K ll ′ ( x , t ; x ′ , t ′ ) d x d x ′ dtdt ′ . (17) Whereas the properties of stochastic uniformity on x , stationarity on t and isotropy takesplace for the field ˜ ϕ ( x , t ) , this correlation function has the form ¯ K ll ′ ( k , ω, k ′ , ω ′ ) = 12 π δ ll ′ δ ( k + k ′ ) ¯ K ( k ) δ ( ω ′ + ω ) , (18)¯ K ( k ) = 1(2 π ) Z R exp h − i ( k , x ) i K ( | x | ) d x . (19) Since the equation system is finite at each fixed k ∈ R , then it is uniquely solvablewhen the initial conditions of the generalized random ˜¯ E ( k , t ) , ˜¯ H ( k , t ) , ˜¯ ρ ( k , t ) realizationsare given. It means that the stochastic model of thermal electromagnetic field described inthis section is complete from the mathematical viewpoint. Then, we may state the following.Since the equation system that determines the generalized ˜¯ E ( k , t ) and ˜¯ H ( k , t ) fields is linearand due to the average value of the field ˜¯ ρ ( k , t ) is zero, the thermal electromagnetic field isthe random Gaussian field with zero average.The initial conditions for calculation of mathematical expectations of various randomfunctions of the ˜¯ E ( k , t ) , ˜¯ H ( k , t ) and ˜¯ ρ ( k , t ) fields become insignificant after the temporalperiod which is much longer than the time κ ¯ L / κ . (We also note that this temporal periodshould be much larger than the characteristic time τ associated with the thermal radiation,so that the value ~ τ − should be of the average temperature order). Then, as the field8 ϕ ( x , t ) is stationary on t , so we may also consider the stochastic fields { ˜¯ E ( k , t ) , ˜¯ H ( k , t ) } which obey Eqs. (12)-(14) as stationary ones. Due to this, we neglect the dependence ontime of the temperature distribution T ( x , t ) in the amplitude a ( x , t ; T ) and, consequently,in sources ˜¯ j ( k , t ) , ˜ ρ ( k , t ) when the transition to the asymptotic values are calculated. Thesuch a disregard of temporal dependencies corresponds to the transition in the asymptoticregion t ≫ κ ¯ L / κ ∼ − sec.When we study the constructed stationary process, it is natural to pass from the evolutionEqs.(12)-(14) to the equations of spectral amplitudes of these fields. They are generalizedfunctions on frequency ω , ˜¯ E ( k , t ) = ∞ Z −∞ ˜ E ( k , ω ) e iωt dω , ˜¯ H ( k , t ) = ∞ Z −∞ ˜ H ( k , ω ) e iωt dω , (20)˜¯ j ( k , t ) = ∞ Z −∞ ˜ ι ( k , ω ) e iωt dω , ˜¯ ρ ( k , t ) = ∞ Z −∞ ˜ ̺ ( k , ω ) e iωt dω (21) where the generalized random field ˜ ι l ( k , ω ) that defines the spectral expansion of fluctuatingcurrent density, is given by the formula ˜ ι ( k , ω ) = 1(2 π ) Z R exp (cid:0) − iωt − i ( k , x ) (cid:1) a ( x , t ; T ) ˜ ϕ ( x , t ) d x dt . (22) Substituting these expansions into Eqs. (12-14) and using the uniqueness of Fourier’simages, we obtain the following complete equations system: iω ˜ E ( k , ω ) + γ ˜ E ( k , ω ) + 4 πε ˜ ι ( k , ω ) = icε [ k , ˜ H ( k , ω )] , (23)˜ H ( k , ω ) = − cµω [ k , ˜ E ( k , ω )] , ( k , ˜ E ( k , ω )) = − πiε ˜ ̺ ( k , ω ) , ( k , ˜ H ( k , ω )) = 0 , (24) iω ˜ ̺ ( k , ω ) + γ ˜ ̺ ( k , ω ) + i ( k , ˜ ι ( k , ω )) = 0 , (25) Solutions of the system are performed by following formulas: ˜ E ( k , ω ) = i πε · (cid:16) ( ω − iωγ )˜ ι ( k , ω ) − ¯ c ( k , ˜ ι ( k , ω )) k (cid:17) ( ω − iγ )( ω − ¯ c k − iωγ ) , (26)˜ H ( k , ω ) = − i πcεµ · [ k , ˜ ι ( k , ω )]( ω − ¯ c k − iωγ ) . (27) Here, Fourier’s images ˜ E ( k , ω ) , ˜ H ( k , ω ) , ˜ ι ( k , ω )) , ˜ ̺ ( k , ω ) are some generalized functions.
5. Energy flux density at the stationary regime.
Let us calculate the averagevalue of the energy flux density S j ( x , t ) , j = 1 , , , if each its component is equal to S j ( x , t ) = c π hh [ E ( x , t ) , H ( x , t )] j ii = ǫ jll ′ c π Z R exp h i ( k − k ′ , x ) + i ( ω − ω ′ ) t i hh ˜ E l ( k , ω ) ˜ H ∗ l ′ ( k ′ , ω ′ ) ii d k d k ′ dωdω ′ (28) where ǫ jll ′ is completely antisymmetric pseudotensor in R (the Levi-Civita symbol).The mathematical expectation in Eq.(28) is calculated on the basis of the explicit expres-sions (26) and (27) of the random fields ˜ E ( k , ω ) , ˜ H ( k , ω ) . These fields are Gaussian, since thefluctuation current ˜ ι ( x , t ) density is the random Gaussian field and due to linear transforma-tions of them. Therefore, the mathematical expectation is expressed by the correlationfunction of the field ˜ ι ( k , ω ) , hh ˜ E l ( k , ω ) ˜ H ∗ l ′ ( k ′ , ω ′ ) ii == − (4 π ) cε µ · ǫ l ′ mm ′ k ′ m (cid:16) ω ( ω − iγ ) δ ln − ¯ c k l k n (cid:17) ( ω − iγ )( ω − ¯ c k − iωγ )( ω ′ − ¯ c k ′ + iω ′ γ ) hh ˜ ι n ( k , ω )˜ ι ∗ m ′ ( k ′ , ω ′ ) ii , (29) hh ˜ ι l ( k , ω )˜ ι ∗ l ′ ( k ′ , ω ′ ) ii = δ ll ′ (2 π ) Z R a ( x , t ; T ) a ( x ′ , t ; T ) K ( | x − x ′ | ) exp h i (cid:16) ( k ′ , x ′ ) − ( k , x ) (cid:17)i d x d x ′ dt . (30) Substituting Eq.(28) into Eq.(29) and using the tensor identity ǫ jll ′ ǫ l ′ mn = δ jm δ ln − δ jn δ lm ,we take into account the expression (30) of the correlation function hh ˜ ι n ( k , ω )˜ ι ∗ m ′ ( k ′ , ω ′ ) ii corresponding to isotropic stochastic fluctuations of the current density. As a result, weobtain the following expression of the energy flux density S j ( x , t ) = Z R R j ( x − y , t − s ; x − y , t − s ) K ( | y − y | ) a ( y , s ; T ) a ( y , s ; T ) d y d y ds , (31) where R j ( x , t ; x ′ , t ′ ) = 1(2 π ) Z R ¯ R j ( k , ω ; k ′ , ω ′ ) exp h i (cid:0) ( k , x ) − ( k ′ , x ′ ) (cid:1) + i ( ωt − ω ′ t ′ ) i d k d k ′ dωdω ′ , (32)¯ R j ( k , ω ; k ′ , ω ′ ) = − R (cid:16) k ′ j (2 ω ( ω − iγ ) − ¯ c k ) + ¯ c k j ( k m k ′ m ) (cid:17) ( ω − iγ )( ω − ¯ c k − iωγ )( ω ′ − ¯ c k ′ + iω ′ γ ) , (33) R = 4 π ¯ c /ε . (34) From formulas (32) and (33), we have the function R j ( x , t ; x ′ , t ′ ) = − R (2 π ) Z R (cid:16) k ′ j (2 ω ( ω − iγ ) − ¯ c k ) + ¯ c k j ( k m k ′ m ) (cid:17) ( ω − iγ )( ω − ¯ c k − iωγ )( ω ′ − ¯ c k ′ + iω ′ γ ) ×× exp h i (cid:0) ( k , x ) − ( k ′ , x ′ ) (cid:1) + i ( ωt − ω ′ t ′ ) i d k d k ′ dωdω ′ , that defines contributions of two radiation sources into the energy flux density at the point x (since the energy flux density is proportional to the square of electromagnetic field).10hese sources are in different spatial points with y and y radius-vectors. So, the function R j ( x , t ; x ′ , t ′ ) should be represented in the form R j ( x , t ; x ′ , t ′ ) = − R h iU ( x , t ) ∇ ′ j V ∗ ( x ′ , t ′ ) ++ ˙ V ( x , t ) ∇ ′ j V ∗ ( x ′ , t ′ ) − i ¯ c ∇ m ∇ j W ( x , t ) ∇ ′ m V ∗ ( x ′ , t ′ ) i (35) where the operators ∇ j and ∇ ′ m , j, m ′ = 1 , , denote gradients on vectors x and x ′ , respec-tively, and the dot denotes the differentiation on t . The scalar fields U ( x , t ) , V ( x , t ) , W ( x , t ) are given by the following integral representations which are some generalized functionscorresponding: U ( x , t ) = 1(2 π ) Z R exp( i ( k , x ) + iωt ) ω − iγ d k dω , (36) V ( x , t ) = 1(2 π ) Z R exp( i ( k , x ) + iωt ) ω − ¯ c k − iωγ d k dω , (37) W ( x , t ) = 1(2 π ) Z R exp( i ( k , x ) + iωt )( ω − iγ )( ω − ¯ c k − iωγ ) d k dω . (38) Besides, functions U ( x , t ) and W ( x , t ) have purely imaginary values, and the function V ( x , t ) has real values.In accordance with Eq.(35), the flux density S j (( x ) , t ) breaks to three parts: S j ( x , t ) = S ( u ) j ( x , t ) + S ( v ) j ( x , t ) + S ( w ) j ( x , t ) (39) where, according to Eq.(31) and Eq.(35), each term of S j (( x ) , t ) has the form: S ( u ) j ( x , t ) = − iR Z R U ( x − y , t − s ) ∇ j V ∗ ( x − y , t − s ) ×× K ( | y − y | ) a ( y , s ; T ) a ( y , s ; T ) d y d y ds , (40) S ( v ) j ( x , t ) = − R Z R [ ˙ V ( x − y , t − s )][ ∇ j V ∗ ( x − y , t − s )] ×× K ( | y − y | ) a ( y , s ; T ) a ( y , s ; T ) d y d y ds , (41) S ( w ) j ( x , t ) = i ¯ c R Z R [ ∇ m ∇ j W ( x − y , t − s )][ ∇ m V ∗ ( x − y , t − s )] ×× K ( | y − y | ) a ( y , s ; T ) a ( y , s ; T ) d y d y ds , (42)
6. Asymptotic expressions of generalized functions
U, V, W . In this section we findasymptotic formulas of generalized functions U ( x , t ) , V ( x , t ) , W ( x , t ) which determine thecontributions S ( p ) j ( x , t ) , p ∈ { u, v, w } to the flux density S j ( x , t ) when the small parameter11 L/ ¯ c → . On the one hand, such a procedure is necessary due to the fact that functions V ( x , t ) , W ( x , t ) are not calculated exactly in terms of standard generalized functions. Itmakes very complicated formulas (41) and (42). On the other hand, we must to calculatethe value S j ( x , t ) so that it may be used for application in the theory of radiation heattransfer in semitransparent semiconductor crystals.For the generalized function U , one can find easily the explicit form U ( x , t ) = δ ( x )2 π Z R e iωt ω − iγ dω = i Θ( t ) δ ( x ) e − γt (43) where we have used the integral representation of three-dimensional δ -function δ ( x ) = 1(2 π ) Z R exp( i ( k , x )) d k and the integral representation of the Heaviside Θ( · ) -function.Functions V ( x , t ) and W ( x , t ) do not have such a simple explicit representation. There-fore, we find their asymptotic representations when the parameter γL/ ¯ c tends to zero. As aresult, we have found the following asymptotic formulas at r > , t > : V ( x , t ) = − Θ( t )4 π ¯ cr e − γt/ δ ( r − ¯ ct ) , (44) W ( x , t ) ∼ − i Θ( t )4 π ¯ c h e − γt r −− r e − γt/ (cid:16) sgn( r + ¯ ct ) + sgn( r − ¯ ct ) − γ c (cid:2) | r + ¯ ct | − | r − ¯ ct | (cid:3)(cid:17) − g ( t ) π ¯ c i (45) where g ( t ) ≡ e − γt/ π/ Z sh (cid:16) γt η (cid:17) dη = π/ Z (cid:16) exp (cid:0) − γt sin η (cid:1) − exp (cid:0) − γt cos η (cid:1)(cid:17) dη . (46)
7. Integral representations S ( u ) j ( x , t ) , S ( v ) j ( x , t ) , S ( w ) j ( x , t ) . Now, we obtain suchasymptotic integral representations of functions S ( u ) j ( x , t ) , S ( v ) j ( x , t ) , S ( w ) j ( x , t ) which do notcontain δ -function singularities when limit transitions γL/ ¯ c → and κ /L ¯ cκ → are made.Further, we denote gradients on variables y j by ∇ ( j ) , j = 1 , respectively. Since allsubintegral expressions of integrals in Eqs.(40-42) contain the gradient of function V ( x , t ) with δ -function singularity, we fulfill integrations on the variable y in them by parts. Then,we calculate the integral on s by means of the δ -function. We use also the asymptotic formula(44). As a result of transformations pointed out, we obtain the following formulas: S ( u ) j ( x , t ) = − i R π ¯ c Z R e − γ | y | / c | y | ∇ (2) j h K ( | y − y | ) a ( x − y , t − s ; T ) a ( x − y , t − s ; T ) i s = | y | / ¯ c × U ( y , | y | / ¯ c ) d y d y , (47) S ( v ) j ( x , t ) = − R π ¯ c Z R e − γ | y | / c | y | ∇ (2) j h K ( | y − y | ) a ( x − y , t − s ; T ) a ( x − y , t − s ; T ) i s = | y | / ¯ c ×× ˙ V ( y , | y | / ¯ c ) d y d y , (48) S ( w ) j ( x , t ) = iR π Z R e − γ | y | / c | y | ∇ (1) j ∇ (1) m ∇ (2) m h K ( | y − y | ) a ( x − y , t − s ; T ) a ( x − y , t − s ; T ) i s = | y | / ¯ c × W ( y , | y | / ¯ c ) d y d y . (49) To obtain asymptotic expressions of these integrals at the limit κ /L ¯ cκ → , we use thatthe temperature distribution T ( x , t ) changes very slowly on spatial coordinates in comparisonwith changing of correlation function K ( · ) . Therefore, we neglect results of operators ∇ (1) and ∇ (2) actions on amplitudes a ( x − y , t ; T ) and a ( x − y , t ; T ) , respectively: S ( u ) j ( x , t ) = − iR π ¯ c Z R e − γ | y | / c | y | U ( y , | y | / ¯ c ) ×× (cid:16) ∇ (2) j K ( | y − y | ) (cid:17) a ( x − y , t − | y | / ¯ c ; T ) a ( x − y , t − | y | / ¯ c ; T ) d y d y , (50) S ( v ) j ( x , t ) = − R π ¯ c Z R e − γ | y | / c | y | ˙ V ( y , | y | / ¯ c ) ×× (cid:16) ∇ (2) j K ( | y − y | ) (cid:17) a ( x − y , t − | y | / ¯ c ; T ) a ( x − y , t − | y | / ¯ c ; T ) d y d y , (51) S ( w ) j ( x , t ) = iR π Z R e − γ | y | / c | y | W ( y , | y | / ¯ c ) ×× (cid:16) ∇ (1) j ∇ (1) m ∇ (2) m K ( | y − y | ) (cid:17) a ( x − y , t − | y | / ¯ c ; T ) a ( x − y , t − | y | / ¯ c ; T ) d y d y , (52)
8. Asymptotic expression of the function S j ( x , t ) at r /L → . Here, we calculateasymptotic expression of the energy flux density S j ( x , t ) when the correlation radius r /L tends to 0.To do such a limit transition, it is necessary to introduce explicitly the parameter r into these expressions. It is done by the following form of the correlation function K ( | x | ) = r − Q ( x / r ) that provides an independence of the integral R K ( | z | ) d z on r . Since thesubintegral function has singularities, the transition to the limit r /L → is not possibleby means of the change of K ( | z | ) to Kδ ( z ) with a positive constant K in the subintegralexpression.After introduction the explicit dependence of correlation function on r into subintegralexpressions in Eqs.(50-52), we make the following changes of integration variables y /r ⇒ y y /r ⇒ y . Then, passing to the limit r → , we calculate main terms of asymptoticexpansion of densities S ( u ) j ( x , t ) , S ( v ) j ( x , t ) , S ( w ) j ( x , t ) , substituting the asymptotic expressionsof functions U ( x , t ) , V ( x , t ) , W ( x , t ) in corresponding integral. As a result, we obtain thefollowing asymptotic formulas S ( u ) j = a ( x , t ; T ) (cid:16) r − R π ¯ c (cid:17) Z R ∇ j Q ( y / | y | d y + r − o (1) (53) where the integral is equal to zero due to the spherical symmetry of the correlation function.Thus, we get finally S ( u ) j = r − o (1) when r → .Further, for the function S ( v ) j ( x , t ) we obtain S ( v ) j ( x , t ) = − R (4 π ¯ c ) Z R e − γr | y | / ¯ c | y || y | (cid:16) ∇ (2) j Q ( | y − y | / (cid:17) × (cid:16) γ c δ ( r ( | y | − | y | )) + δ ′ ( r ( | y | − | y | )) (cid:17) a ( x − r y , t ; T ) a ( x − r y , t ; T ) d y d y (54) where we have taken into account that δ -function δ ( | y | + | y | ) does not give a contributioninto the integral.After the introduction of dimensionless integration variables, the first term is proportionalto the small parameter γL/ ¯ c . However, it is necessary to ascertain that the correspondingintegral has the order of r − and we may neglect this term. Using spherical coordinates, thisintegral is obtained in the form Z R e − γr | y | / ¯ c | y || y | a ( x − r y , t ) a ( x − r y , t ) h ∇ (2) j Q (( y − y ) / i δ ( r ( | y | − | y | )) d y d y == r − ∞ Z ξ exp h − γr ξ/ ¯ c i dξ ×× Z n = n =1 a ( x − r ξ n , t ; T ) a ( x − r ξ n , t ; T )( n − n ) j Q ′ ( ξ | n − n | / d Ω( n ) d Ω( n ) where we introduce integrations on d Ω( n ) and d Ω( n ) that means integrations on unitspheres of vectors y and y respectively. The last integral is equal to zero, since thesubintegral expression of its internal integral is antisymmetric comparatively to the permuta-tion n ⇔ n . Consequently, we may really neglect the term with δ -function.Now, we calculate the contribution into asymptotic expression of the function S ( v ) j ( x , t ) connected with the integral that contains δ ′ ( · ) -function. To calculate this contribution, it isconvenient to return to the integral variable r y j ⇒ y j , j = 1 , , ¯ R = R, r − / (4 π ¯ c ) , − ¯ R Z R e − γ | y | / ¯ c | y || y | a ( x − y , t ; T ) a ( x − y , t ; T ) h ∇ (2) j Q (( y − y ) / r ) i δ ′ ( | y | − | y | ) d y d y = − r − ¯ R Z R e − γ | y | / ¯ c | y | a ( x − y , t ; T ) (cid:16) C k ( y ) n + C ⊥ ( y ) (cid:17) d y , (55) where n = y / | y | and C k ( y ) = Z R | y | − ( n , y ) | y | Q ′ (( y − y ) / r ) δ ′ ( | y | − | y | ) a ( x − y , t ; T ) d y , C ⊥ ( y ) = − Z R y − ( y , n ) n | y | Q ′ (( y − y ) / r ) δ ′ ( | y | − | y | ) a ( x − y , t ; T ) d y so that ( C ⊥ , n ) = 0 .Passing to the limit r → , we find the limiting expression of introduced integrals C k ( y ) = − πQ | y | r a ( x − y , t ; T )(1 + O ( r )) (56) where it is assumed that Q = Q (0) < ∞ . Similarly, we find that C ⊥ ( y ) = o ( r ) .Substituting asymptotic expressions C k ( y ) and C ⊥ ( y ) into Eq.(55) and after that intoEq.(54), we find the final asymptotic formula for S ( v ) j ( x , t ) , S ( v ) j ( x , t ) = r − RQ π ¯ c Z R y j e − γ | y | / ¯ c | y | a ( x − y , t ; T ) d y . (57) The obtained formula shows that we may neglect the function S ( u ) j ( x , t ) when the main termof the asymptotic flux density S j ( x , t ) is calculated.Now, we pass to calculation of asymptotic formula of S ( w ) j ( x , t ) . Substituting the asympto-tical expression (45) of the function W ( x , t ) into Eq.(52) and producing the replacements ofthe integration variables r y j ⇒ y j , j = 1 , , we find S ( w ) j ( x , t ) = − r − R (4 π ¯ c ) Z R e − γ | y | / ¯ c | y || y | (cid:16) ∇ (1) j ∆ (1) Q (( y − y ) / r ) (cid:17) a ( x − y , t ; T ) a ( x − y , t ; T ) ×× h e − γ | y | / c − (cid:16) | y | − | y | ) (cid:17)i d y d y ≡ S ( w, j ( x , t ) + S ( w, j ( x , t ) (58) where we have neglected the term connected with the function g ( t ) because of its smallness.Let us consider the first summand. We replace the integration variables according toformulas y = ( y + y ) / , z = y − y which have the jacobian equal to 1. After that wereplace z /r ⇒ z . As a result, we obtain S ( w, j ( x , t ) ≡ − r − R (4 π ¯ c ) Z R exp (cid:2) − γ | y − r z / | / c (cid:3) | y − r z / || y + r z / | h ∇ ( z ) j ∆ ( z ) Q ( z / i × a ( x − y − r z / , t ; T ) a ( x − y + r z / , t ; T ) d y d z . From here, passing to the limit r → , we find that the asymptotic term of the function S ( w, j ( x , t ) is proportional to r − , is equal to zero, since it is equal to zero the limitingexpression of the following integral, Z R e − γ | y | / c y a ( x − y , t ; T ) d y Z R h ∇ ( z ) j ∆ ( z ) Q ( z / i d z = 0 , due to the fact that replacement of the integration variable z ⇒ − z in the subintegralexpression of last integral changes its sign to the opposite one.At last, we consider the second term in Eq.(58). Taking into account that [1 + sgn( | y | −| y | )] / θ ( | y | − | y | ) , we have S ( w, j ( x , t ) ≡ r − R (4 π ¯ c ) Z R : | y | > | y | e − γ | y | / ¯ c | y || y | h ∇ (1) j ∆ (1) Q (( y − y ) / r ) i ×× a ( x − y , t ; T ) a ( x − y , t ; T ) d y d y . Producing replacements of integration variables which are analogous to ones producedat the analysis of the function S ( w, j ( x , t ) , we obtain the following expression: S ( w, j ( x , t ) ≡ r − R (4 π ¯ c ) Z R :( y , z ) > exp (cid:2) − γ | y − r z / | / ¯ c (cid:3) | y − r z / || y + r z / | h ∇ ( z ) j ∆ ( z ) Q ( z / i ×× a ( x − y − r z / , t ; T ) a ( x − y + r z / , t ; T ) d y d z . Further, passing to the limit r → , we obtain the main asymptotic term of analyzedfunction in the following form: S ( w, j ( x , t ) ≡ r − R (4 π ¯ c ) Z R e − γ | y | / ¯ c y a ( x − y , t ; T ) d y Z R :( y , z ) > h ∇ ( z ) j ∆ ( z ) Q ( z / i d z . (59) We transform the internal integral, using the formula of Gaussian type Z R :( y , z ) > h ∇ ( z ) j ∆ ( z ) Q ( z / i d z = − n j Z R ∆ ( z ) Q ( z / d Σ( z ) (60) where n = y / | y | and the integration is fulfilled over the plane that is perpendicular to n and contains the point z = 0 . It is assumed that ∆ ( z ) Q ( z / tends by sufficiently rapid wayto zero in R when | z | → ∞ . Using polar coordinates h η, α i on the integration plane in thelast integral, we obtain the following result: Z R ∆ ( z ) Q ( z / d Σ( z ) = 2 π ∞ Z η Q ′ ( η / dη = − πQ . r − in the asymptotic expression of S ( w, j ( x , t ) vanishes. Then, weobtain the asymptotic expression of the function: S ( w ) j ( x , t ) = r − RQ π ¯ c Z R y j e − γ | y | / ¯ c | y | a ( x − y , t ; T ) d y . (61) The sum of the expression (61) and the main asymptotic term of the function S ( v ) j ( x , t ) (57) which have the same order of r → , we find the final asymptotic expression of theenergy flux density of fluctuating electromagnetic field S j ( x , t ) = r − RQ π ¯ c Z R y j e − γ | y | / ¯ c | y | a ( x − y , t ; T ) d y . (62)
9. Conclusion.