Kirchhoff-Planck law for freely radiating bodies and fluctuation-dissipation theorem

Physica 128A (1984) 467-485 North-Holland, Amsterdam KIRCHHOFF-PLANCK LAW FOR FREELY RADIATING BODIES AND FLUCTUATION-DISSIPATION THEOREM W. ECKHARDT

Abteilung flir Mathematische Physik, UniversitiitUlm, Ulm, Fed. Rep. Germany Received 8 May 1984

The interrelation between the Kirchhoff-Planck law (KPL) and the fluctuation--dissipation theorem (FDT) is discussed. It is known that the KPL is valid for freely radiating atoms if (i) the occupation probability of the atomic levels is the Boltzmann equilibrium probability and if (ii) the induced emission is treated as negative absorption. It is shown that the assumption (ii) is also inherent in the definition of the generalized susceptibility. The susceptibility is directly connected with the corresponding correlation function (FDT). It is proofed that the attached dipole fluctuation spectrum in its normally ordered form and the classical Maxwell equations reproduce the KPL. This result serves as the guideline for the extension of this procedure to macroscopic systems (susceptibility e(to)). The theory can be used to describe radiative heat transfer in inhomogeneous systems. The validity of this macroscopic theory implies the same assumptions as the validity of the KPL for freely radiating atoms. The freely radiating dielectric half-space is treated as a simple example.

1. Introduction In its original f o r m u l a t i o n the K i r c h h o f f - P l a n c k law r e p r e s e n t s the principle of d e t a i l e d b a l a n c e . T h e r m a l e q u i l i b r i u m d e m a n d s that in a closed v a c u u m m a t t e r system the a b s o r b e d r a d i a t i o n e n e r g y equals the e m i t t e d r a d i a t i o n energy. T h e spectral emissive p o w e r (specific intensity) of the b l a c k b o d y is described by Max P l a n c k ' s f a m o u s r a d i a t i o n law,

ibb(w, T ) _ C 09___~2 47r 7r2c3 hto

[

hto exp k ~ -

]-1 1

,

(1.1)

a n d c o n s e q u e n t l y , the K i r c h h o f f - P l a n c k law can be w r i t t e n in the following form: I(o~, T ) = y(w, T)Ibb(W, T ) . 0378-4371/84/$03.00 © Elsevier Science P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g Division)

(1.2)

468

W. E C K H A R D T

The specific intensity I(o, T) is defined by the relation: dE = I(w, T) dt do dO (If cos 0,

(1.3)

where dE denotes the energy in the frequency range [w, w +dw] which is emitted in the time dt by the surface element d F in the solid angle dO around the direction s; 0 denotes the angle between the unit normal to dF and the direction s; y(o, T) represents the spectral absorption coefficient with respect to s, dO and dF. Some years ago the problem was discussed if the Kirchhoff-Planck law is also valid for bodies which are not in equilibrium with their surrounding radiationS-3). (Further references can be found in the review article by Baltes4)). For simple model systems it was shown by Weinsteinl), Bauer 2) and Burkhard et al. 4) that (1.2) should be valid for freely radiating bodies if the suited definition of absorption is used, i.e. y(w, T) is considered as the difference of two parts: y(w, T ) = 7(w, T) (~b~)- 7(o, T) (era) .

(1.4)

In (1.4) y(abs) describes the pure absorption processes, i.e. processes by which the matter makes transitions from lower to higher energy eigenstates. On the contrary, y(em)describes the induced emission processes, i.e. induced transitions from higher to lower eigenstates. Induced emission is treated as negative absorption. Consequently, I(o, T) only contains the spontaneous emission processes. The so defined emission and absorption are the suited quantities for the description of measuring processes3). Besides these definitions, the validity of (1.2) for freely radiating bodies demands the crucial assumption that the matter is in thermal equilibrium irrespective of the state of the radiation field. It is obvious that (1.2) can be used (freely radiating bodies) to describe radiative heat transfer in the frame of the phenomenological theory (transport equation)~). This application is based on an intuitive photon picture which should be considered with careS-7). The interpretation of induced downward transitions as negative upward transitions is also inherent in the definition of the dissipative part of the generalized susceptibility of a macroscopic systemS). For instance, the absorption (dissipation) of electromagnetic (EM) energy by a quantummechanically treated macroscopic system is described in the sense of the definition (1.4). The dissipative part of the susceptibility is connected with the equilibrium fluctuations of the considered quantities (fluctuation--dissipation theorem, FDT); in our case with the polarisation fluctuations of the system. In

KIRCHHOFF-PLANCKLAW AND FLUCTUATION-DISSIPATIONTHEOREM 469 some way these fluctuations will be related with the emissive power (1.2). To find this connection the appropriate form of the polarization fluctuations must be coupled with the Maxwell equations. In this paper we will reveal and discuss the relation between the KirchhoffPlanck law and the FDT. In section 2 the calculations leading to (1.2) are presented (first order perturbation theory in quantum electrodynamics). In section 3 we apply the FDT to atomic dipole fluctuations. The demand for correspondence with the results of section 2 leads us to the appropriate form of the dipole fluctuation spectrum. In section 4 we extend our considerations to macroscopic systems and we obtain a macroscopic description of stationary radiative heat transfer in the frame of the vectorial Maxwell theoryg'l°). On this way we will make contact with the results of ref. 9 and with the herein discussed concepts for the description of EM flucutations in equilibrium 1L12) and non-equilibrium 13'14) systems.

2. K i c h h o f f - P l a n c k

law

The kets In} denote the stationary energy eigenstates of an atomic system. In thermal equilibrium this system is described by the diagonal density matrix (ij o ~ exp(-lglo/kBT))

. : <,,laol,,): exp(_

[Xexp(_ E. ]11 \

kBTIJ

(2.1) •

Under the influence of an applied EM field transitions between energy eigenstates are induced. The quantized EM field will be referred to the quantization volume V. In first order perturbation theory and in the dipole approximation we obtain the following transition probabilities for the induced emission (em) and absorption (abs) processes and the spontaneous (spon) emission processes (we assume that these transitions do not change the distribution (2.1), i.e. there must be a radiationless mechanism which keeps this distribution unaltered):

(m
-'-

^

~ P. ---~ to.,. nk,,l(mld" ~,~ln)128(,,,k - ,,,n,,), n, ra (n>m)

(2.2)

470

W. E C K H A R D T

4~ 2 p. -~-~ o~,..nk.l(m ld " e,,.In)l=,~(wk - Wm.) ,

(2.3)

n, m

(n
n

m

n~ra

(n>m) 477-2

= ~

,o. ~

o~,,,,I(m

Id- ~,,An)l=,~(o-,,,-

o.,,,,).

(2.4)

/1, m

(n>m)

In (2.2) and (2.3) nk~ denotes the averaged photon number in the mode which is characterized by the wavenumber k, the polarization vector e,, (~r = 1, 2) and the frequency Wk; d denotes the dipole operator; s represents the unit vector in the direction of k. Furthermore, we defined the transition frequencies to,,, = (E, - Em)/h. The vectors e,~, e,2 and s are mutually orthogonal. The difference of absorption and induced emission determines the energy per time (o~abs)) which is absorbed and dissipated by the system. Interchanging n and m in (2.2) we find (we use the Einstein summation convention) Q (abs)

o.)k 1

: 2- V 8~-,,,,,.h,,,,,(,~,,.),(,~,.)j 77"

x -

h

~,

(0. - om)(a,..),(a.,.)ja(,,,k

- ,.,,..)

(2.5)

n, m (n
In (2.5) we introduced the notation

[(m Id" e,,,ln)l 2 = (dmn)i(d,,,n)j(es,,)i(es,~)j.

(2.6)

The mode characterized by k, o- and nk~ corresponds to a classical electric field: E = Re{E0(k, ~r) exp(ioJkt)} ,

(2.7)

where Eo(k, or) = e,,,Eo(k , o').

We notice the time averaged EM energy density

(2.8)

KIRCHHOFF-PLANCK LAW AND FLUCTUATION-DISSIPATION THEOREM

1 ~-~

W¢m = 47r

1 [E0l z

471

(2.9)

8~"

and perform the replacement V-

1 8"n'nk~rho)k~so.~,~

,

"-}

IE012~s:s~ = Eo(k,~r)E~(k, ~r).

(2.10)

We insert (2.10) in (2.5). We put o)k = ~o and write (2.5) as an even function of o):

Q~abs) =o)(Eo),(eDj 2-

~

(p.--pm)(a,.,)i(a.m)j

n, m

(n
x {O(o))6(o) - o ) m , ) - 0(--O))6(--00 -- O),,,)}.

(2.11)

In (2.11) O(o)) denotes the Heaviside step function. Due to the 6-functions the requirement m > n for both cases, w > 0 and o) < 0, is automatically fulfilled. Therefore, we may omit the restriction m > n. We interchange n and m in the second term of (2.11) (second term with respect to the second 6-function in the brace) and we find: o)

oca~) = 7 (E0(k, o0),(E0(k, o0);,~0(o))

(2.12)

with

a°(o)) = -h- 1 - exp -

Y~ p n ( d m n ) i ( d m n ) ~ 6 ( o )

- O)mn) .

(2.13)

?1, rtl

It is well known s) that ai)(o) ) is the imaginary part of the generalized susceptibility, which is here represented in the basis of the.energy eigenstates* (see footnote list before the reference list). We define the absorption coefficient as

O ~'bs~= CWom3'(O),S, O').

(2.14)

From (2.8), (2.9) and (2.12) we obtain the relation between y and a "o* * ( s e e footnote list before the reference list) O)

tt

y(o), s, o ' ) = 4zr - (E~,),(e,,, ) joqj(o)) . c

(2.15)

472

W. E C K H A R D T

In (2.13) and (2.15) we can distinguish the absorption part and the induced emission part (see (2.11) and (2.1)):

{[O(to)-exp(-h~--T)O(-to)J q (abs)

O ,j(to) =

q (em))

[exp ( - k--~-T) 0 ( ~ ) - O(-to)]

x ~ p.(d",.),(d,".)*j 8(to

-

J

(2.16)

to.,.).

.,, n

It is now obvious that (2.15) can be represented as the difference of two (positive) parts: the absorption part and the induced emission part. Up to now we discussed the net absorption. For the averaged spontaneously emitted energy per time we obtain (see (2.4)): 1" (s~.)

(2.17)

(,,1<,,)

We take the limit to the k-continuum and we find the energy which is spontaneously emitted per time in the solid angle dO around the direction s:

B-~-~--- ~' (e,.),(e,.)j ~

~ --~-o.(d",.)i(d,..)/.

(2.18)

Or

(n>m)

We want to relate (2.18) with (2.13) and (2.15). To this aim we write (2.18) as an integral over a frequency spectrum: o0 Q (spon)

f

dto

to 4 _

• 7"r

C

,

= J Z--'-O(-to)-3 Z (e,or),(e.or)j~ p.(d",.),(d",.)/~( ~J"~

,7

to

- to,..).

m, n

(2.19) The restriction to the negative frequency part includes the condition that n > m. Therefore, the summation restriction can be omitted in (2.19). The comparison of (2.19) with (2.13) and with (2.15) yields the desired emission-absorption relations:

KIRCHHOFF-PLANCK LAW AND FLUCUTATION-DISSIPATION THEOREM 8 Q(spon)

~/2

- f dto [exph]to, ~-~2.2hO(-w) k--~-l]

-1

473

,, c to4 a,,(lwl)~--~-~Z(e,iT),(e,iT),,,, (2.20)

and

8a

=

)'(to, s, o'),

do.)Ibb(tO, T ) ~

(2.21)

0

respectively. The equation (2.21) represents the Kirchhoff-Planck law for the atomic system** (see footnote list before the reference list). We remember the crucial assumptions for the validity of (2.21): (i) Induced emission must be treated as negative absorption and (ii) the atomic system can be described by the density matrix (2.1) irrespective of the state of the radiation field. We easily can rewrite (2.19) in a form which reveals the dipole character of the radiation. We define an averaged angle 0 between s and the dipole matrix elements din.* (see footnote list before the reference list): -1

sin20 = ~] p.ldm.12sin = O,,.6(w- w,.;)[~.~ p.ld,..126(w - w,.,)]

.

(2.22)

n, m

In (2.22) 0m. denotes the angle between s and dr... If we note that ~'~ [dm," e,~l 2= sin 20m,,Id,,,,I2 ,

(2.23)

IT

and if we compare (2.22) with (2.13), we find: •

"

2

,,

= sm 0 T r aij((o )

(2.24)

IT

and oo

8Q (w°") ( dto - -~/2 - J

k--j-

_-1_ c to4

}

sin 0

-oo

The equation (2.25) can be compared with the classical formula for dipole

474

W. E C K H A R D T

radiation15). A classical dipole which is characterized by a stationary stochastic process radiates according to the formula (this formula is based on the calculation of the Poynting vector) o~

8Q(d) r do9 W4 8.0 = j ~ {(d. d ~(c')~'.,oc_~4~.--c4 sin2 0 ,

(2.26)

where 0 is the angle between the direction of observation s and the dipole axis d. The expression in the curly brackets of (2.26) represents the spectrum of the stationary dipole--dipole correlation function:

(d.

= I

d r ei~'(d(r) • d(O)) (~) .

(2.27)

The classical result (2.26) coincides with the Q E D result (2.25) if we make the substitution -1

(d . d)~')--+ 2heq~(Iw[) [ e x p h }Wl kBT

l

t

.

(2.28)

At the end of this chapter we note that for simple models the angle 0 in (2.25) corresponds to the classical angle between s and the dipole moment d: i) For the charged harmonic oscillator (charge e, mass m, frequency too) we find: 7r he 2 a','j(w) = 6 o h 2mo9o{6(o9 - o9o)- 6(o9 + o9o)} •

(2.29)

The direction d = er must be defined by an infinitesimal constant external field. We emphasize that (2.29) is independent of the temperature. The linear response is the exact response and consequently the dipole-dipole commutator is a c-number and % is independent of T* (see footnote list before the reference list). ii) For the two level atom (transition frequency too) there is only one angle (between d~2 and s), and a 0 depends on the temperature: .

~"

hogo

aq(o9) = -h--(d12) (d21)j tanh 2 - ~ T a {6(w - o9o) - 6(o9 + o9o)} •

(2.30)

KIRCHHOFF-PLANCK LAW AND FLUCTUATION-DISSIPATION THEOREM

3. Fluctuation-dissipation

475

theorem s't6,17)

In this chapter we will express the spectra of differently defined dipoledipole correlation functions by a'i~(to) (FDT's). We will find the appropriate spectrum which reproduces the Q E D result (2.25). To begin with, we define the spectrum of the symmetrized dipole--dipole correlation function:

(d.d.~ ! ] ] t o(sym)=

f dr e ~to"E P. (nl~{d,(r)d~(O)+ dj(O)d,(r)}ln)

(3.1)

n

We use the completeness relation 1 = E,, Im) (m] and note the time dependence of the Heisenberg operators. We find

(d.d.'~(sy m) = (didj)~bs) + ( d i 4 1 ~ m) , -'1-'1/to

(3.2)

where

]

(d,4)~"~ = ~- [0(,o)+ exp -k---~ O(-oJ) Z

p.(d.,.)~(d,..)j6(w

- co,.,)

m, n

(3.3/ and

(d._,_j,to d.~(°m)= rr [exp (hk----~r) O(w) + o(-,o)]

~ p.(a.,.),(a,..)js(o., - ,,,,.,). h i , ~'1

(3.4/ Similar to (2.16) we have split up (3.2) in an absorptive part (fluctuation contribution due to upward transitions) and in an emissive part (fluctuation contribution due to downward transitions). The upper and the lower states (upper and lower with respect to the reference state) must be interpreted as virtual intermediate states. We assume that the real operator d(t) can be split up into two parts:

d(t) = il(+)(t) + d(-)(t) .

(3.5)

This splitting should have the property that d (+) cannot induce any upward transitions and d (-) cannot induce any downward transitions:

(nld(+)lm) =

0

for m ~< n,

(3.6)

476

W. E C K H A R D T


0

for m ) n.

(3.7)

Therefore, d (+) can be interpreted as an annihilation operator which only contains positive frequency parts and d (-) can be interpreted as a creation operator which only contains negative frequency parts ~ (see footnote list before the reference list) (for the analogous definitions in the case of electric and magnetic fields see e.g. refs. 18 and 19). Furthermore, the annihilation and creation operators form an adjungate pair:

[dr+)] + = d (-) .

(3.8)

After these preliminaries we can define the spectrum of a normal-ordered correlation function (for the corresponding definition in the case of EM fields see refs. 20 and 21):

(didj),o(,) _- ~ dr ei.,, Z

p.(n Idl-)(r)dJ+)(O)ln)

(3.9)

n

We note (3.5)-(3.8) and find =

Z

-

(3.10)

tl, m

The splitting (3.5) enables us to define the normally ordered form of the symmetrized correlation function (3.1): In (3.1) all annihilation operators are written on the right-hand side of the creation operators. The corresponding spectrum which we denote with the expression (:dflj:), o is in close connection with the spectrum of the normal-ordered correlation function (3.10) and with the emissive part (3.4). It is easily checked that the following relations are valid: (:d,dj:)~ = (did,)~) + (d.d.~ (n) 1 ", ] 11-o9 (3.11)

= 2(didj)~ m) .

The comparison of the spectra (3.2), (3.4) and (3.10) with (2.13) leads to fluctuation-dissipation theorems:

d d, j , o~(sym)= h coth

hfJ0

LICB I

"

(3.12)

KIRCHHOFF-PLANCK LAW AND FLUCTUATION-DISSIPATION THEOREM

r

hl,ol

(d, dj)~ )= O(-to)2h [exp k--~-

r

hl,,,I

1 -~

(d, dj)~m)= h [exp-7--~-ka.l1]

1

]-' a,i([to[), ,,

,, , 41,ol).

477

(3.13)

(3.14)

The comparison of (2.20) and (2.25) with the classical formula (2.26) reveals which of the spectra reproduce the QED result: ( 4 4 ) ~ ' ) ~ 2(d,4 )~)

(3.15)

(44),~ l)~ (:d,4 :),o = 2 ( 4 g ) ~ m) ,

(3.16)

and

respectively. It was to be expected from the results of section 2 that only the (virtual) downward transitions (or the emissive parts) of the fluctuations can contribute to the radiation. We see however that these parts are doubly weighed with respect to the classical formula (2.26) a~ (see footnote list before the reference list).

4. Macroscopic polarization fluctuations We will extend the results of the preceding sections to macroscopic systems. We define the dipole density/~(r): tb(r) = ~ d(rU))8(r

-

r°)).

(4.1)

i

The vector r °) denotes the place of the atom with the dipole operator d(rU)). For a great enough dipole density we may average over an infinitesimal volume element A V which contains many dipoles (atoms). With respect to the emitted radiation we demand the inequality A -> ~/A'-ff

(4.2)

for all relevant wave lengths A. We introduce the density n(r) of the atoms and rewrite (4.1):

478

W. E C K H A R D T

P(r, w) = d(r, o~) ~

6 ( r - r c0) =

n(r)d(r, ~o).

(4.3)

i

The influence of an external electric field leads to the perturbation Hamiltonian* (see footnote list before the reference list):

tgI1 = - ~ d3rP(r)E(r,

t),

(4.4)

and we obtain the linear response equation (Pi(r, w)),~ = f d3r/3ij(r, r', co)Ej(r', ¢o),

(4.5)

where (see (2.13)) /3ij(r, r', w) --- aii(co)n(r')6(r

- r').

(4.6)

Now we can identify (4.5) with the linear constitutive equation of macroscopic electrodynamics:

P~(r, ~o)= f d3r'-~-~ [eij(r, r', w)- 6~j6(r - r')]Ej(r', ¢o) .

(4.7)

This identification allows us to extend our model to anisotropic and non-local media which are characterized by the electric susceptibility eij(r, r', w). Instead of (4.6) we may write

~i~(r, r', w) = -~ [eo(r, r', w) -

61j6(r - r')].

(4.8)

(By this generalization we get rid of the restriction to pure dipole radiation.) The linear response function (4.8) can be represented by the equilibrium commutator of the polarization operators* (see footnote list before the reference list): 1

-4-~[eo(r,r',w)-6,j6(r-r')]

= ~. drei='h O(r)([P,(r'z)'PJ(r"O)])" (4.9)

KIRCHHOFF-PLANCK LAW AND FLUCTUATION-DISSIPATION THEOREM

479

Completely analogous to (3.12) and (3.11) we obtain the FDT's

( Pi(r)~(r'))~ ym)= h coth

h~o 1 e ij(r, r, oJ) 2kBT 4rr tt

t

(4.10)

and p , Ihl¢°l ]-1 1 ,, (:Pi(r) j ( r ) : ) , o = 2 h [ e x P k - - ~ - i 4--~eo(r,r',loJ[),

(4.11)

respectively. The physical interpretation of (4.9) is crucial for all further discussions. We suppose that the measurable electric susceptibility is independent of a radiation field by which the considered body is eventually surrounded. This assumption implies that the statistical average on the right-hand side of (4.9) refers only to the material part of the system and that the temperature is only defined in the material part. Therefore, the average with respect to t~0exp(-fflo/ksT ) does not contain the long wave (non-additive) components of the EM field. These components must be irrelevant for the microscopic calculation of the electric susceptibility since otherwise % would depend on the surrounding radiation. Consequently, (4.10) and (4.11) describe polarization fluctuations due to the thermal motion of the atoms. The thermal motion leads to a long wave thermal radiation. The back reaction of this radiation onto the thermal motion of the atoms is negligible and cannot influence the equilibrium distribution of the material states. Similar to formula (2.26) which is based on the application of the free Maxwell equations (e ~-/z =- 1) to a pointlike fluctuating dipole source we now combine the macroscopic polarization fluctuations in the normally ordered form (4.11) with the macroscopic Maxwell equations (/z-= 1). The Maxwell equations have to be considered as Langevin equations: io)

I7 x E(r, w) = - - B(r, w),

(4.12)

c

V × B(r, to) . . . . i¢° { J d3r' ~(r, r ' , co)E(r', w)+ 47rP(r, ¢o)} ,

(4.13)

c

V. { f d3r' "~(r, r', ~o)E(r', ~o)+ 4zrP(r, oJ)} = O,

(4.14)

g. B(r, w) = 0.

(4.15)

480

W. E C K H A R D T

The particular solutions of (4.12)-(4.15) can be expressed by the Green functions

E(r,

o~) =

B(r, ~o) = f

d3r ' GEE(r,

r', og)P(r', o9),

(4.16)

d3r ' GHE(r,

r', oJ)P(r', o~).

(4.17)

The comparison of (4.13) with (4.7) and (4.16) reveals the relation between the imaginary part of e and the inverse of ~EE:

1 eii(r, ,,

4rr

r ,, w) = - I m tr rUEE(r, r ,, o~)]~l

(4.18)

The formula (4.11) allows us to calculate the electric, the magnetic and the mixed correlation tensors of the EM field. For instance, we can calculate the EM energy density '°) and the Poynting vector 9) in the vacuum regions of inhomogeneous systems. Up to now, we have been led by the procedure of sections 2 and 3. The concrete application of (4.11) to inhomogeneous systems shows some peculiarities on which we now have to focus9). Let us assume that our system consists of material regions and of vacuum regions. These regions are separated by sharp boundaries on which the EM boundary conditions have to be fulfilled (the special peculiarities with respect to the Poynting vector which appear at the sharp boundaries of non-local media were treated by Bishop and Maradudin22)). The expression (4.11) vanishes if r or r' lie in the vacuum regions. We distinguish three different cases: i) If the system is bounded (i.e. finite; e.g. the system is bounded by walls with an infinite electric conductivity) we assume that e " = 0 in the vacuum parts: A priori, there is no temperature attached to these parts; formally, in (4.11) the limit e " ~ 0 corresponds to the limit T ~ 0 . It is obvious that in this case the solution of the Maxwell equations (4.12)-(4.15) has to result in an equilibrium radiation in the vacuum regions, because stati0narity and perfect boundaries demand equilibrium between matter and radiation (see example 2 in section 4 of ref. 23). If there are infinitely extended vacuum regions, the system can be treated as a closed one or, alternatively, as an open one: ii) We demand that the system is closed. T o this aim we formally put e" different from zero also in the vacuum parts and we postulate a global temperature T. It can be seen from (4.16) and (4.17) that the calculation of

KIRCHHOFF-PLANCK LAW AND FLUCTUATION-DISSIPATION THEOREM

481

correlation functions includes integrations over the volume of the total system. The limit e " ~ 0 in the vacuum parts has to be taken after these integrations have been performed (see refs. 12, §89 and example 1 in section 4 of ref. 23). iii) The definition of an open system (freely radiating body) includes that e" is put to zero in the vacuum parts from the beginning (~- T = 0). In this way we can describe radiation transfer problems9). It is desirable to incorporate the Q E D zero point contributions in the above sketched semi-classical theory. For that purpose we add the T = 0 contribution of (4.10) to the normally ordered form (4.11):

(: P , ( r ) e j ( )r : t) .

r

+

hl,,,I

0

=2h Lexpk--~-l]

1-11

., -~-~eii(r, r', lw[, T ) + h 1 e#(r, r,l~ol, T 41r 0

-

t

-

=

O)

,

(4.19)

It is clear that the zero point contribution must be treated in the sense of the closed system: With respect to this part there always must be "equilibrium" between matter and radiation. Now we make use of the fact that e " = 0 corresponds to T = 0 in the vacuum regions and that in the linear regime of the Maxwell equations the linear response is the exact response. Consequently, the commutator in (4.9) is a c-number and e'i~ is independent of the temperature (for comparison see (2.29) and (2.30)). Therefore, (4.19) leads back to (4.10); but now the application of (4.10) has to be supplemented with the prescription that for freely radiating bodies the temperature has to be put zero in the vacuum parts and that the limit e" = 0 in the vacuum parts has to be taken after the spatial integrations have been performed. On the contrary, the treatment of systems which are in global thermal equilibrium demands the definition of an a priori temperature at every point of the system # (see footnote list before the reference list). Finally, we want to make contact with the Kirchhoff-Planck law (1.2) and (2.21). Starting from formula (4.10) and following the above prescriptions we calculated in ref. 9 the Poynting vector for the freely radiating isotropic dieclectric half-space z < 0. We found oo

(S)=ez f dw 0

f (O<~r/2)

dalbb(oJ, T)cosO~[y,l(oJ, O)+y±(oJ, O)].

(4.20)

482

W. E C K H A R D T

($) denotes the averaged Poynting vector along the z-axis; YII and Yi polarized parallel and perpendicular which impinges with angle 0 on the

vector in the region z > 0 ; e z is the unit describe the absorption of light which is to the plane of incidence, respectively, and plane z = 0:

sin 2 0] 1/2_ rtl(,o, 0) = 1 -

yl(w, O) = 1 -

E (¢.o) COS

0

[ e ( t o ) - s i n 2 0 ] m + e(t0)cos tl

2

(4.21)

'

[ e ( w ) - sin 2 O]m - cos O] 2 [e-~)-- sin z O]m + cos 0

(4.22)

The analogy of (4.20) with (2.21) gets even more distinct if we write (4.20) in a slightly modified form: ~c

(S)=fd¢o o

f dg2I(w,T)sO(2-O),

(4.23)

(4~r)

where

I(w, T) dF± -: Ibb(~O,T)~[TII(o, 0) + yl(w, 0)] dF_.

(4.24)

In (4.24) we introduced the surface element dF+ which is perpendicular to the light ray: d F ± = d F cos 0. Therefore, the quantity coincides in dimension and in its physical meaning with the integrand in (2.21).

I(to, T)dF±

5. Summary We have seen that the concept of treating induced emission as negative absorption leads to the validity of the Kirchhoff-Planck law for freely radiating atomic systems. This validity implied the crucial assumption that the occupation probability of the energy eigenstates is determined by the Boltzmann distribution. The dissipative properties of macroscopic systems are determined by the generalized susceptibility. The representation of this quantity in the basis of the undisturbed stationary energy eigenstates revealed that the emissive transitions are considered as negative absorptive transitions. Due to these facts the Kirchhoff-Planck law could be formulated as a FDT. We saw that the spectrum of the normally ordered form of the symmetrized

KIRCHHOFF-PLANCKLAW AND FLUCTUATION-DISSIPATIONTHEOREM 483 dipole-dipole correlation function is the proper spectrum which reproduces the QED result. We used these insights for the extension of our considerations to macroscopic systems. Therefore, the validity of the results which are obtained by the polarization fluctuations (4.11) and the Maxwell equations (4.12)-(4.15) principally implies the same assumptions as the Kirchhoff-Planck law for a freely radiating atomic system. We had to demand that in the density operator only the additive (short wave) components of the EM field are relevant, i.e. the long wave (non-additive) parts can be neglected. The c-number character of the polarization commutator in the linear regime made it possible to include the T = 0 contribution in the semi-classical theory. In this paper we have not discussed the limits of the above made assumptions. We only remark, that-dependent on the atomic structure of the condensed m a t t e r - t h e r e could be surface states which cannot be described by an equilibrium density operator (temperature T) if the body is not in equilibrium with a surrounding radiation. These microscopic surface effects can only be handled by a dynamical non-equilibrium theory. Such a theory is far out of the scope of this paper.

Footnotes

*The real part a~/(to) can be calculated by use of the Kramers-Kronig dispersion relation. The generalized susceptibility aii(to) = a~(to)+ ia'i~(to) represents the linear response of the dipole moment d to the action of an applied external electric field: tt

~,j = (d, Cto))ool~ (to).

(di(t))ne denotes a non-equilibrium expectation value which is based on the density operator iS- exp [ - k ~1 (/_~0..~./~tl)] , where Ht describes the influence of the external field:

H, - d . ~(t). =

We assumed that the expectation value vanishes in the undisturbed system.

484

W. ECKHARDT

% ( w ) can be expressed by the equilibrium expectation value of the dipoledipole commutator s) ~e

ao(w ) = ~ dr e i~" ~-i 00") ([d,(~'), dAo)]).

The representation in the basis of the energy eigenstates can symbolically be written in the form:

('" ") =

p.(nl

"

" In).

n

** We emphasize that the definition of the absorption coefficient (2.14) is different from the definition which was used in (1.2): In (1.2) the dimensionless quantity 7(w, T) is defined as the ratio of the absorbed specific intensity and the incident one. In (2.14), 7(w, s, o') represents the ratio of the absorbed energy per time and the incident energy flux. In a rarified ensemble of identical atoms instead of (2.1) one could use the occupation numbers of the energy eigenstates divided by the volume of the ensemble. In this case the dimension of T((o, $, or) is cm -1. In ref. 2 the so defined quantity represents the absorption coefficient in the phenomenological equation of radiative heat transfer. t dmn can be considered as a real quantity if the energy eigenstates are not degenerate. An arbitrary expectation value of d (÷) can be expanded in terms of the energy eigenstates:

(~laC+)(t)l~0) = ~

a'am exp(iWmnt) (nld~+)lm)

n, m

= ~,a*.a,n

~-~e-i~'8(w+CO,m)(nld(+)lm).

n, m

Due to (3.6) this expectation value must vanish for W,m > 0. For n < m (i.e. for w,,, < 0 ) the 6-function only contributes if co > 0. Therefore, the operator d(+)(t) only contains positive frequency parts. Analogously a (-) only contains negative frequency parts.

KIRCHHOFF-PLANCK LAW AND FLUCTUATION-DISSIPATION THEOREM

485

~~There is another way to express this fact. We note that the spectrum (3.11) can also be written in the form:

(:didj :)o, = (didj)~m) + exp ( - h~BT) tdd )Cabs) •, - - i - ' 1 j aJ

"

Therefore, the reproduction of the Q E D result demands that the absorptive part (virtual upwards transitions) is weighted with the factor exp(-hto/kBT ) while the emissive part is unaltered. In the classical limit kaT >>hto the spectra (3.11) and (3.12) are identical and the emissive part (3.3) equals the absorptive part (3.4). # The qualitative properties which were stated above are presented in ref. 9 in a more detailed manner. Furthermore, in ref. 9 the relations between (4.10) and the theory of EM equilibrium fluctuations are discussed.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

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