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On fluctuations and coherence of radiation in classical and semi-classical plasmas
Felderhof, B. U. 1965
Physica 31 295-316
ON FLUCTUATIONS AND COHERENCE OF RADIATION IN CLASSICAL AND SEMI-CLASSICAL PLASMAS by B. U. FELDERHOF*) Department of Chemistry, MassachusettsInstitute of Technology, Cambridge, Mass., U.S.A.
synopsis The theory of thermal equilibrium fluctuations in plasmas, developed in a previous paperl), is extended to include radiation. Expressions are derived for the correlation tensors of the electromagnetic field.
9 1. Intro&ctio~~. In this paper the theory of thermal equilibrium fluctuations in plasmas, presented in a previous paper I), hereafter referred to as I, is extended to include radiation. Explicit expressions are derived for the correlations in space and time of several quantities of interest. In particular the correlation tensors for the electromagnetic field, which characterize the coherence properties of the radiation, are calculated; in principle these quantities may be experimentally investigated. For simplicity we have restricted ourselves to the electron gas with positive background, but the theory may without difficulty be extended to many-component plasmas. Almost all work on the subject seems to have been done by Russian scientists. Most of their work2) 3) is based on a suggestive, though in our opinion not completely satisfactory theory of Rytov4) on electromagnetic fluctuations in general media. Silin and Klimontovich have taken several other approaches, but these are either open to questions) 6) or require ad hoc assumptions to arrive at results7). Recently, Dupre es) has pursued Klimontovich’s work but he does not give explicit expressions for the electromagnetic field correlations. The present theory is based on the canonical ensemble of equilibrium statistical mechanics. Canonical variables are introduced for the degrees of freedom of the radiation field. The continuum approximation is used to derive a probability ensemble for the simultaneous probability of a given distribution f(r, p) of particles in 6-dimensional p-space, a given vector potential A(r) and a given transversal electric field S(r) (section 2). Subsequently it is shown that this probability ensemble exhibits a maximum *)
On leave
of absence
from
the Institute
-
for Theoretical
295 -
Physics,
Rijksuniversiteit,
Utrecht.
296
B. U. FELDERHOF
at the thermal equilibrium configuration corresponding to a MaxwellBoltzmann distribution of the particles and vanishing electric and magnetic fields leads
(section 3). Expansion of the ensemble about thermal equilibrium to a Gaussian probability distribution for fluctuations {fi(r, p), set ion 4) ; this distribution is the basis of all further calculations. A(r), 8(r)> ( t Like in I, it appears that the exponent of the Gaussian distribution is
diagonalized by the normal mode solutions of the set of linearized Vlasov and Maxwell’s equations (sections 5 to 11). This permits explicit derivation of joint probability distributions of quantities which depend linearly on fr(r, P), A(r) and b(r). In low density, high temperature plasmas collisions may be neglected and the time behaviour of a fluctuation {jl(r, p), A(r), &f’(r)}is governed by the set of linearized Vlasov and Maxwell equations. In this case the formalism is particularly adapted to calculate correlations not only in space, but also in time. In section 12 a general expression for the correlation of any pair of quantities taken in two different points of space and time is presented. In section 13 the most important correlations are calculated. The density correlations and the correlations of the electrostatic potential are found to be not affected by the presence of the radiation, but the fluctuations in the current density are, of course, influenced by the radiation fields. Next the correlation tensors of the transverse electric and magnetic fields, which characterize the coherence of radiation in the plasma, are derived. It is shown that the spatial correlations at equal times are the same as those in vacuum. Remarkably, the long range part of the longitudinal electric field correlations just cancels against the transverse field correlations, so that the total electric field shows no long range correlation. This might affect transport coefficients. In section 14 the range of validity of the above results is investigated. Also the average energy density of a fluctuation is calculated and the use of classical statistics for the radiation fieldis discussed. In section 15 the theory is extended to the semi-classical case, in which Fermi-Dirac statistics of the electrons is taken into account; otherwise the treatment is classical. It is shown to what extent the correlations are modified by exclusion effects in section 16. § 2. Probability ensemble. We shall consider a gas of N electrons of charge e, mass m in a volume Q with a positive neutralizing background of charge density --nee, where no = N/Q. There are also electric and magnetic fields E and B which may be derivedfrom a vector potential A and a scalar potential Qi,
E=--
1
c
i3A - grad@,
at
We choose the Coulomb-gauge,
B = curl A.
P-1)
i.e., div A = 0.
(2.2)
FLUCTUATIONS
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COHERENCE
OF RADIATION
IN PLASMAS
297
From Maxwell’s equations it then follows that A and @ satisfy AA---
1
82A
c2
at2
= - f
- igrad:
j
A@ = - 4-cp,
cw
where j is the current density and p the charge density. In this gauge the potential @ does not enter as a separate degree of freedom but only serves to transmit the instantaneous Coulomb interaction between the electrons
4-J 4 =
e2 ,r
_
r,,
.
(24
In a canonical formalism the momentum conjugate to the field variable A is given by -&‘6/4zzc, where d is the rotational part of the electric field &=--
1 aA c at’
div 8 = 0.
(2.5)
In thermal equilibrium the canonical ensemble determines the probability that the particles have positions between ri and ri + dri, momenta between pr and pi + dpg and the vector potential and transverse electric field at I values between A and A + dA and & and I + d& respectively. In order to perform actual calculations we wish to use a smeared-out description to which purpose r-space and p-space are divided into cells; the cross-sectional cells in (r, p)-space should be large enough to contain many particles and so small that the kinetic energy and the interaction do not vary appreciably over the cells. One now asks for the probability that the cells in (r, p)-space contain given numbers of particles and that the potential and field have given average values in the cells in r-space. Or, circumventing the explicit introduction of cells one asks for the probability that the distribution function /(r, p) assumes values between f and f + df and potential and field values between A and A + dA and & and 8 + db; here it should be kept in mind that f, A and & only have a coarse-grained meaning. The division of particles over cells gives rise to a combinatorial factor of entropic nature exp[-
SfPt P) log f(r, P) dr +I
(2.6)
and at a temperature T there is an energy factor of the form exP[-BVf(r,
P)7 A(f), W)Il,
(2.7)
where /I = 1/kBT and U { } is the energy of a configuration {f(r, p), A(r), 8(r)}. According to the canonical ensemble the probability that a configuration assumes values between {f(r, p), A(r), 8(r)) and {f + df, A +
29%
B. U. FELDERHOF
+ dA, 6’ + d&Z’}is therefore given by df dA d& =
P{f(r, P), A(r), g(r))
= exp !P{f(r, p), A(r), 8(r)} df dA dI/JP
df dA d&,
(2.8)
where
-S[ -
-f(r, p) log f(r,p) - B
-+p/zu(r, --it3
s
(’ -2iA’c’2 f(r, p)]dr dp -
r’) f(r,p)f(r’,p’)
drdpdr’dp’
+ B/m(r, r’) f(r,p)
w(r, r’) sz dr dr’ -
$-s[(curl
A(r))2 + &‘(r)2] dr.
no drdpdr’-
(2.9)
In the expression for the energys) the interaction of the electrons with the background and of the background with itself has been taken into account. $3. Thermal equilibrium configuration. The probability ensemble (2.8) will exhibit a maximum at the thermal equilibrium configuration. At the maximum the first variation of Y must vanish for arbitrary variations sf(r, p) subject to the condition that /f(r, p) dr dp = N remains constant, i.e.
and for arbitrary
solenoidal
dA(r) = curl g(r), M(r)
(3.1)
variations
dA(r) and &T(r),
= curl q(r),
where g(r) and q(r) are arbitrary fields. Assuming the variations to vanish at the boundary and taking the condition (3.1) into account by a Lagrange multiplyer y, one obtains that Y/ is stationary for solutions of
1 - 1%f(r, P) - B
(p - eAM2 2m
- @ 1 w(r, r’) f(r’, P’) dr’ dp’ + + ,9 /w(r,
curI[$J(p-$)f(r,p)
f
dp -
r’) 1~0dr’ - y = 0
$curlcurlA(r)]
= 0
(3.2a) (3.2b)
curl &t(r) = 0. One easily finds from (3.2a) that for a stationary point the current density,
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
IN PLASMAS
299
given by
j(r)=$
p-- em9 SC >
f(r,
c
vanishes identically. Equation (3.2b) therefore which together with div B = 0 leads to
PI dP7 yields
curl
curl B = 0,
LlB = 0. Equation
(3.3)
(3.2~) together with div &’ = 0 yields 8 = const..
A possible solution of (3.2~~)satisfying f(r, p) = no Since the second functional
(&),
cv!P Sfp, p) df(r’, P’)
d(r = -
(3.1) is given by
exp [ -
derivative
(3.4)
@ (’
-2c’c)2].
(3.5)
of Y with respect to f,
r’) S(p - p’) f (r, P)
-
@(r,
4,
(3.6)
is easily shown to be negative definite, it follows that (3.5) is the only solution of (3.2~) for given A. Equations (3.2) yield an infinity of configurations for which Kis stationary for variations vanishing at the boundary, and given by d = const., solutions B of (3.3) with corresponding vector potential, and f(r, P) given by (3.5). It will be shown in section 11 that one of these, the thermal equilibrium configuration feq(r, P) = fiofM&)
= no (&)t
exp [ - $J
Aeg = 0 &,
= O,
(3.7)
is a maximum. (Actually A,, is determined up to a constant by B,, = 0, but a constant in A may always be eliminated by redefining the momenta). Now all terms of (2.9) except the last one are the same for all stationary configurations, whereas for any configuration for which the electric and magnetic fields do not vanish the last term in (2.9) is smaller than it is for the configuration (3.7). Hence the thermal equilibrium configuration (3.7) is the absolute maximum of P. $4. Fluctuations. In order to establish a probability ensemble for fluctuations {fl(r, P), A(r), d?(r)} about the equilibrium configuration (3.7), the quantity Y must be expanded to second order in these deviations from
300
B. U. FELDERHOF
equilibrium.
The second functional
62Y
derivatives
of Y are, besides (3.6),
=g(p-+)6(r-r’)
dfh PI Wr’) 6‘?P
PP
sf(r, p) M(r’)
= O’ 6A(r) M’(r’)
fwf
0
=
(4.1)
- J?? &+?(r - r’) f(r, p) dp mc2 s
GAi(r)SAj(r’) =
$
(
.-.
454
82
axg245> d(r -
d?P
- -&
&Tt(r) SEj(r’) = Calculating
&5 d(r -
r’)
r’).
the difference ~{~o~MB(P)
fl(r, p), A(r), g(r)} - !+o~MB(P),
+
to second order one obtains for the probability
0, O}
ensemble of the fhctzcations
P{fl(r, P), A(r), b(r)} dfl dA dg = =exp
[
+ & --
+
s
fk
P)
nOfMB(p)
/p-A(r)
dr dp -
t
s
w(r,
fl(r,p) dr dp - g
r’) fl(r,p) fl(r’,p’)
dr dp dr’ dp’ +
/A(r)2 dr -
' 1 [(curl A(r))2 + E(r)21 dr] dfl an
dA d&‘/SF’
dfi dA d&.
It is of interest to remark that now the energy part of the exponent is no longer negative definite, as in the electrostatic case. The complete exponent in (4.2), however, is negative definite, as will be shown in section 11. The time behaviour of a fluctuation {fl(r, p), A(r), &S’(r)} will be assumed to be governed by the linearized Vlasov equation and Maxwell’s equations. Consequently, the probability ensemble (4.2) enables one to calculate joint probability distributions of quantities, expressible in fl, A and 8, taken at different points of space and time. It will be shown that, in analogy to the electrostatic casel), the ex$onent in (4.2) is diagonalized by the normal mode solutions of the set of linearized Vlasov and Maxwell equations. 9 5. Normal modes. function reads ;+
P - eA(r)/c
The potentials
m
Vlasov’s
at ‘F
equation
a ar [
satisfy Maxwell’s
for the one-particle
(P - eA(r)w 2m equations
+e@].-$
distribution
= 0.
(5.1)
(2.3) with charge and current
FLUCTUATIONS
AND
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OF RADIATION
IN PLASMAS
301
densities
p(r, t) = e[Jf(c p, t) dp - m01
A time-independent
eA(r, 4
S[
j(r, t) = f
p -
solution
c
1f(r)
of these equations
f(r, p) =
no/o(P),
A
=
0,
(5.2)
t) dP.
P#
is given by @
=
0.
(5.3)
Here /a($) is normalized to unity and will only be assumed to be an isotropic function of p ; this enables us to cover also the semi-classical case. The linearized set of equations for deviations from this time-independent solution may be written
+ +
-
ad5 at
=
s,:_::,,
aoe2
fl(r',
4ne -_cdA+?!&-C m
p’, t) dr’ dp']
p'fdw',
4 dp’
A(r, t) b(r, t)
(5.4
> I’
where the index J_ indicates that the rotational part taken and where wp = (4nrtees/m)* is the plasma potential @ has been eliminated by solving Poisson’s One now looks for normal modes, i.e. stationary the form
fl(r, p, t) = g&p)
a$ = 0
of the vector is to be frequency; the scalar equation. periodic solutions of
ez(k’r--wt)
= A,,, e”(k”-“t) =
(5.5)
cYk,@ei(k*r-ot).
For fixed k we shah always take the z-axis in the direction of k. Introducing the variable s = ma/K one finds from (5.4) that the normal modes must satisfy
sA1,* = -
SdZkS=
;
imc k
f%s
W2c2 + $1 A,,, -
F
( /P%~,.(P’)
dp’>i
(5.6)
302
B. U. FELDERHOF
where sp = mc+Jk. The index J_ now means projection cular to k. In the following two sections a complete obtained for these equations. 5 6. Reduced equations.
in the plane perpendiset of solutions will be
In order to solve equations (5.6) the same method
will be used as employed in ref. 10. On account of the isotropy of fe equations (5.6) easily separate for the three directions of polarization. Define
dPz dP?J %(Pz) =JYP&,AP) dPx dPw 4, G,APz) =flPv&,AP) dPx dPw 4, G,.s(Pz) =/J&s(P)
= k%,s)z?4,s = mz,s)z = (Ak,& c,, = Wk,s)Y. (6.1)
For h.O(&) one obtains from (5.6) (s where
(6.2)
Pz) %,,(Pz) = -s;~‘(Pz)_%,,(P6) dPL>
(6.3)
F(Pz) = JYfdP) dPx dPv*
(We employ the same notation as in ref. 1 and 10, but since we here work in momentum space the corresponding quantities differ by factors m.) Equation (6.2) has a solution for any real value of s,
+
4 d(s- pz),
Xo(k
(6.4)
where P stands for principal value, and where Xa and Ye are the real and imaginary part of *)
Zo(k, s) = 1 - s;
s
F’(Pz) p, - S -
iE
dPz>
(6.5)
so that
Xo(k, s) = 1 - s;P
s
F’(Pz) pz - s dpz
____
(6.6)
Yo(k, s) = -ns;F’(s). The solution (6.4) is normalized to unity. Completeness of the set of solutions (6.4) has been proved by van Kampenlr) ; any function uniquely written as a superposition of these solutions. For the second set of quantities in (6.1) one obtains from
b-
PJ G,,(Pz) sA’ k,s
= -
(5.6)
4,
PAPZ)
imc =
SC& * = *) See appendix.
F
of p, may be
-
-
k
G,,
-& (m2c2 +
si)
Ai,s -
4nie K
s
G,AP2 dPL* (6.7)
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
303
IN PLASMAS
Again there is a solution for any real value of s. With the normalization J-h;,,(&)
dp, = s2 -
mW
-
s;
Xl@,
s) qs
(6.8)
one obtains h&J+,) = ; A’
k.5
7”;;)
+
-
pz)
4cemc
=_-
(6.9)
k2 4nies -3 k
& =k,s
where X1 and Yr are the real and imaginary Zl(k, s) = s2 -
w&9 + ss;
so that X1(k, s) = 9 -
part of *)
s
F(Pz)
p, - s - ie dPZ
w&2 + ss;P
s
F(Pz) p, -s dpz
~
(6.10)
(6.11)
Y,(k, s) = ns;sF(s). The proof of completeness of the solutions (6.9) is entirely analogous to that given in ref. 10 ; any set of quantities {f(&), A, d?} may be uniquely written as a superposition of these solutions. The third set of quantities in (6.1) leads to a set of equations identical to (6.7) ; the solutions are the same as (6.9) and need only be distinguished by an index 2. Note that Zl(k, s) = Zz(k, s). 5 7. Solzltions of eqzGatiolzs (5.6). There are, of course, solutions of (5.6) which reduce to the solutions of the preceding section when the appropriate integration procedure is carried out. Solutions corresponding to (6.4) are
d,,(P)=
afo/aPz qp,)
h:,,(pz),A:,, = 0, &:,a= 0.
Similarly, the solutions corresponding to (6.9) and the analogous for the other direction of polarization are readily found to be
(7.1) solutions
(7.2)
where use has been made of the isotropy l)
See appendix.
of
fo.
304
B. U. FELDERHOF
There are, however, also solutions of (5.6) with vanishing charge and current densities and vanishing potential and field. The dependence on p, of these solutions g(p) must be given by a delta function and the solutions vanish identically-when
the reduction
procedures
of the preceding
section
are applied. We shall consider a series of such solutions
afom
G,,(P)= ____ F’(p
z
A;,
J&*(P)
P?A S(s- Pd
(7.3) (V = 3, 4, . ..).
is a polynomial
G,,(Pz, Pd
PUP,,
= 0
CF;,, = 0 where
)
in p, and 9, with
dP = JPa&,AP) dP =/P,&,,(P)
dP = 0.
(7.4)
Moreover, the polynomials will be chosen in such a manner that the following orthonormality relations hold
s
afow
~‘(p
Pb(Pzt
Pd P;,s(Pz> Pd
d(s-
PA
(7.5)
dp = 4w
2
If one extends the series of polynomials so that (7.5) also holds for PO = 1, Pl = const. .p5, P2 = const. *p,, which is possible on account of (7.4) and the isotropy of fo, then the series may be constructed in a unique fashion from (7.5) by successively including higher powers of 9, and 9,. In section 10 this procedure will be carried out explicitly for the case that fo(p) is the Maxwell-Boltzmann distribution. Obviously the solutions (7.1) and (7.3) cannot be used if there are regions in p-space for which F’(p Z) vanishes identically. Although the analysis could be extended to include this case, we shall for the sake of simplicity assume that F’(p,) vanishes only at isolated points. A similar assumption will be made about afo/ap,, since otherwise the solutions would clearly not suffice to expand an arbitrary function of p. In the cases which are of interest to us, viz. the Maxwell-Boltzmann and Fermi-Dirac case, these assumptions are evidently
satisfied (except
for T = 0).
3 8. Adjoint sohtiorts. For expansion purposes it will be convenient to have the solutions of the system adjoint to (5.6) at our disposal. The adjoint system of equations reads
(S- Pd g;,,,(P) = h,s s&ks=
= $, -
TP*&,,
- s$&
(m2c2 + s;, &,, + 7 imc A”,,,. k
&,&p’) dp’ (sp’
f$
,&(p’)
dp’)l
(8.1)
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
305
IN PLASMAS
A first set of solutions is E:,,(P) = @,,(PZ)>
4,s
= 2i.S = 0,
(8.2)
where
Q,s(Pz) =
,zo(;, s),2
[;
y;;
X0@,4
+
qs-
pz)]
is the solution of the equation adjoint to the reduced equation (s -
P,) @,,(P,)
= -
s;SF’(PS)
The solution (8.3) has been normalized orthonormality relation holds
&,(PL)
dP;.
(8.3)
(6.2) Is), (84
in such a way that the following
(8.5)
/Gi,,(Pz) @,APz,dPz= 0 - 0
This may be verified by direct calculation with the aid of the PoincarCBertrand relation I (9.2). It follows also from the solution of a singular integral equation 11). A second and third set of solutions of (8.1) are given by
(8.6)
the other components
of A and 2 vanish. Here 1
%h
=
,Zl(h,
p q,2
[
y
Yl(k s _
s) p,
+
Xl&
4
Sb
-P.,]
together with the nonvanishing components of A1 and & of the system adjoint to the reduced equations (6.7)
(s -
Pz)
&,(fJz)= sA;,, =
F
$
SC@-& = -
imc
is the solution
&,,
(mW +
K
(8.7)
s;,&, -
y/PlF(Pl)
@,,(P2dp;: (8.8)
so that the following
orthonormality
A:,.
This solution has been normalized relation holds
306
B. U. FELDERHOF
This may again be verified by direct calculation; it follows also from the solution of a singular integral equationlo). For the third set of solutions G,,(PZ) = @,,(P z1, and a relation similar to (8.9) holds. As follows from (7.3) and (7.4) a final set of solutions of (8.1) is given by
!i%,s(P) = K,,(P) 4s - Pz) P*,, = S-L,, =
(8.10)
(aJ= 3, 4, . ..).
0
with an orthonormality relation which follows from (7.5). 3 9. Expansion in normal mode sol&ions. From the orthonormality relations (7.5), (8.5) and (8.9) and the isotropy offa it follows that the following orthonormality relation holds between the solutions of section 7 and their adjoints of section 8 J&(P)
&,(P):dP
+ A:,;&,,
+ &,&,a,
= &W
- s’).
(9.1)
This relation will be used to determine expansion coefficients. The following expansion theorem may now be stated: Any perturbed configuration {fl(r, p), A(r), 8’ (r)} may be written
dsC,(k, s) AL,, eik.r
A(r) = &yJ k,v d?(r) = 7
C’SdsC,(k. k,v
(9.2)
s) Rk,, eik’r,
where the expansion coefficients C,Jk, s) are given by
G(k, s) =
& ~dremik”[ /g;,,(P) fl(c p) dp +
&,s-A(r) +
&,;+j]. P-3)
Furthermore, the solution of the linearized Vlasov equation (5.4) with given initial configuration (9.2) at t = 0 is obtained by multiplying every coefficient C,(k, s) by an harmonic factor exp[-&St/m]. Since the completeness of the solutions (6.4) and (6.9) has been proved in another connection, the completeness of the solutions {g;,,(pj, A;,, c%‘;,,} hinges upon the completeness of the polynomial functions (7.3). From the construction it is clear that the normal modes constitute a basis for a large class of configurations (fl(r, p), A(r), 8(r)}. The prime on the summation sign in (9.2) pertains to the k = 0 component. From the restriction/fr(r, p) dr dp = 0 it follows that the k = 0 component of fr vanishes when integrated over p. Furthermore, in order to avoid
FLUCTUATIONS
unnecessary
AND
complications
COHERENCE
we exclude
OF RADIATION
solutions
307
IN PLASMAS
for which A(r)
or d?‘(r)is
homogeneous in space by imposing the condition that A and d vanish at the boundary. This restriction will not affect the final results. It is then readily seen that the functions g:,,(p) for Y = 3, . . . for some chosen k may be taken as the basic set for the k = 0 component of the distribution function; the prime on the summation sign in (9.2) therefore the terms v = 0, 1, 2 do not occur.
indicates
that for k = 0
3 IO. The Maxwell-Boltzmann case. For the case that f&5) is the MaxwellBoltzmann distribution (3.7), the explicit expressions for the distribution function parts of the normal modes of section 7 are
&8(P)=&exP
[
-B
PZ+-Pi 2m
I[ ;
p YfE(k Pz) s _p,
+Xyv, s)4s- Pz)]
(v = 3, 4, . ..) (10.1) where v represents the pair (m, n) = (2, 0), (1, I), (0.21, etc.. The functions H, are the usual Hermite polynomials. The adjoint solutions of section 8 are given
%8(p)
=
by 1 IZfB(k,
s),z
&:,2(P) = Px, Y * ,z”&
192
b%,,(P)=
Hm(Pz~
P Yf”“(k, s) s _ p, + XifB(k, 7c P
s),2 IIT
Y?i?(k,
s’_ p,
s)
PA],
+ Xfy(kJ 4 S(s- P.)]9
Hn(Pd/B/W S(s- P,), (v =
d2mrn ! 2% !
In this case completeness functions.
s) S(s -
follows from well-known
3,4, . ..).
theorems
(10.2)
on Hermite-
§ Il. Probability distribution of the coefficients C,(k, s). With the aid of the expansion (9.2) the probability ensemble (4.2) for the fluctuations may now be transformed into a probability distribution for the coefficients Cv(k, s); since the transformation (9.2) is linear the phase-space factor is a constant which drops out against the normalization. If (9.2) is inserted into (4.2)
308
B. U. FELDERHOF
one obtains explicitly P{Cv(k, s)} = exp .
z’j+ds
- q
s
g:,,(P)g&(P)
dp
ds’ [ yT:(k,
s) Cv(k, s’) *
+
2’ytOfMAP)
+ 7
(C;(k
*[(s2 + s’2 -
S) Co@, s’) + [C;(k, s) Cl&, s’) + C;(k, s) Cz(k 2vnw - 24
+ s; + mw
+ ss’])
s')l.
I/S P,
(11.1)
where the different terms correspond to those in (4.2) in the same order. If one now inserts the expressions (10.1) in the integral over p in the first term and uses the Poincare-Bertrand relation I (9.2) one obtains
*(s2 + ss’ + s’2 - m3c3 -
s
si)
( 11.2)
(aJ= 3, 4, . ..).
whereas for v # v’ the integral finally obtains P{C,(k,
s)) = exp [ -
vanishes.
7
Inserting
(11.2)
C’ l ds ~~~f~(~‘:) k,v Y )
into (Il. 1) one
I//P,
(11.3)
where
Llpyk, s) = Llfyk,
120
83~3 IZfB(k,
s) = LqB(k,
d,MB(k,s) =
FMB(s) s)13 mFMB(s)
s) = 2% 8723 Bj.ZFB(k, s)13
$$ FMB(s)
)
(11.4)
(v = 3, 4, . ..).
The prime on the summation sign in (11.3) indicates that the terms k = 0, v = 0, 1, 2 are absent (cf. section 9). The exponent in (11.3) is not only diagonal but also negative definite, which shows that P has a maximum at the thermal equilibrium configuration (3.7).
FLUCTUATIONS
Since
according
AND
COHERENCE
to linearized
Vlasov
fluctuation is found by multiplying factor exp[-&t/m], the ensemble this theory. $ 12. Joint probability
OF RADIATION
theory
309
IN PLASMAS
the time
behaviour
of a
each coefficient C,(k, s) by an harmonic (11.3) is obviously a stationary one in
distributions.
As the exponent
of the
Gaussian
probability ensemble (11.3) is diagonal in the coefficients C,(k, s) it is an easy matter to derive probability distributions for quantities which may be expressed in terms of these coefficients. The joint probability distribution of quantities which are linear in the Cp(k, s) will, of course, again be Gaussian. In I the expressions for the probability distributions of one and two quantities have been derived. Sufficient information is contained in the expression for the correlation of any two quantities linear in the coefficients. In particular we shall consider (real) quantities of the form
Q(r, t) =
7
F: jdsC,(k,
s) av(k, s) eik’r--ikst’m.
For example, for the vector potential A(r, t) one has a,(k, s) = A;,, similarly for the electric field. For quantities of the form
Q(r, t) = /p(r
-
(12.1) and
r’, p’) fl(r’, p’, t) dr’ dp’
(12.2)
p) &(p)
(12.3)
one has a#,
s) = /p[r,
e--ik’r dr dp.
One must be somewhat careful in taking averages over the ensemble (11.3), since complex quantities have been used. In averaging the product of two quantities Qi and Qs one can either explicitly take into account that they are real and consider real and imaginary part of the coefficients C,(k, s) to be independent, or one can average the product QiQi, but then dependence of real and imaginary part of C, must be accounted for, which yields a factor
8. One thus obtains
= Re alv(k, s) a&(k) s) eik’(r-r’)-iks(t-t’)‘~~YMB(K, s) ds.
(12.4)
The correlation depends only on the differences r - r’ and t - t’ in accordance with the translation invariance in space and time of the system. 3 13. Density, currertt density, potential and field correlations. One of the advantages of our method is that equation (12.4) clearly exhibits the contributions from the different modes to the fluctuations. Thus, for example, only the Y = 0 modes contribute to the density fluctuations, because (12.3)
310
B. U. FELDERHOF
yields
here
V(Y - r’, p’) = 6(r - r’), Therefore
(13.1)
ar(k, s) = dP.0.
one finds the same result as in I; in our present notation
=
7
/dfB(k, s) cos(k-r
:’
- kst/m) ds, (13.2)
--Kr <12i(O, 0) 121(r, 0)) = wad(r) where K2 integral
4cn&$9
=
z
e,
r
is the square inverse of the Debye length and where the
s
Film(S)
IZyB(k,
ds=
s)12
k2 k2
+
(13.3) K2
is performed by separating the denominator and making Similarly one finds for the electrostatic potential an3 16nses @(O, 0) @(r, t)> = 7 ZR4 k
s
dfB(k,
s) cos (k-r -
use of (6.6).
kst/m) ds (13.4)
1 _ em”? <@(O, 0) @(r, 0)) = By
.
Since the calculation of the correlations is quite straightforward and explicit examples have been given in I, we shall for the sake of brevity merely quote results. For the current density correlations one obtains
kik’s2doMB(k, s) +
x2-
+
(&,- Z)
In particular
(9 -
dFB(k, s)
m%2)2
1
cos(k.r
- kst/m) dk ds.
(13.5)
for t = 0 <~c(O,0) ij(r, 0)> = 12062 W(r) mP
(13.6)
a
where use has been made of the following integrals
s
S2FMB(S)
I.ZrB(k, s)12 ds = +ib’ SF(s)
s
1
s IZl(k, s)12 ds = 3’
F(s)
IZl(k, s)j2 ds = s4F(s)
s
IZl(k, s)12 ds =
1 WA%;
(13.7)
FLUCTUATIONS
For the correlations
AND
COHERENCE
OF RADIATION
in the vector potential
.@“(k,
s) cos(k. r -
where the tensor JS?‘~,may conveniently
one finds
M/m) dk ds = A?‘&, t).
be written
(13.8)
(see also appendix)
. cos (k-r which clearly exhibits the contribution sion equation. For t = 0
311
IN PLASMAS
&t/m) dk ds,
(13.9)
from the roots of the plasma disper-
(13.10) The correlations of the transverse expressed in the tensor A?U,
where &#jjrkis the These correlations For t = 0 <8$(0,
electric
and magnetic
fields may also be
completely antisymmetric unit tensor characterize the coherence of radiation
0) 8&,
O)> =
1
of Levi-Civita. in the plasma.
0)) =
45&b(r)
+
a2 WY’
1
1
(13.12)
0)) = 0: The correlations (13.11)
in the total
= - ~
a2
ax&v,
electric
field may be found from (13.4)
and
<@(O,0) @(r, t)> ---
1
a2
c2
at2
<&(O, 0) A+,
g>.
(13.13)
312
B. U. FELDERHOF
In particular for t = 0
0) = -
B1 [
that is, the long-range
a2
+ ~
4?z6&9
e-or --
axtax, I
part of the Coulomb-field
1t .
correlations
(13.14) disappears
when the transverse field correlations are taken into account. This is probably of importance in relation to transport coefficients. The range of validity of the above results will be discussed in the next section. Finally, we note that for the pressure fluctuations (11.3) yields the same result as derived in I. 9 14. Average energy density of the flwtuations. It is of interest to calculate the average energy density of the fluctuations {fl(r, p), A(r), g(r)}. The averages of the different energy terms in the exponent of (4.2) will be calculated separately. In (11.1) these terms were transformed into expressions in normal mode coefficients, which may immediately be averaged with the aid of the ensemble (11.3). One thus obtains (see the remark preceding (12.4))
(i++, r’) fl(r,
p) fl(r’, p’) dr dp dr’ d$)
= ds =
p.A(r)
.
s
fl(r,p)
(9
-
drdp)
m2c2
-
m _.
= 5’ T
SE)
FMB(s) Iz?“(k,
A(r)2 dr) = x’ 7
B
T/s:
ds = -2kBT 4 I2
,,~~k(s~),2
k
b(r)2
&)
=
F’
ds=kaT&&
>
(curl A(r))2 dr) = 5’ F
y
-F /w&2
519
k%2
FiwB(S) ds=knTx’l Iz?% s) I2
zF(‘)
12,
(14.1)
c’ o,“, & k%2
(k, s) I2
k
ds=kBTx;‘l. k
For each of the terms the average energy density is obtained by dividing by a; one could also directly average &f(r)2 etc., or employ the results of the preceding section. It is remarkable that the total energy density of the long transverse waves with k -=c o,/cd2 is on the average negative. Equations (14.1) clearly exhibit how the energy is divided between the different frequencies w = KS/m. For any k most of the energy goes in frequencies near the corresponding roots of the dispersion equation (cf. (13.7)).
FLUCTUATIONS
It is noteworthy
AND
COHERENCE
that for every
IN PLASMAS
313
electromagnetic
energy
OF RADIATION
k the average
density is the same as in vacuum. This is quite understandable: Though the frequency kc of every field oscillator is renormalized (in our case to a spectrum of new frequencies), the average energy must still be kBT. All terms in (14.1) yield a divergence for small wavelengths. On the one hand this is because our smeared-out description is not correct for too small dimensions. On the other hand one cause of divergence must be our classical treatment of the radiation gas. As the characteristic frequencies of the transverse waves are of the order cc)= (of + kVJ)*, we may expect our classical treatment to be correct for wave numbers up to k = kBT/fic, because in a classical plasma in any case fiiwP<< kBT. This expectation is borne out by the fact that the spatial correlations of the electric and magnetic fields, given by (13.12), are the same as in vacuum Is), where the expressions (13.12) are valid for distances larger than fic/kBT; hence one may expect that in the plasma these results are exact over the same range. The behaviour in time of the correlations, given by (13.1 l), can, however, not be expected to be correct for too small wavelengths, owing to the limitations of the continuum treatment. 0 15. The semi-classical case. The theory may without much difficulty be extended to the semi-classical case, where Fermi-Dirac statistics of the electrons is taken into account, but where the equations of motion are still classical; the radiation gas is treated classically. A measure of the exclusion effects is given by the parameter no&, where &B is the de Broglie-wavelength h/d2nmkBT. The radiation gas may be treated classically if the parameter fio+/kBT is small. Since the latter may be written ficop/kBT = = K&B/1/27C, it is an order of magnitude smaller than n?jjZd~for low density, high temperature plasmas. For metals it is essential to use quantum statistics. For classical plasmas, for which the theory is valid, the corrections are, of course, quite small. The present case is therefore of rather academic interest. The difference from the classical case arises from the entropy factor (2.6). If one accounts for the indistinguishability of the electrons and for the fact that a cell of volume h3 in phase space can contain at most one electron, one finds that the entropy factor is now given by exp(-
S V(C P) log f(~ P) + [l/h3 -
f(~
~)llog[l /h3- f(c1411 drW. (15.1)
Equation -
(3.2a) is replaced by t(r,
P) _ B (P - eW2 2m 1/h3 - t(r) PI
log -
B /w(T,
-
r’) f(r’, p’) dr’ dp’ + /I / ~(r, r’) no dr’ -
CL= 0.
(15.2)
B. U. FELDERHOF
314 Instead
of (3.7) this yields the equilibrium f&9
P) = %fm@)
The second functional
= no
derivative
6(r -
and the considerations of (4.2) becomes
-
= -
1/noh3
[l -
(15.3)
exp(a + P$2/2m) +
of Y with respect
@!P W(r, P) W’> P’)
distribution
1
to f is now given by
r’) S(p - p’) h3f(r> P)I f(ft P)
@(r,
-
r’),
(15.4)
of section 3 still apply. The first term in the exponent
’s
P-
f;(rJ P) h3fiofm($)1 fiofFo($)
(15.5)
dr dp*
The time behaviour of a fluctuation {fl(r, p), A(r), b(r)} is again given by (5.4) where for fo one must use the Fermi-Dirac distribution. The modified exponent (4.2) is diagonalized by the normal mode solutions of this set of equations ; instead of the normal mode solutions of section 10 the corresponding solutions for the Fermi-Dirac case must be used. The same straightforward calculation yields for the probability ensemble of normal mode coefficients
where
(15.7)
9 16. Semi-classical
correlations.
The probability
ensemble
(15.6) results
in expressions for the correlations which are slightly different from the The integrals classical ones in that d,MB is everywhere replaced by A”. corresponding to (13.3) and the first of (13.7) are now given by -1 s
120
a(noFFo(s))/aa IZ:D(k
ds =
m 2@s,
s)l2
Y:o(k
=--
-1 n0
s -1
%O
s2a(fiohD(s))la~ IzfD(k, 4 I2 ds
=
m +J
2
s)
s lZfD(k, s)12 ds =
s
ano aa
k2 k2 + /c(a)2
s2YFD(k, s) IZ;D(k,
s)\2 ds = $’
(16.1)
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
315
IN PLASMAS
where ~(a)2 = -47ce2@zo/&~ In the classical limit K(E)~ reduces to K2, since for a Maxwell-Boltzmann gas no = e -“&$. For a highly dilute FermiDirac gas K(c~)~
=
hCe2@o(l
-
(16.2)
?Zo&/2’).
For the spatial density correlations one obtains in the semi-classical case
0) n1p, O)>=
7
;’
jdfD(k, s) cos k-r =_-
an0
ds = ~(42
w-
&
7
e-x(a)r
___
Y
1 -
(16.3)
On account of (16.2) the density correlations have a slightly longer range than in the classical case. Similarly one finds for the correlations in the potential 1 _ e-4a-
(16.4)
<@(O, 0) @(r, O)> = By
*
The other spatial correlations for current density, (13.6), vector potential, (13. lo), and electric and magnetic fields, (13.12), remain unaltered, because the last three integrals in (13.7) yield the same result for any /e(p). Acknowledgement. The author wishes to thank Professor I. Oppenheim for the hospitality extended to him at the Institute. He also gratefully acknowledges the grant of a NATO Science Fellowship by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.). APPENDIX
In order to facilitate comparison with other theories we note the following identities. Zo(k, ma/k) = &L(k,co) = 1 - 0;
s
Zl(k, ma/k) = G
k%2 w2
e(k, 0) -
fo(?)
(UJ- k-p/m + i&)2 dp (A’1)
(A.21
,
where Et(k, co) =
1-
3
0 s
fdP) co-k-p/m
dp j-i&
’
(A.3)
In the literature the tensor
(A-4) is often called the generalized dielectric constant.
316
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
IN PLASMAS
The tensor JZIUmay be written cd&,
t) = kBT
27~3~3
la&
w)
k%‘+o212 ’
-
‘cos (k*r Received
&) dk dw.
(A.5)
l-9-64
REFERENCES 1) Felderhof, 2)
Akhiezer,
3)
Silin,
V. P., Rukhadze,
Moskva 4)
B. U., Physica
1771.
I. A. and Sitenko,
Rytov,
S. M., Teoriya
Press (1960),
Chapter
elektricheskikh
fluktuatsii
and E. M. Lifshitz,
plazmy
14 (1962) 462.
i plazmopodobnykh
sred,
Yu. L. and Silin,
7)
Klimontovich,
Yu.
8)
Dupree,
T. H., Phys. of Fluids 8 (1963)
Heitler,
W.,
Felderhof, Kampen,
izlucheniya,
of continuous
V. P., Sov. Phys. Doklady
L., Sov. Phys. JETP
9)
The
i teplovogo
Electrodynamics
Moskva media,
(1953) ;
Pergamon
14 (1962) 689.
Klimontovich,
11) Van
svoistva
XIII.
V. P., Sov. Phys. JETP
6)
10)
A. G., Sov. Phys. JETP
A. A., Elektromagnitnye
(1961).
see also L. D. Landau 5) Silin,
30 (1964)
A. I., Akhiezer,
Quantum
B. U., Physica
Theory
7 (1958)
1714.
of Radiation,
Third
29 (1963) 293.
N. G., Physica
7 (1963) 698.
119.
21 (1955) 949.
12) Case, K. M., Ann. Physics 7 (1959) 349. 13) Mehta, C. L. and Wolf, E., Phys. Rev. 134 (1964) A 1149.
edition,
Oxford
(1954),
Chapter
1.
Physica 31 295-316
ON FLUCTUATIONS AND COHERENCE OF RADIATION IN CLASSICAL AND SEMI-CLASSICAL PLASMAS by B. U. FELDERHOF*) Department of Chemistry, MassachusettsInstitute of Technology, Cambridge, Mass., U.S.A.
synopsis The theory of thermal equilibrium fluctuations in plasmas, developed in a previous paperl), is extended to include radiation. Expressions are derived for the correlation tensors of the electromagnetic field.
9 1. Intro&ctio~~. In this paper the theory of thermal equilibrium fluctuations in plasmas, presented in a previous paper I), hereafter referred to as I, is extended to include radiation. Explicit expressions are derived for the correlations in space and time of several quantities of interest. In particular the correlation tensors for the electromagnetic field, which characterize the coherence properties of the radiation, are calculated; in principle these quantities may be experimentally investigated. For simplicity we have restricted ourselves to the electron gas with positive background, but the theory may without difficulty be extended to many-component plasmas. Almost all work on the subject seems to have been done by Russian scientists. Most of their work2) 3) is based on a suggestive, though in our opinion not completely satisfactory theory of Rytov4) on electromagnetic fluctuations in general media. Silin and Klimontovich have taken several other approaches, but these are either open to questions) 6) or require ad hoc assumptions to arrive at results7). Recently, Dupre es) has pursued Klimontovich’s work but he does not give explicit expressions for the electromagnetic field correlations. The present theory is based on the canonical ensemble of equilibrium statistical mechanics. Canonical variables are introduced for the degrees of freedom of the radiation field. The continuum approximation is used to derive a probability ensemble for the simultaneous probability of a given distribution f(r, p) of particles in 6-dimensional p-space, a given vector potential A(r) and a given transversal electric field S(r) (section 2). Subsequently it is shown that this probability ensemble exhibits a maximum *)
On leave
of absence
from
the Institute
-
for Theoretical
295 -
Physics,
Rijksuniversiteit,
Utrecht.
296
B. U. FELDERHOF
at the thermal equilibrium configuration corresponding to a MaxwellBoltzmann distribution of the particles and vanishing electric and magnetic fields leads
(section 3). Expansion of the ensemble about thermal equilibrium to a Gaussian probability distribution for fluctuations {fi(r, p), set ion 4) ; this distribution is the basis of all further calculations. A(r), 8(r)> ( t Like in I, it appears that the exponent of the Gaussian distribution is
diagonalized by the normal mode solutions of the set of linearized Vlasov and Maxwell’s equations (sections 5 to 11). This permits explicit derivation of joint probability distributions of quantities which depend linearly on fr(r, P), A(r) and b(r). In low density, high temperature plasmas collisions may be neglected and the time behaviour of a fluctuation {jl(r, p), A(r), &f’(r)}is governed by the set of linearized Vlasov and Maxwell equations. In this case the formalism is particularly adapted to calculate correlations not only in space, but also in time. In section 12 a general expression for the correlation of any pair of quantities taken in two different points of space and time is presented. In section 13 the most important correlations are calculated. The density correlations and the correlations of the electrostatic potential are found to be not affected by the presence of the radiation, but the fluctuations in the current density are, of course, influenced by the radiation fields. Next the correlation tensors of the transverse electric and magnetic fields, which characterize the coherence of radiation in the plasma, are derived. It is shown that the spatial correlations at equal times are the same as those in vacuum. Remarkably, the long range part of the longitudinal electric field correlations just cancels against the transverse field correlations, so that the total electric field shows no long range correlation. This might affect transport coefficients. In section 14 the range of validity of the above results is investigated. Also the average energy density of a fluctuation is calculated and the use of classical statistics for the radiation fieldis discussed. In section 15 the theory is extended to the semi-classical case, in which Fermi-Dirac statistics of the electrons is taken into account; otherwise the treatment is classical. It is shown to what extent the correlations are modified by exclusion effects in section 16. § 2. Probability ensemble. We shall consider a gas of N electrons of charge e, mass m in a volume Q with a positive neutralizing background of charge density --nee, where no = N/Q. There are also electric and magnetic fields E and B which may be derivedfrom a vector potential A and a scalar potential Qi,
E=--
1
c
i3A - grad@,
at
We choose the Coulomb-gauge,
B = curl A.
P-1)
i.e., div A = 0.
(2.2)
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
IN PLASMAS
297
From Maxwell’s equations it then follows that A and @ satisfy AA---
1
82A
c2
at2
= - f
- igrad:
j
A@ = - 4-cp,
cw
where j is the current density and p the charge density. In this gauge the potential @ does not enter as a separate degree of freedom but only serves to transmit the instantaneous Coulomb interaction between the electrons
4-J 4 =
e2 ,r
_
r,,
.
(24
In a canonical formalism the momentum conjugate to the field variable A is given by -&‘6/4zzc, where d is the rotational part of the electric field &=--
1 aA c at’
div 8 = 0.
(2.5)
In thermal equilibrium the canonical ensemble determines the probability that the particles have positions between ri and ri + dri, momenta between pr and pi + dpg and the vector potential and transverse electric field at I values between A and A + dA and & and I + d& respectively. In order to perform actual calculations we wish to use a smeared-out description to which purpose r-space and p-space are divided into cells; the cross-sectional cells in (r, p)-space should be large enough to contain many particles and so small that the kinetic energy and the interaction do not vary appreciably over the cells. One now asks for the probability that the cells in (r, p)-space contain given numbers of particles and that the potential and field have given average values in the cells in r-space. Or, circumventing the explicit introduction of cells one asks for the probability that the distribution function /(r, p) assumes values between f and f + df and potential and field values between A and A + dA and & and 8 + db; here it should be kept in mind that f, A and & only have a coarse-grained meaning. The division of particles over cells gives rise to a combinatorial factor of entropic nature exp[-
SfPt P) log f(r, P) dr +I
(2.6)
and at a temperature T there is an energy factor of the form exP[-BVf(r,
P)7 A(f), W)Il,
(2.7)
where /I = 1/kBT and U { } is the energy of a configuration {f(r, p), A(r), 8(r)}. According to the canonical ensemble the probability that a configuration assumes values between {f(r, p), A(r), 8(r)) and {f + df, A +
29%
B. U. FELDERHOF
+ dA, 6’ + d&Z’}is therefore given by df dA d& =
P{f(r, P), A(r), g(r))
= exp !P{f(r, p), A(r), 8(r)} df dA dI/JP
df dA d&,
(2.8)
where
-S[ -
-f(r, p) log f(r,p) - B
-+p/zu(r, --it3
s
(’ -2iA’c’2 f(r, p)]dr dp -
r’) f(r,p)f(r’,p’)
drdpdr’dp’
+ B/m(r, r’) f(r,p)
w(r, r’) sz dr dr’ -
$-s[(curl
A(r))2 + &‘(r)2] dr.
no drdpdr’-
(2.9)
In the expression for the energys) the interaction of the electrons with the background and of the background with itself has been taken into account. $3. Thermal equilibrium configuration. The probability ensemble (2.8) will exhibit a maximum at the thermal equilibrium configuration. At the maximum the first variation of Y must vanish for arbitrary variations sf(r, p) subject to the condition that /f(r, p) dr dp = N remains constant, i.e.
and for arbitrary
solenoidal
dA(r) = curl g(r), M(r)
(3.1)
variations
dA(r) and &T(r),
= curl q(r),
where g(r) and q(r) are arbitrary fields. Assuming the variations to vanish at the boundary and taking the condition (3.1) into account by a Lagrange multiplyer y, one obtains that Y/ is stationary for solutions of
1 - 1%f(r, P) - B
(p - eAM2 2m
- @ 1 w(r, r’) f(r’, P’) dr’ dp’ + + ,9 /w(r,
curI[$J(p-$)f(r,p)
f
dp -
r’) 1~0dr’ - y = 0
$curlcurlA(r)]
= 0
(3.2a) (3.2b)
curl &t(r) = 0. One easily finds from (3.2a) that for a stationary point the current density,
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
IN PLASMAS
299
given by
j(r)=$
p-- em9 SC >
f(r,
c
vanishes identically. Equation (3.2b) therefore which together with div B = 0 leads to
PI dP7 yields
curl
curl B = 0,
LlB = 0. Equation
(3.3)
(3.2~) together with div &’ = 0 yields 8 = const..
A possible solution of (3.2~~)satisfying f(r, p) = no Since the second functional
(&),
cv!P Sfp, p) df(r’, P’)
d(r = -
(3.1) is given by
exp [ -
derivative
(3.4)
@ (’
-2c’c)2].
(3.5)
of Y with respect to f,
r’) S(p - p’) f (r, P)
-
@(r,
4,
(3.6)
is easily shown to be negative definite, it follows that (3.5) is the only solution of (3.2~) for given A. Equations (3.2) yield an infinity of configurations for which Kis stationary for variations vanishing at the boundary, and given by d = const., solutions B of (3.3) with corresponding vector potential, and f(r, P) given by (3.5). It will be shown in section 11 that one of these, the thermal equilibrium configuration feq(r, P) = fiofM&)
= no (&)t
exp [ - $J
Aeg = 0 &,
= O,
(3.7)
is a maximum. (Actually A,, is determined up to a constant by B,, = 0, but a constant in A may always be eliminated by redefining the momenta). Now all terms of (2.9) except the last one are the same for all stationary configurations, whereas for any configuration for which the electric and magnetic fields do not vanish the last term in (2.9) is smaller than it is for the configuration (3.7). Hence the thermal equilibrium configuration (3.7) is the absolute maximum of P. $4. Fluctuations. In order to establish a probability ensemble for fluctuations {fl(r, P), A(r), d?(r)} about the equilibrium configuration (3.7), the quantity Y must be expanded to second order in these deviations from
300
B. U. FELDERHOF
equilibrium.
The second functional
62Y
derivatives
of Y are, besides (3.6),
=g(p-+)6(r-r’)
dfh PI Wr’) 6‘?P
PP
sf(r, p) M(r’)
= O’ 6A(r) M’(r’)
fwf
0
=
(4.1)
- J?? &+?(r - r’) f(r, p) dp mc2 s
GAi(r)SAj(r’) =
$
(
.-.
454
82
axg245> d(r -
d?P
- -&
&Tt(r) SEj(r’) = Calculating
&5 d(r -
r’)
r’).
the difference ~{~o~MB(P)
fl(r, p), A(r), g(r)} - !+o~MB(P),
+
to second order one obtains for the probability
0, O}
ensemble of the fhctzcations
P{fl(r, P), A(r), b(r)} dfl dA dg = =exp
[
+ & --
+
s
fk
P)
nOfMB(p)
/p-A(r)
dr dp -
t
s
w(r,
fl(r,p) dr dp - g
r’) fl(r,p) fl(r’,p’)
dr dp dr’ dp’ +
/A(r)2 dr -
' 1 [(curl A(r))2 + E(r)21 dr] dfl an
dA d&‘/SF’
dfi dA d&.
It is of interest to remark that now the energy part of the exponent is no longer negative definite, as in the electrostatic case. The complete exponent in (4.2), however, is negative definite, as will be shown in section 11. The time behaviour of a fluctuation {fl(r, p), A(r), &S’(r)} will be assumed to be governed by the linearized Vlasov equation and Maxwell’s equations. Consequently, the probability ensemble (4.2) enables one to calculate joint probability distributions of quantities, expressible in fl, A and 8, taken at different points of space and time. It will be shown that, in analogy to the electrostatic casel), the ex$onent in (4.2) is diagonalized by the normal mode solutions of the set of linearized Vlasov and Maxwell equations. 9 5. Normal modes. function reads ;+
P - eA(r)/c
The potentials
m
Vlasov’s
at ‘F
equation
a ar [
satisfy Maxwell’s
for the one-particle
(P - eA(r)w 2m equations
+e@].-$
distribution
= 0.
(5.1)
(2.3) with charge and current
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
IN PLASMAS
301
densities
p(r, t) = e[Jf(c p, t) dp - m01
A time-independent
eA(r, 4
S[
j(r, t) = f
p -
solution
c
1f(r)
of these equations
f(r, p) =
no/o(P),
A
=
0,
(5.2)
t) dP.
P#
is given by @
=
0.
(5.3)
Here /a($) is normalized to unity and will only be assumed to be an isotropic function of p ; this enables us to cover also the semi-classical case. The linearized set of equations for deviations from this time-independent solution may be written
+ +
-
ad5 at
=
s,:_::,,
aoe2
fl(r',
4ne -_cdA+?!&-C m
p’, t) dr’ dp']
p'fdw',
4 dp’
A(r, t) b(r, t)
(5.4
> I’
where the index J_ indicates that the rotational part taken and where wp = (4nrtees/m)* is the plasma potential @ has been eliminated by solving Poisson’s One now looks for normal modes, i.e. stationary the form
fl(r, p, t) = g&p)
a$ = 0
of the vector is to be frequency; the scalar equation. periodic solutions of
ez(k’r--wt)
= A,,, e”(k”-“t) =
(5.5)
cYk,@ei(k*r-ot).
For fixed k we shah always take the z-axis in the direction of k. Introducing the variable s = ma/K one finds from (5.4) that the normal modes must satisfy
sA1,* = -
SdZkS=
;
imc k
f%s
W2c2 + $1 A,,, -
F
( /P%~,.(P’)
dp’>i
(5.6)
302
B. U. FELDERHOF
where sp = mc+Jk. The index J_ now means projection cular to k. In the following two sections a complete obtained for these equations. 5 6. Reduced equations.
in the plane perpendiset of solutions will be
In order to solve equations (5.6) the same method
will be used as employed in ref. 10. On account of the isotropy of fe equations (5.6) easily separate for the three directions of polarization. Define
dPz dP?J %(Pz) =JYP&,AP) dPx dPw 4, G,APz) =flPv&,AP) dPx dPw 4, G,.s(Pz) =/J&s(P)
= k%,s)z?4,s = mz,s)z = (Ak,& c,, = Wk,s)Y. (6.1)
For h.O(&) one obtains from (5.6) (s where
(6.2)
Pz) %,,(Pz) = -s;~‘(Pz)_%,,(P6) dPL>
(6.3)
F(Pz) = JYfdP) dPx dPv*
(We employ the same notation as in ref. 1 and 10, but since we here work in momentum space the corresponding quantities differ by factors m.) Equation (6.2) has a solution for any real value of s,
+
4 d(s- pz),
Xo(k
(6.4)
where P stands for principal value, and where Xa and Ye are the real and imaginary part of *)
Zo(k, s) = 1 - s;
s
F’(Pz) p, - S -
iE
dPz>
(6.5)
so that
Xo(k, s) = 1 - s;P
s
F’(Pz) pz - s dpz
____
(6.6)
Yo(k, s) = -ns;F’(s). The solution (6.4) is normalized to unity. Completeness of the set of solutions (6.4) has been proved by van Kampenlr) ; any function uniquely written as a superposition of these solutions. For the second set of quantities in (6.1) one obtains from
b-
PJ G,,(Pz) sA’ k,s
= -
(5.6)
4,
PAPZ)
imc =
SC& * = *) See appendix.
F
of p, may be
-
-
k
G,,
-& (m2c2 +
si)
Ai,s -
4nie K
s
G,AP2 dPL* (6.7)
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
303
IN PLASMAS
Again there is a solution for any real value of s. With the normalization J-h;,,(&)
dp, = s2 -
mW
-
s;
Xl@,
s) qs
(6.8)
one obtains h&J+,) = ; A’
k.5
7”;;)
+
-
pz)
4cemc
=_-
(6.9)
k2 4nies -3 k
& =k,s
where X1 and Yr are the real and imaginary Zl(k, s) = s2 -
w&9 + ss;
so that X1(k, s) = 9 -
part of *)
s
F(Pz)
p, - s - ie dPZ
w&2 + ss;P
s
F(Pz) p, -s dpz
~
(6.10)
(6.11)
Y,(k, s) = ns;sF(s). The proof of completeness of the solutions (6.9) is entirely analogous to that given in ref. 10 ; any set of quantities {f(&), A, d?} may be uniquely written as a superposition of these solutions. The third set of quantities in (6.1) leads to a set of equations identical to (6.7) ; the solutions are the same as (6.9) and need only be distinguished by an index 2. Note that Zl(k, s) = Zz(k, s). 5 7. Solzltions of eqzGatiolzs (5.6). There are, of course, solutions of (5.6) which reduce to the solutions of the preceding section when the appropriate integration procedure is carried out. Solutions corresponding to (6.4) are
d,,(P)=
afo/aPz qp,)
h:,,(pz),A:,, = 0, &:,a= 0.
Similarly, the solutions corresponding to (6.9) and the analogous for the other direction of polarization are readily found to be
(7.1) solutions
(7.2)
where use has been made of the isotropy l)
See appendix.
of
fo.
304
B. U. FELDERHOF
There are, however, also solutions of (5.6) with vanishing charge and current densities and vanishing potential and field. The dependence on p, of these solutions g(p) must be given by a delta function and the solutions vanish identically-when
the reduction
procedures
of the preceding
section
are applied. We shall consider a series of such solutions
afom
G,,(P)= ____ F’(p
z
A;,
J&*(P)
P?A S(s- Pd
(7.3) (V = 3, 4, . ..).
is a polynomial
G,,(Pz, Pd
PUP,,
= 0
CF;,, = 0 where
)
in p, and 9, with
dP = JPa&,AP) dP =/P,&,,(P)
dP = 0.
(7.4)
Moreover, the polynomials will be chosen in such a manner that the following orthonormality relations hold
s
afow
~‘(p
Pb(Pzt
Pd P;,s(Pz> Pd
d(s-
PA
(7.5)
dp = 4w
2
If one extends the series of polynomials so that (7.5) also holds for PO = 1, Pl = const. .p5, P2 = const. *p,, which is possible on account of (7.4) and the isotropy of fo, then the series may be constructed in a unique fashion from (7.5) by successively including higher powers of 9, and 9,. In section 10 this procedure will be carried out explicitly for the case that fo(p) is the Maxwell-Boltzmann distribution. Obviously the solutions (7.1) and (7.3) cannot be used if there are regions in p-space for which F’(p Z) vanishes identically. Although the analysis could be extended to include this case, we shall for the sake of simplicity assume that F’(p,) vanishes only at isolated points. A similar assumption will be made about afo/ap,, since otherwise the solutions would clearly not suffice to expand an arbitrary function of p. In the cases which are of interest to us, viz. the Maxwell-Boltzmann and Fermi-Dirac case, these assumptions are evidently
satisfied (except
for T = 0).
3 8. Adjoint sohtiorts. For expansion purposes it will be convenient to have the solutions of the system adjoint to (5.6) at our disposal. The adjoint system of equations reads
(S- Pd g;,,,(P) = h,s s&ks=
= $, -
TP*&,,
- s$&
(m2c2 + s;, &,, + 7 imc A”,,,. k
&,&p’) dp’ (sp’
f$
,&(p’)
dp’)l
(8.1)
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
305
IN PLASMAS
A first set of solutions is E:,,(P) = @,,(PZ)>
4,s
= 2i.S = 0,
(8.2)
where
Q,s(Pz) =
,zo(;, s),2
[;
y;;
X0@,4
+
qs-
pz)]
is the solution of the equation adjoint to the reduced equation (s -
P,) @,,(P,)
= -
s;SF’(PS)
The solution (8.3) has been normalized orthonormality relation holds
&,(PL)
dP;.
(8.3)
(6.2) Is), (84
in such a way that the following
(8.5)
/Gi,,(Pz) @,APz,dPz= 0 - 0
This may be verified by direct calculation with the aid of the PoincarCBertrand relation I (9.2). It follows also from the solution of a singular integral equation 11). A second and third set of solutions of (8.1) are given by
(8.6)
the other components
of A and 2 vanish. Here 1
%h
=
,Zl(h,
p q,2
[
y
Yl(k s _
s) p,
+
Xl&
4
Sb
-P.,]
together with the nonvanishing components of A1 and & of the system adjoint to the reduced equations (6.7)
(s -
Pz)
&,(fJz)= sA;,, =
F
$
SC@-& = -
imc
is the solution
&,,
(mW +
K
(8.7)
s;,&, -
y/PlF(Pl)
@,,(P2dp;: (8.8)
so that the following
orthonormality
A:,.
This solution has been normalized relation holds
306
B. U. FELDERHOF
This may again be verified by direct calculation; it follows also from the solution of a singular integral equationlo). For the third set of solutions G,,(PZ) = @,,(P z1, and a relation similar to (8.9) holds. As follows from (7.3) and (7.4) a final set of solutions of (8.1) is given by
!i%,s(P) = K,,(P) 4s - Pz) P*,, = S-L,, =
(8.10)
(aJ= 3, 4, . ..).
0
with an orthonormality relation which follows from (7.5). 3 9. Expansion in normal mode sol&ions. From the orthonormality relations (7.5), (8.5) and (8.9) and the isotropy offa it follows that the following orthonormality relation holds between the solutions of section 7 and their adjoints of section 8 J&(P)
&,(P):dP
+ A:,;&,,
+ &,&,a,
= &W
- s’).
(9.1)
This relation will be used to determine expansion coefficients. The following expansion theorem may now be stated: Any perturbed configuration {fl(r, p), A(r), 8’ (r)} may be written
dsC,(k, s) AL,, eik.r
A(r) = &yJ k,v d?(r) = 7
C’SdsC,(k. k,v
(9.2)
s) Rk,, eik’r,
where the expansion coefficients C,Jk, s) are given by
G(k, s) =
& ~dremik”[ /g;,,(P) fl(c p) dp +
&,s-A(r) +
&,;+j]. P-3)
Furthermore, the solution of the linearized Vlasov equation (5.4) with given initial configuration (9.2) at t = 0 is obtained by multiplying every coefficient C,(k, s) by an harmonic factor exp[-&St/m]. Since the completeness of the solutions (6.4) and (6.9) has been proved in another connection, the completeness of the solutions {g;,,(pj, A;,, c%‘;,,} hinges upon the completeness of the polynomial functions (7.3). From the construction it is clear that the normal modes constitute a basis for a large class of configurations (fl(r, p), A(r), 8(r)}. The prime on the summation sign in (9.2) pertains to the k = 0 component. From the restriction/fr(r, p) dr dp = 0 it follows that the k = 0 component of fr vanishes when integrated over p. Furthermore, in order to avoid
FLUCTUATIONS
unnecessary
AND
complications
COHERENCE
we exclude
OF RADIATION
solutions
307
IN PLASMAS
for which A(r)
or d?‘(r)is
homogeneous in space by imposing the condition that A and d vanish at the boundary. This restriction will not affect the final results. It is then readily seen that the functions g:,,(p) for Y = 3, . . . for some chosen k may be taken as the basic set for the k = 0 component of the distribution function; the prime on the summation sign in (9.2) therefore the terms v = 0, 1, 2 do not occur.
indicates
that for k = 0
3 IO. The Maxwell-Boltzmann case. For the case that f&5) is the MaxwellBoltzmann distribution (3.7), the explicit expressions for the distribution function parts of the normal modes of section 7 are
&8(P)=&exP
[
-B
PZ+-Pi 2m
I[ ;
p YfE(k Pz) s _p,
+Xyv, s)4s- Pz)]
(v = 3, 4, . ..) (10.1) where v represents the pair (m, n) = (2, 0), (1, I), (0.21, etc.. The functions H, are the usual Hermite polynomials. The adjoint solutions of section 8 are given
%8(p)
=
by 1 IZfB(k,
s),z
&:,2(P) = Px, Y * ,z”&
192
b%,,(P)=
Hm(Pz~
P Yf”“(k, s) s _ p, + XifB(k, 7c P
s),2 IIT
Y?i?(k,
s’_ p,
s)
PA],
+ Xfy(kJ 4 S(s- P.)]9
Hn(Pd/B/W S(s- P,), (v =
d2mrn ! 2% !
In this case completeness functions.
s) S(s -
follows from well-known
3,4, . ..).
theorems
(10.2)
on Hermite-
§ Il. Probability distribution of the coefficients C,(k, s). With the aid of the expansion (9.2) the probability ensemble (4.2) for the fluctuations may now be transformed into a probability distribution for the coefficients Cv(k, s); since the transformation (9.2) is linear the phase-space factor is a constant which drops out against the normalization. If (9.2) is inserted into (4.2)
308
B. U. FELDERHOF
one obtains explicitly P{Cv(k, s)} = exp .
z’j+ds
- q
s
g:,,(P)g&(P)
dp
ds’ [ yT:(k,
s) Cv(k, s’) *
+
2’ytOfMAP)
+ 7
(C;(k
*[(s2 + s’2 -
S) Co@, s’) + [C;(k, s) Cl&, s’) + C;(k, s) Cz(k 2vnw - 24
+ s; + mw
+ ss’])
s')l.
I/S P,
(11.1)
where the different terms correspond to those in (4.2) in the same order. If one now inserts the expressions (10.1) in the integral over p in the first term and uses the Poincare-Bertrand relation I (9.2) one obtains
*(s2 + ss’ + s’2 - m3c3 -
s
si)
( 11.2)
(aJ= 3, 4, . ..).
whereas for v # v’ the integral finally obtains P{C,(k,
s)) = exp [ -
vanishes.
7
Inserting
(11.2)
C’ l ds ~~~f~(~‘:) k,v Y )
into (Il. 1) one
I//P,
(11.3)
where
Llpyk, s) = Llfyk,
120
83~3 IZfB(k,
s) = LqB(k,
d,MB(k,s) =
FMB(s) s)13 mFMB(s)
s) = 2% 8723 Bj.ZFB(k, s)13
$$ FMB(s)
)
(11.4)
(v = 3, 4, . ..).
The prime on the summation sign in (11.3) indicates that the terms k = 0, v = 0, 1, 2 are absent (cf. section 9). The exponent in (11.3) is not only diagonal but also negative definite, which shows that P has a maximum at the thermal equilibrium configuration (3.7).
FLUCTUATIONS
Since
according
AND
COHERENCE
to linearized
Vlasov
fluctuation is found by multiplying factor exp[-&t/m], the ensemble this theory. $ 12. Joint probability
OF RADIATION
theory
309
IN PLASMAS
the time
behaviour
of a
each coefficient C,(k, s) by an harmonic (11.3) is obviously a stationary one in
distributions.
As the exponent
of the
Gaussian
probability ensemble (11.3) is diagonal in the coefficients C,(k, s) it is an easy matter to derive probability distributions for quantities which may be expressed in terms of these coefficients. The joint probability distribution of quantities which are linear in the Cp(k, s) will, of course, again be Gaussian. In I the expressions for the probability distributions of one and two quantities have been derived. Sufficient information is contained in the expression for the correlation of any two quantities linear in the coefficients. In particular we shall consider (real) quantities of the form
Q(r, t) =
7
F: jdsC,(k,
s) av(k, s) eik’r--ikst’m.
For example, for the vector potential A(r, t) one has a,(k, s) = A;,, similarly for the electric field. For quantities of the form
Q(r, t) = /p(r
-
(12.1) and
r’, p’) fl(r’, p’, t) dr’ dp’
(12.2)
p) &(p)
(12.3)
one has a#,
s) = /p[r,
e--ik’r dr dp.
One must be somewhat careful in taking averages over the ensemble (11.3), since complex quantities have been used. In averaging the product of two quantities Qi and Qs one can either explicitly take into account that they are real and consider real and imaginary part of the coefficients C,(k, s) to be independent, or one can average the product QiQi, but then dependence of real and imaginary part of C, must be accounted for, which yields a factor
8. One thus obtains
(12.4)
The correlation depends only on the differences r - r’ and t - t’ in accordance with the translation invariance in space and time of the system. 3 13. Density, currertt density, potential and field correlations. One of the advantages of our method is that equation (12.4) clearly exhibits the contributions from the different modes to the fluctuations. Thus, for example, only the Y = 0 modes contribute to the density fluctuations, because (12.3)
310
B. U. FELDERHOF
yields
here
V(Y - r’, p’) = 6(r - r’), Therefore
(13.1)
ar(k, s) = dP.0.
one finds the same result as in I; in our present notation
7
/dfB(k, s) cos(k-r
:’
- kst/m) ds, (13.2)
--Kr <12i(O, 0) 121(r, 0)) = wad(r) where K2 integral
4cn&$9
=
z
e,
r
is the square inverse of the Debye length and where the
s
Film(S)
IZyB(k,
ds=
s)12
k2 k2
+
(13.3) K2
is performed by separating the denominator and making Similarly one finds for the electrostatic potential an3 16nses @(O, 0) @(r, t)> = 7 ZR4 k
s
dfB(k,
s) cos (k-r -
use of (6.6).
kst/m) ds (13.4)
1 _ em”? <@(O, 0) @(r, 0)) = By
.
Since the calculation of the correlations is quite straightforward and explicit examples have been given in I, we shall for the sake of brevity merely quote results. For the current density correlations one obtains
kik’s2doMB(k, s) +
x2-
+
(&,- Z)
In particular
(9 -
dFB(k, s)
m%2)2
1
cos(k.r
- kst/m) dk ds.
(13.5)
for t = 0 <~c(O,0) ij(r, 0)> = 12062 W(r) mP
(13.6)
a
where use has been made of the following integrals
s
S2FMB(S)
I.ZrB(k, s)12 ds = +ib’ SF(s)
s
1
s IZl(k, s)12 ds = 3’
F(s)
IZl(k, s)j2 ds = s4F(s)
s
IZl(k, s)12 ds =
1 WA%;
(13.7)
FLUCTUATIONS
For the correlations
AND
COHERENCE
OF RADIATION
in the vector potential
.@“(k,
s) cos(k. r -
where the tensor JS?‘~,may conveniently
one finds
M/m) dk ds = A?‘&, t).
be written
(13.8)
(see also appendix)
. cos (k-r which clearly exhibits the contribution sion equation. For t = 0
311
IN PLASMAS
&t/m) dk ds,
(13.9)
from the roots of the plasma disper-
(13.10) The correlations of the transverse expressed in the tensor A?U,
where &#jjrkis the These correlations For t = 0 <8$(0,
electric
and magnetic
fields may also be
completely antisymmetric unit tensor characterize the coherence of radiation
0) 8&,
O)> =
1
of Levi-Civita. in the plasma.
0)) =
45&b(r)
+
a2 WY’
1
1
(13.12)
in the total
a2
ax&v,
electric
field may be found from (13.4)
and
<@(O,0) @(r, t)> ---
1
a2
c2
at2
<&(O, 0) A+,
g>.
(13.13)
312
B. U. FELDERHOF
In particular for t = 0
0) = -
B1 [
that is, the long-range
a2
+ ~
4?z6&9
e-or --
axtax, I
part of the Coulomb-field
1t .
correlations
(13.14) disappears
when the transverse field correlations are taken into account. This is probably of importance in relation to transport coefficients. The range of validity of the above results will be discussed in the next section. Finally, we note that for the pressure fluctuations (11.3) yields the same result as derived in I. 9 14. Average energy density of the flwtuations. It is of interest to calculate the average energy density of the fluctuations {fl(r, p), A(r), g(r)}. The averages of the different energy terms in the exponent of (4.2) will be calculated separately. In (11.1) these terms were transformed into expressions in normal mode coefficients, which may immediately be averaged with the aid of the ensemble (11.3). One thus obtains (see the remark preceding (12.4))
(i++, r’) fl(r,
p) fl(r’, p’) dr dp dr’ d$)
= ds =
p.A(r)
.
s
fl(r,p)
(9
-
drdp)
m2c2
-
m _.
= 5’ T
SE)
FMB(s) Iz?“(k,
A(r)2 dr) = x’ 7
B
T/s:
ds = -2kBT 4 I2
,,~~k(s~),2
k
b(r)2
&)
=
F’
ds=kaT&&
>
(curl A(r))2 dr) = 5’ F
y
-F /w&2
519
k%2
FiwB(S) ds=knTx’l Iz?% s) I2
zF(‘)
12,
(14.1)
c’ o,“, & k%2
(k, s) I2
k
ds=kBTx;‘l. k
For each of the terms the average energy density is obtained by dividing by a; one could also directly average &f(r)2 etc., or employ the results of the preceding section. It is remarkable that the total energy density of the long transverse waves with k -=c o,/cd2 is on the average negative. Equations (14.1) clearly exhibit how the energy is divided between the different frequencies w = KS/m. For any k most of the energy goes in frequencies near the corresponding roots of the dispersion equation (cf. (13.7)).
FLUCTUATIONS
It is noteworthy
AND
COHERENCE
that for every
IN PLASMAS
313
electromagnetic
energy
OF RADIATION
k the average
density is the same as in vacuum. This is quite understandable: Though the frequency kc of every field oscillator is renormalized (in our case to a spectrum of new frequencies), the average energy must still be kBT. All terms in (14.1) yield a divergence for small wavelengths. On the one hand this is because our smeared-out description is not correct for too small dimensions. On the other hand one cause of divergence must be our classical treatment of the radiation gas. As the characteristic frequencies of the transverse waves are of the order cc)= (of + kVJ)*, we may expect our classical treatment to be correct for wave numbers up to k = kBT/fic, because in a classical plasma in any case fiiwP<< kBT. This expectation is borne out by the fact that the spatial correlations of the electric and magnetic fields, given by (13.12), are the same as in vacuum Is), where the expressions (13.12) are valid for distances larger than fic/kBT; hence one may expect that in the plasma these results are exact over the same range. The behaviour in time of the correlations, given by (13.1 l), can, however, not be expected to be correct for too small wavelengths, owing to the limitations of the continuum treatment. 0 15. The semi-classical case. The theory may without much difficulty be extended to the semi-classical case, where Fermi-Dirac statistics of the electrons is taken into account, but where the equations of motion are still classical; the radiation gas is treated classically. A measure of the exclusion effects is given by the parameter no&, where &B is the de Broglie-wavelength h/d2nmkBT. The radiation gas may be treated classically if the parameter fio+/kBT is small. Since the latter may be written ficop/kBT = = K&B/1/27C, it is an order of magnitude smaller than n?jjZd~for low density, high temperature plasmas. For metals it is essential to use quantum statistics. For classical plasmas, for which the theory is valid, the corrections are, of course, quite small. The present case is therefore of rather academic interest. The difference from the classical case arises from the entropy factor (2.6). If one accounts for the indistinguishability of the electrons and for the fact that a cell of volume h3 in phase space can contain at most one electron, one finds that the entropy factor is now given by exp(-
S V(C P) log f(~ P) + [l/h3 -
f(~
~)llog[l /h3- f(c1411 drW. (15.1)
Equation -
(3.2a) is replaced by t(r,
P) _ B (P - eW2 2m 1/h3 - t(r) PI
log -
B /w(T,
-
r’) f(r’, p’) dr’ dp’ + /I / ~(r, r’) no dr’ -
CL= 0.
(15.2)
B. U. FELDERHOF
314 Instead
of (3.7) this yields the equilibrium f&9
P) = %fm@)
The second functional
= no
derivative
6(r -
and the considerations of (4.2) becomes
-
= -
1/noh3
[l -
(15.3)
exp(a + P$2/2m) +
of Y with respect
@!P W(r, P) W’> P’)
distribution
1
to f is now given by
r’) S(p - p’) h3f(r> P)I f(ft P)
@(r,
-
r’),
(15.4)
of section 3 still apply. The first term in the exponent
’s
P-
f;(rJ P) h3fiofm($)1 fiofFo($)
(15.5)
dr dp*
The time behaviour of a fluctuation {fl(r, p), A(r), b(r)} is again given by (5.4) where for fo one must use the Fermi-Dirac distribution. The modified exponent (4.2) is diagonalized by the normal mode solutions of this set of equations ; instead of the normal mode solutions of section 10 the corresponding solutions for the Fermi-Dirac case must be used. The same straightforward calculation yields for the probability ensemble of normal mode coefficients
where
(15.7)
9 16. Semi-classical
correlations.
The probability
ensemble
(15.6) results
in expressions for the correlations which are slightly different from the The integrals classical ones in that d,MB is everywhere replaced by A”. corresponding to (13.3) and the first of (13.7) are now given by -1 s
120
a(noFFo(s))/aa IZ:D(k
ds =
m 2@s,
s)l2
Y:o(k
=--
-1 n0
s -1
%O
s2a(fiohD(s))la~ IzfD(k, 4 I2 ds
=
m +J
2
s)
s lZfD(k, s)12 ds =
s
ano aa
k2 k2 + /c(a)2
s2YFD(k, s) IZ;D(k,
s)\2 ds = $’
(16.1)
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
315
IN PLASMAS
where ~(a)2 = -47ce2@zo/&~ In the classical limit K(E)~ reduces to K2, since for a Maxwell-Boltzmann gas no = e -“&$. For a highly dilute FermiDirac gas K(c~)~
=
hCe2@o(l
-
(16.2)
?Zo&/2’).
For the spatial density correlations one obtains in the semi-classical case
0) n1p, O)>=
7
;’
jdfD(k, s) cos k-r =_-
an0
ds = ~(42
w-
&
7
e-x(a)r
___
Y
1 -
(16.3)
On account of (16.2) the density correlations have a slightly longer range than in the classical case. Similarly one finds for the correlations in the potential 1 _ e-4a-
(16.4)
<@(O, 0) @(r, O)> = By
*
The other spatial correlations for current density, (13.6), vector potential, (13. lo), and electric and magnetic fields, (13.12), remain unaltered, because the last three integrals in (13.7) yield the same result for any /e(p). Acknowledgement. The author wishes to thank Professor I. Oppenheim for the hospitality extended to him at the Institute. He also gratefully acknowledges the grant of a NATO Science Fellowship by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.). APPENDIX
In order to facilitate comparison with other theories we note the following identities. Zo(k, ma/k) = &L(k,co) = 1 - 0;
s
Zl(k, ma/k) = G
k%2 w2
e(k, 0) -
fo(?)
(UJ- k-p/m + i&)2 dp (A’1)
(A.21
,
where Et(k, co) =
1-
3
0 s
fdP) co-k-p/m
dp j-i&
’
(A.3)
In the literature the tensor
(A-4) is often called the generalized dielectric constant.
316
FLUCTUATIONS
AND
COHERENCE
OF RADIATION
IN PLASMAS
The tensor JZIUmay be written cd&,
t) = kBT
27~3~3
la&
w)
k%‘+o212 ’
-
‘cos (k*r Received
&) dk dw.
(A.5)
l-9-64
REFERENCES 1) Felderhof, 2)
Akhiezer,
3)
Silin,
V. P., Rukhadze,
Moskva 4)
B. U., Physica
1771.
I. A. and Sitenko,
Rytov,
S. M., Teoriya
Press (1960),
Chapter
elektricheskikh
fluktuatsii
and E. M. Lifshitz,
plazmy
14 (1962) 462.
i plazmopodobnykh
sred,
Yu. L. and Silin,
7)
Klimontovich,
Yu.
8)
Dupree,
T. H., Phys. of Fluids 8 (1963)
Heitler,
W.,
Felderhof, Kampen,
izlucheniya,
of continuous
V. P., Sov. Phys. Doklady
L., Sov. Phys. JETP
9)
The
i teplovogo
Electrodynamics
Moskva media,
(1953) ;
Pergamon
14 (1962) 689.
Klimontovich,
11) Van
svoistva
XIII.
V. P., Sov. Phys. JETP
6)
10)
A. G., Sov. Phys. JETP
A. A., Elektromagnitnye
(1961).
see also L. D. Landau 5) Silin,
30 (1964)
A. I., Akhiezer,
Quantum
B. U., Physica
Theory
7 (1958)
1714.
of Radiation,
Third
29 (1963) 293.
N. G., Physica
7 (1963) 698.
119.
21 (1955) 949.
12) Case, K. M., Ann. Physics 7 (1959) 349. 13) Mehta, C. L. and Wolf, E., Phys. Rev. 134 (1964) A 1149.
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Oxford
(1954),
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1.