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Radiative transfer and irreversibility
J. Quant. Specrrosc. Radial. Transfer. Vol. 6, pp. 65-91. PergamonPressLtd., 1966. Printedin Great Britain
RADIATIVE
TRANSFER AND IRREVERSIBILITY* J.
Joint Institute
OXENIUSt
for Laboratory
$
Astrophysics,
Boulder, Colorado
(Received 12 July 1965) Abstract-The irreversibility of the interaction of radiation with matter is investigated by considering a stationary isothermal gas that emits the resonance line of atoms with only two discrete energy levels. In order to describe such a situation microscopically, the steady-state balance of the occupation numbers of the atomic levels and the equation of radiative transfer have to be taken into account simultaneously. The explicit expressions for the non-equilibrium entropies and temperatures of the radiation field and of the gas of the two-level atoms are derived in a systematic way. In the framework of the model considered, the kinetic temperature is an upper limit for both the radiant temperature and the excitation temperature. The local production of radiant entropy, integrated over the whole spectral line, is always positive, but, as a result of non-coherent scattering, it can be negative in certain frequency intervals. The radiant entropy produced by the entire atmosphere is, however, positive for every frequency emitted. Likewise, the total local entropy production is, in agreement with the Second Law, always positive. It is further shown that an anisotropic radiation field has a smaller entropy density and a greater production of radiant entropy than the isotropic radiation field of equal spectral distribution of energy. As an example, an atmosphere is treated numerically that emits a spectral line of large optical thickness under extreme non-equilibrium conditions. I.
INTRODUCTION
PLANCK(~) was the first to investigate the interaction of light and matter with respect to its irreversibility. Quite in contrast to the situation of thermodynamic equilibrium, the study of non-equilibrium phenomena requires that a model of the matter and its interaction with radiation is adopted. Planck chose a set of independent linear harmonic oscillators with parallel axes and could show that in the case of an interaction of these oscillators with a given radiation field the entropy of the total system, oscillators plus radiation, never decreases. As was pointed out already by Planck himself, the most severe limitation of this model lies in its restriction to a strictly monochromatic situation: Irreversibility due to changes of the spectral energy distribution of the radiation field remains necessarily outside of its scope. Thermodynamic aspects of radiative transfer processes were later studied by WILDT(~) who was mainly interested in atmospheres in radiative equilibrium, i.e. in situations in which the divergence of the radiant energy flux vanishes. The transfer equation used by him was formulated assuming “true” absorption and isotropic coherent scattering and hence excluding the effect of.non-coherent scattering. Wildt derived a number of inequalities for the radiant entropy that follow from macroscopic considerations alone, and was the first to formulate the transfer equation for the radiant entropy.
* Work supported in part by the Advanced Research Projects Agency (Project Defender) Contract DA-31-124-ARO-D-139. t On leave from Association Euratom-CEA, Fontenay-aux-Roses (Seine), France. $ Present address: Institut fiir Theoretische Physik der Universitlt Bonn, Germany. 65
under
66
J. OXENIUS
The present paper treats the irreversibility of the interaction of a radiation field with matter by means of a well-defined microscopic model. It considers the statistically steadystate of a self-excited gas that contains atoms with only two discrete energy levels. The irreversibleprocessto bestudied is the emission of the resonance line of the two-level atoms, In order to maintain a steady-state, the energy loss due to radiation has then to be counterbalanced by some energy-releasing reaction taking place inside the gas. The model chosen avoids two limitations of Planck’s oscillator model. First, the radiation field is not prescribed arbitrarily but follows in a unique way from the physical properties of the system and the boundary conditions. And second, the restriction to one single frequency is removed since line broadening effects are taken into account; i.e. non-coherent scattering is treated instead of the monochromatic coherent scattering studied by Planck. The main limitation of the present model consists in its restriction to only one atomic transition, excluding any coupling with other bound states and with the continuum. In general, the radiation of hot gases deviates strongly from that of a black body, i.e. from the radiation in thermodynamic equilibrium. In an investigation of irreversible radiation processes, the possibility of large deviations from thermal equilibrium has therefore to be taken into consideration from the outset in order to avoid a restriction to cases that are of only little interest. On the other hand, non-equilibrium thermodynamics has considered until now only processes that take place in the neighborhood of thermodynamic equilibrium. Section II of this paper describes the model chosen in greater detail and puts together some results of former investigations that will be used in the subsequent sections. Planck’s general definition of entropy, which is applicable also to non-equilibrium states, is used in Section III to derive in a short and systematic way the entropies, together with the corresponding temperatures, of the gas of the two-level atoms and of the radiation field. Some general theorems on temperatures, entropies, and entropy production, valid for the model treated, are listed in Section IV. It is shown that the kinetic temperature of the gas is an upper limit for both the radiant temperature and the excitation temperature of the two-level atoms. The production of radiant entropy has two sources: inelastic collisions and non-coherent scattering. While the local production of radiant entropy due to collisions is positive for every frequency, that due to non-coherent scattering can be negative within certain frequency intervals; the radiant entropy produced by the entire atmosphere is, however, positive for every frequency emitted. Likewise, the local production of radiant entropy, integrated over the whole spectral line, is positive. Furthermore, the total local entropy production that contains in addition to the production of radiant entropy a contribution due to the steady-state energy flux from an energy-releasing reaction, is, in agreement with the Second Law, positive. Finally, it is shown that an anisotropic radiation field has a smaller entropy density and a greater entropy production than the isotropic radiation field of equal spectral distribution of energy. The expressions derived in the former sections are then specialized in Section V to the practically important limiting case of low temperature by means of which a plane parallel atmosphere is treated numerically that emits a spectral line of large optical thickness under extreme non-equilibrium conditions. The radiation field used for this example has been taken from the literature.
Radiative transfer and irreversibility II.
THE
67
MODEL
In this paper, an isothermal gas in a statistically steady-state will be treated that contains atoms with only two discrete energy levels and no continuum. It will be assumed that the mass motion of each gas component vanishes, that no external electromagnetic fields are present, and that no radiation is incident on the gas from outside. The process to be investigated is the formation and emission of the resonance line of the two-level atoms when the corresponding radiant energy loss is counterbalanced by some energy-releasing reaction inside the gas such that a steady-state obtains. In order to determine the radiation field of the considered spectral line, the equation of radiative transfer and the stationarity condition for the occupation numbers of the two atomic energy levels have to be taken into account simultaneously. The microscopic form of the time-independent equation of radiative transferC3)
-Wn) = ds describes the change of the specific intensity Z,(n) of an unpolarized light pencil of frequency v over a distance ds in a direction that is specified by the unit vector n. The frequency vO refers to the center of the spectral line, nl and n2 are the number densities of two-level atoms in the ground state 1 and the excited state 2, and &a, B,,, and Aa, are the Einstein-Mime coefficients for absorption, induced emission, and spontaneous emission of the atomic transition 1 t) 2. Line broadening effects are taken into consideration by the absorption profile &(n) and the emission profile -q,(n); more precisely, 4,(n) dv dQ/4rr is that fraction of non-excited atoms that can absorb photons with a frequency in the range dv centered about v, and a direction inside the solid angle dQ centered about n, and T,(n) dv dCl2/4r is correspondingly that fraction of excited atoms that can emit these photons. Both profiles are therefore normalized in the following way: (2.2)
In (2.1), use has been made of the fact that the profiles for spontaneous emission and for induced emission are identical. (*) Furthermore, a constant energy hv,, has been ascribed to all photons of the spectral line. The simplifying assumption will now be made that the emission profile and the absorption profile are equal: TV(n) = +,(n). (2.3) It can be shown(*) that (2.3) holds exactly if the following two conditions are satisfied: First, the frequency distribution of scattered photons is in the atom’s rest frame given by the frequency dependence of the atomic absorption coefficient, and their angular distribution is isotropic; by scattering, the absorption of a photon followed by reemission of another photon is here to be understood. And second, the velocity distributions of the non-excited and excited atoms, respectively, are equal, i.e. Xl(v) = fi(v) where theflv)
denote normalized
distribution
functions.
(2.4)
68
J.
OXENIUS
A situation in which these two conditions are fulfilled obtains when the elastic collisions of the two-level atoms with one another or with other particles are so frequent that the corresponding relaxation time is smaller than the mean lifetime of the excited level.‘4’ In the opposite limiting case of vanishing elastic collisions, eq. (2.3) often holds approximately. (3.5.6b.7) It will be assumed that the gas of the two-level atoms has a kinetic temperature T, i.e. that the total velocity distribution function of all atoms, irrespective of whether they are excited or not, is Maxwellian. Furthermore, the kinetic temperature T is supposed to be equal to the electron temperature T,: T=
T,.
(2.5)
In view of (2.4), this means that the velocity distributions of the non-excited atoms, of the excited atoms, and of the free electrons are all Maxwell distributionsf,(v) corresponding to the same temperature T: fi(V)
= fi(v)
(2.6)
= L?(v) = fif(v)-
Since the velocity distributionf,(v) of the two-level atoms in the ground state is Maxwellian and hence isotropic, and since no electromagnetic fields are present, the absorption profile 4, is isotropicC4’ and satisfies because of (2.2) the normalization condition i
$,dv = 1.
(2.7)
Writing the transfer equation in its usual macroscopic
form
dl,,(n)/ds = K,(S* -IV(n)),
(2.8)
one gets by comparison with its microscopic form (2.1) and taking (2.3) into account an isotropic absorption coefficient Kv = h%(R&
(2.9)
--QJ-%l)(~V/4~)
and a source function S* = %&,l(R&, which is isotropic and independent relations Azl/B,,
(2.10)
-R&?&1)
of the frequency V. For because of the well-known
= 2hv;/c2,
(2.11)
g, BI, = g,Bn
where g,, g, are the statistical weights of the two energy levels, S* depends only on the frequency v0 that characterizes the energy difference of the two levels. The absorption coefficient (2.9) and the source function (2.10) depend for a given total density nl +n, of the two-level atoms only on the ratio n2/n, of the occupation numbers. This ratio follows from the stationarity equationC3) n,(&zj-%v
dv +
42)
=
nz(A21+
B,,/&+,
dv +
a,,)
(2.12)
where (2.3) has been taken into account. i,, denotes the mean intensity, 7, =
i
I,(n) dL2/4r,
(2.13)
69
Radiative transfer and irreversibility
and Q12,QR,,are the rate coefficients for excitation collisions and de-excitation collisions, respectively. For the sake of simplicity, it will be assumed that inelastic collisions only with free electrons are important. Both !& and !&, are then proportional to the electron density and are related to each other through %,/%
=
k&h)e-nY~‘kT
(2.14)
since the electron gas has a kinetic temperature T. Inserting the ratio nz/nl from (2.12) in the source function (2.10) and taking (2.11) and (2.14) into account now leads toC3’ 1 s* = &$, dv f &L(T) .E lfr s
(2.15)
BY&T) = (2hv~/c2)/(ehvofkT- 1)
(2.16)
where is the Planck function and l
= (QzJA,,)(l
-e-nvJkT)
= Q,,/B,,B,,(
T)
(2.17)
is a dimensionless quantity that depends on the electron density and the kinetic temperature. At low temperatures (e-h”oikT < l), the parameter E is simply the ratio of electronic collisional de-excitation to spontaneous radiative de-excitation of the upper atomic level and is therefore a measure for whether the atomic occupation numbers are determined by collisional processes (E B 1) or by radiative ones (E < 1). The steady-state source function (2.15) is composed of two parts : The first term is due to non-coherent scattering whereas the second term describes the “real” source of radiation due to collisional excitation. This is most easily seen from the divergence of the total flux of radiant energy (cf. Appendix A, eq. (A.8)) div F = [k,~/(l +E)][&,( T) -/7+$, dv] where k0 = hv,(n,B,, -nzBzl). that
(2.18)
It follows from Z,(n) < i&(T) (cf. Appendix B, eq. (B.2)) div F 2 0
(2.19)
which expresses the fact that one is dealing with a self-excited gas. In the approximation used that ascribes to all photons of the spectral line the same energy hvo, the energy flux F is proportional to the photon flux. For E = 0, div F vanishes so that no photons are created; the limiting case E = 0 thus corresponds to a purely scattering atmosphere where the source function is given by J~,c$,,dv. On the other hand, in a collisionally dominated gas (C = co) the source function is simply the Planck function B,.(T). In order to calculate the radiation field, the equation of radiative transfer (2.8) and the stationarity condition (2.15) have to be solved simultaneously, taking the boundary condition into account that no external radiation is incident on the gas. Supposing the absorption coefficient K, to be known, one can eliminate one of the two unknown quantities S* and I,(n) leading either to an integral equation for S* or to an integro-differential equation for I,(n). The solution of this problem has been discussed extensively in the literature.(6,8,9~10,11)
J. OXENIUS
70
III. ENTROPY
AND
TEMPERATURE
The entropy S of an arbitrary, equilibrium or non-equilibrium macroscopically, is according to PLANCK~) given by S=klnW
state that is defined (3.1)
where W denotes the number of independent microstates that are compatible with the macrostate under consideration. If the entropy can be written as a function of the energy U, the volume V, and the number of particles N, the temperature T of the macrostate is defined by l/T = (as/au),,,. (3 *I) For an ideal gas, to which the discussion will be limited, a macrostate can be characterized in the following way (12): One divides the quantum states of the gas into groups of neighboring states with almost identical physical properties. Be G(j) the number of quantum states in the group j and N(j) the occupation number of this group, one has for the mean occupation number of a quantum state of the group j j$j) = N(j)/G(f).
(3.3)
For a given division of the quantum state into groups, a macrostate can thus be characterized by specifying either the N(j) or the ti(j). Is W(j) the number of possibilities to distribute No’ particles over G(j) quantum states, one has
and hence for the entropy
W = RW’j’ i
(3.4)
S = k 2 In W(j).
(3.5)
1. Classical ideal gas The gas of the two-level atoms will be considered as an ideal gas in the classical limit. In the case of Boltzmann statistics one haso2) W(i) = G(i)N’t’/N(j)l.
(3.6)
Using Sterling’s formula, eq. (3.5) now takes the form S = kN_k
c G(j)j$j) ln j$j) j
(3.7)
where N = CNCi) is the total number of particles. i The total entropy So of the gas of two-level atoms is the sum of the entropies of the non-excited and of the excited particles: &=&fS2=
2 si. 1=1.2
(3.8)
Denoting again by h(v) the normalized velocity distribution function of two-level atoms in the excitation state i, one has for a division of the translational quantum states into groups corresponding to velocity ranges d3u N’:’ = N,(v)
d3v = N&(v) d3v
(3.9)
Radiative
71
transfer and irreversibility
and G’f = G,(v) d3u = g, VM3 d3v/h3
(3.10)
where M is the mass of a two-level atom and V an arbitrary, and (3.10), equation (3.7) leads to an entropy density
small volume. Using (3.9)
S,/ V = kni[ln (eg,M3/nrh3)
-JA(v)
In_&(v) d3u],
(3.11)
i.e. except for an additive constant, to the negative H-function. The local number densities n, = Ni/V have been introduced, and e is the base of the natural logarithms. In the model considered, the distribution functions A(v) are Maxwell distributions, h(v) = fM(v) = (M/27& T)3’2e- M”2’2kT, so that (3.11) transforms
(3.12)
into the formula of Sackur and Tetrode (3.13)
A!?,/V = kni[g + In (g,/Q3)] with the thermal de Broglie wavelength x = h(2aMkT)-1’2.
(3.14)
From (3.8) and (3.13) follows the total entropy density of the gas of two-level atoms S,/ V = k[n, ln (s&Q
+n2 ln
k2/n2>l-t~~o[~-3 ln Al
(3.15)
with the total number density of two-level atoms Izo = n,+ns.
(3.16)
The total entropy density (3.15) can now be written as the sum of the entropy density of the translational degrees of freedom S trans/ v = kn,[P - ln
(d3)1
(3.17)
and the entropy density of the excitation degrees of freedom S,,,/ v = k[fi, ln (gln&)
+n2 ln k2no/n2)l.
(3.18)
S trans corresponds to the entropy of a gas of N, = n, V non-localized mass-points of a thermal de Broglie wavelength A, and S,,, to the entropy of a system consisting of No = (nl +nJ Vlocalized particles that have two quantum states 1 and 2 with statistical weights gl and g2 and occupation probabilities nJn,, and n2/no. It should be stressed that such a decomposition of the total entropy is possible only when the respective velocity distributions of particles in the different excitation states are equal. Starting from (3.18) and the general temperature definition (3.2), the excitation temperature T* is given by l/T*
= (aS.Aa
Uexc)~,~o = (a(&,,/
V/%&J)n,
for U,,, = N2hvo is the excitation energy. Carrying out the differentiation or
(3.19) leads to
1/ T* = (k/&J ln (gZnl /gl%)
(3.20)
rt2/lz1 = (g2/g,)e-nYJkT* .
(3.21)
Expressing in (3.20) the ratio n2/n1 by the source function (2.10) and taking (2.11) into account, one gets T* = (hv,fk)/ln[l +(2hvE/c2P)] (3.22)
72
J. OXENIUS
so that the source function is the Planck function corresponding
to the temperature
T* :
S* = (2hv~/c2)/(e WkTX- 1) = B,& F). (3.23) 2. Radiation field The radiation field can be considered as an ideal photon gas. In the case of Bose statistics one haso’) J+‘(j)= (G’i’ + No’ _ 1) I /[( G’i’ -l)!W’!]. (3.24)
Using Sterling’s formula, equation (3.5) now takes the form s = k 2 G(j)[(l -+.j@) ln(1 +@)
-i(j) ln $)I.
Dividing the quantum states into groups corresponding elements of solid angle dQ, one has
(3.25)
to frequency
intervals dv and
No) = N,(n) dv da = VZ,(n) dv dQ/hv,c
(3.26)
because an unpolarized light pencil Z,(n) dv dQ of the considered spectral line has a photon density Z,,(n) dv dL?/hv,c; V denotes again an arbitrary, small volume. Since the number of quantum states of this group is G(j) E G,,(n) dv dQ = V2vz dv d!J/c3, the mean occupation
(3.27)
number of a quantum state of this group is given by z(j) = n,(n) = c2Z,(n)/2hv~.
(3.28)
A comparison with (3.25) now shows that an unpolarized light pencil of intensity Z,(n) has an entropy density per unit frequency and unit solid angleoJ3J4) y
=k:[(l+z)
ln[l+$)-sin%].
(3.29)
= c%OQ/ v
(3.30)
The specific entropy intensity is then(l) L(n)
and the local entropy density of the radiation field (3.31) The temperature T,(n) of a light pencil of specific intensity Z,(n) is now introduced means of the general relation (3.2)(l): l/T,(n)
= (aS/aQ
= aL,(n)/aZy(n)
by
(3.32)
since the energy U = VZJn) dv dQ/c corresponds to the entropy S = V&,(n) dv dQ/c and since the number of photons N has not to be held constant here because there is no conservation law for it. Carrying out the differentiation, one gets TV(n) =
(hv,/k)/ln
[l +(2hv~/c2ZV(n))]
so that the specific intensity is the Planck function corresponding
T,(n):
Z,(n) = (2hv~/c2)/(ehv~‘kT~cn) - 1) = BYo(T,(n)).
(3.33) to the temperature (3.34)
13
Radiative transfer and irreversibility
Finally, it follows from the equation intensity obeys the transfer equation(2) dL(n)lds
of radiative
= (‘G/T,(n))(S*
transfer
(2.8) that the entropy (3.35)
-Z,(n))
for the energy exchange of a light pencil with a material system is reversible if it takes place at the temperature T,,(n). Since the source function and the intensity can be written as Planck functions (cf. (3.23) and (3.34)), it follows from the transfer equation (2.8) that the energy exchange between a light pencil and the system of the excitation degrees of freedom satisfies the usual convention that the energy flows from the higher to the lower temperature: For instance, if T* > T,(n), then S* > Z,(n) and hence dZ,(n)/ds > 0, i.e. energy flows from the atoms to the light pencil; eq. (3.35) shows that then also dl,(n)/ds > 0 obtains. IV.
GENERAL
THEOREMS AND
ON
ENTROPY
TEMPERATURES,
ENTROPIES,
PRODUCTION
For the derivation of some theorems on temperatures, entropies, and entropy production, valid for the model considered, it is useful to introduce dimensionless quantities. Instead of the frequency v, one defines a dimensionless frequency x = (v-v,,)/AvD
(4.1)
with the Doppler width Av, = (~,/c)(2kT/M)~‘~.
(4.2)
The source function S* and the intensity Z,,(n) will be expressed by their values relative to the Planck function B,,(T): s* = S*/&,(T),
.L(n) = L(WG,(
T);
(4.3)
eq. (2.15) then takes the form s* = ( _,&dx++(l s
+E)
with the dimensionless mean intensity (4.5) and the dimensionless absorption
profile
A = &$v, Instead of the temperatures E = hvo/k T,
I
+,dx
= 1.
T, T*, and T,,(n), the dimensionless f* = hv,JkT*,
(4.6) quantities
E,(n) = hvdkT,(n)
(4.7)
will be introduced. Entropies are made dimensionless by dividing them by Boltzmann’s constant. For the entropy density tsexcof the excitation degrees of freedom one gets from (3.18) S,,,lk V = =exc = n1 ln (glno/nl) +n2 ln (g,n,/n,). (4.8)
74
J. OXENIUS
The entropy density per unit frequency and unit solid angle o,(n) and the entropy intensity A,(n) that correspond to the specific intensity j,(n) are according to (3.29) and (3.30) given by AvDS,(n)/k V = o,(n) = (2vgAvD/c3)pz(n) (4.9) and where the dimensionless
(4.10)
quantity
p,(n) has been introduced.
(2@vD/c2>4Q
= h,(n) =
Av,L,(n)/k
ln (I+$!$!)
= (1 +s,j
Finally,
the local entropy
-gins density
(4.11)
of the radiation
field o,,~ is (cf.
(3.31)) S&k 1. Inequalities
V s o,,~ =
SI
ox(n) dx da.
(4.12)
for the temperatures
Since the Planck function B,,,(T) is an upper limit for the source function and the specific intensity (cf. Appendix B, eqs. (B. 1) and (B.2)), one has the following two inequalities: s* i jz(n) On the other hand, be written
eqs. (3.23) and (3.24)
1,
(4.13)
I 1.
(4.14)
respectively,
show that the source function
s* = (eS - l)/(er* - 1)
can
(4.15)
and the intensity jz(n) = (er - l)/(erJ”’ The comparison
with (4.13) and (4.14) thus leads to the inequalities E* 2 6,
T” I
T,
E,(n) 2 E,
T,(n)
I
(4.17) (4.18)
T,
i.e. both the excitation temperature T* and the radiant exceed the value of the kinetic temperature T. 2. Production
(4.16)
- 1).
temperature
T,(n)
can never
of radiant entropy
The production of radiant entropy is given by the divergence of the flux of radiant entropy (cf. Appendix A). The local production of radiant entropy of frequency x is therefore (cf. (A.13)) div(Av,E,/k)
= s&=(n) dQ = x$,
(4.19)
where #, = +zs(s* -j&W,(n) x =
n142&,(T)(1
dQi4r
(4.20)
-ewr*>,
(4.21)
Radiative
and the total local production
transfer and irreversibility
15
of radiant entropy is given by (cf. (A.14)) div(E/k)
(4.22)
= Grad = x#
where (4.23) In the framework of the model considered, the following theorem holds: The local production of radiant entropy, integrated over the whole spectral line, is always positive, (4.24)
* 20.
In order to prove this theorem, one inserts the source function (cf. (4.4) and (4.5)) s* = (J‘J’j,(n’)d,dydn’/4n)/(l
+E)+E/(~ +E)
(4.25)
in (4.23) and writes the expression thus obtained in the form # = #‘O’/(l +E) +#“‘/(1
+E)
of a purely scattering where I+P) corresponds to the entropy production and zjCrn)to that of a collisionally dominated (E = co) gas. Explicitly one has
#Cm1= jj(l
-j&0)5,(n)+,
dx dV4n.
Eq. (4.24) is proved if it can be shown that both #(‘) and #(O”)are positive. The validity of the inequality $GQ 2 0
(4.26) (C = 0)
(4.28)
(4.29)
can immediately be seen from (4.14). Moreover, since the integrand of (4.28) is positive for every x, it follows that the radiant entropy locally produced by inelastic collisions is positive for every frequency. In order to prove the inequality $P) r 0, (4.30) one multiplies the second term of the right-hand side of (4.27) by jJ+, dy dCX/4v = 1 so that one can write
=
sss.i
=
liz(n)t,(n’) SSlS
[_@>E,(n’)-~l(n’>~z(n’)l~z+y dx dy dQ dQ’/(W2
--.AW4,(n’)WAv dx dy dQ dQ’/(W2
(4.3 1)
16
J. OXENIUS
where the last three integrals follow from the first by a simple exchange of the integration variables. Addition of the four integrals in (4.31) now yields
+ (jy(n) --_i,(n’)Ns(n’) -tyb))lMy
dx dy dQ dQ’l(4nY.
(4.32)
As can be seen from (4.16), both terms of the integrand are positive: (j, -j,)([, - tl) B 0. The inequality (4.30) and hence the theorem (4.24) are thus proved. An analogous theorem does not hold for the monochromatic entropy production #,, i.e. JI, can be negative. Since the radiant entropy produced by collisions is positive for every frequency, a negative local entropy production can only be due to non-coherent scattering. Physically, a negative & simply means that more photons of frequency x are scattered into other frequency ranges than are created by inelastic collisions and by scattering from other frequencies. However, the production of monochromatic radiant entropy, integrated over the whole volume of the gas, is positive for every frequency: s x$, d V 2 0.
(4.33)
By means of Gauss’ theorem, one can write Jx+,d V = Jdiv(Av,E,/k) =
=
$ $
n . (Av,E,/k)
dV dA
dA h,(n’)n .n’d!CI’ i
(4.34)
where (A.9) has been used and the integration dA is over the surface of the atmosphere whose outer normal is n. Since, by assumption, no radiation is incident on the surface of the gas from outside, one has*
Un’)
LO
if
n.n’>O
= 0
if
n.n’-cO
1
because h,(n) is never negative and vanishes only when the specific intensity vanishes (cf. (4.10) and (4.11)). From (4.34) and (4.35) follows (4.33) at once. 3. Total entropy production
In order to maintain a steady state, the radiant energy loss of the gas has to be compensated by a steady energy flux from some energy releasing reaction to the electrons (cf. Fig. 1). Quantitatively, the energy per unit volume and unit time hv,(n,!&, -nzC&) * For simplicity, an everywhere concave surface is assumed. The results do not depend on this assumption.
Radiative transfer and irreversibility
hv, ( n, a,,-
Electrons T
t
FIG. 1. Schematic
n2 a,,
1
Reaction T’
I
energy flow diagram for the model treated. The values indicated energies per unit volume and unit time.
has to flow from the reaction to the electron gas, giving rise to a production per unit time and unit volume S’jk V = 6’ = -(hv,,/kT’)(nlL2,,
are
of entropy (4.36)
-n2i22,)
if the reaction takes place at the temperature T’. Such energy flux is possible only when the reaction temperature is higher than the electron temperature: T’ 2 T
(4.37)
hvJkT’ s 6’ 5 6.
(4.38)
or Using (2.14), (2.17), (4.15), and (4.21), the entropy production algebra the form c? = -xc(l
(4.36) takes after some (4.39)
-s*)f’.
The total local entropy production is the sum of the production and the entropy production due to the energy flux just mentioned,
of radiant entropy
. . Qtot = %$,+K
(4.40)
since the entropy of the electron gas and the system of the excitation degrees of freedom, respectively, production vanishes, . Qe1
. =
The Second Law in its local formulation
Qexo
=
(4.41)
0.
now requires thaP5)
. dtot 2 0.
By means of (4.22), (4.26), and (4.40), the total entropy production
(4.42) can be written
itot= “[&O)“g,p”) -(l -s’)E’)]
(4.43)
18
J.
OXENIUS
where #(O) and #cm) are given by (4.27) and (4.28), respectively. Since I/(O) 3 0 has already been shown (cf. (4.30)), the inequality (4.42) will be proved if A = #‘“‘/(l can be shown.
Using
the inequality A=$
+E) -(l
t,(n)
-s*)[’
(4.44)
3 E 3 6’ (cf. (4.18) and (4.38)),
(1 -j,(n))E,(n)+,
Ess
2 0
dxg
-(I
one has
-s*>Y
1 (1 -j,(n))E’+,
2-
dxz
-(l
-s*)E’
l+E = 5’[(1 -si;$z because
of (4.4). The relations
dx)/(l
+E> -(l
-$*>I
(4.45)
= 0
(4.44) and hence (4.42) are therefore
proved.
4. Isotropy and anisotropy An arbitrary field j;, of equal
radiation fieldj,(n) spectral distribution
will now be compared with the isotropic radiation of energy, i.e. j,(n) is connected with J5 through
s
j,(n)
for every frequency are equal, too:
x so that according
dQ/4rr
= 3, = j^,
(4.46)
to (4.4) the two corresponding
source functions
s* = s”’
(4.47)
The following theorems hold for every frequency x: The mean local entropy density of an anisotropic radiation of the isotropic radiation field of equal energy density: J‘
field is smaller than that
o,(n) dQ2/4rr = 0, 5 ox.
The local entropy production of an anisotropic the corresponding isotropic radiation field :
radiation
(4.48) field is greater
than that of
(4.49)
VL 2 8,. The mean inverse temperature of an anisotropic of the corresponding isotropic radiation field: J These theorems
will be proved
Ss(n) dQ/4rr in Appendix
= i,
radiation
2 6,.
field is greater
than
that
(4.50)
C.
V. EXAMPLE
As an illustration, some numerical values of entropies, temperatures, and the production of radiant entropy will be calculated in this section. First, approximate formulas will
Radiative transfer and irreversibility
19
be derived for the limiting case of low temperature that almost always obtains in problems of line radiation. They will then be applied to an example of an atmosphere that emits a spectral line of large optical thickness under extreme non-equilibrium conditions. The radiation field used for the numerical calculations will be taken from the literature. 1. The limiting case of low temperature
Line radiation of atoms usually takes place at low temperatures, i.e. at temperatures that are small compared with the excitation energy of the spectral line, because at higher temperatures ionization is prevailing. The limiting case of low temperature means physically that the induced emissions are negligible compared with the spontaneous ones. Quantitatively, it is defined by erzco B 1,
et* 3 1,
erB 1,
(5.1)
where the last two inequalities are consequences of the first because of (4.17) and (4.18). Before the corresponding approximation can be carried out, all quantities have to be expressed by the temperatures. By means of (3.16) and (3.21) the densities of the non-excited and the excited atoms are given by rz2 = n,g,e-r*/(g, +gzemr*). nl = nogAg, +gze-r*), (5.2) Inserting these expressions in (4.8) leads to an entropy density of the excitation degrees of freedom %xc = no[gz%eer*/(g, +gzeer*) + ln(g, +gzeer*)l. (5.3) Using (4.16), the quantity p=(n) (cf. (4.11)) that determines the behavior of the radiant entropy takes the form p2(n) = .!J,(n)erJn)/(eTJn) - 1) - ln(e’J”) - 1).
(5.4)
Finally, the source function s* and the specific intensity j,(n) have already been written as functions of the temperatures in (4.15) and (4.16), respectively. The state of thermodynamic equilibrium is characterized by the equality of all temperatures: [ = [* = [Jn). The expressions for thermodynamic equilibrium can thus be obtained from the general expressions by replacing there all temperatures by the kinetic temperature. In particular, the source function (4.15) and the intensity (4.16) are then Planck functions, s* = j,(n) = 1, so that according to (4.20) the entropy production vanishes for all frequencies: #, = 0. The transition to the limiting case of low temperature, i.e. the application of the inequalities (5. l), now leads to the following approximate formulas: n1 =
no,
n2 =
~ok2/gl)e-t*~
%xc = no(g2/g1)5*e-z* am
= e-rJo(l se = eT-r* j,(n)
+no lng,,
+4,(n)), 9
= ee-rAn).
(5.5) (5.6) (5.7) (54 (5.9)
80
J. OXENIUS
One can now rewrite the occupation numbers, entropies, and temperatures in terms of the source function and the specific intensity because these latter quantities are usually calculated. It is useful to divide at the same time all quantities by their values in thermodynamic equilibrium. In this way, one gets for the density of excited particles n,jnih = s*,
(5.10)
i.e. the spatial distribution of excited atoms is directly given by the source function. Further, one obtains for the temperatures [*/t = 1 -(l/Q tZ(n)/E = 1 -(l/4)
Ins*,
(5.11)
lnjZ(n),
(5.12)
for the variable part of the excitation entropy (%XC--no
lngd/(~~%
-no Ing,)
= s*[l -(l/Q
Ins*],
(5.13)
In_idn)l
(5.14)
and for the radiant entropy s,(n)/3””
= pz(n)/pth
= j,(n)U
-(l/(1
+5))
where eth = 02~147~. 2. Numerical example The actual behavior of the entropies and temperatures will be illustrated by an example of an atmosphere of large optical thickness whose radiation field deviates strongly from that of thermal equilibrium. The numerical values of the source function and of the intensity have been taken from a paper by AVRETTand HUMMER~~). The atmosphere considered has the following characteristics: The line broadening is entirely due to the thermal Doppler effect; the normalized absorption profile is hence +I = ,-1/2e-~2. The atmosphere is homogeneous, plane parallel and of a total optical thickness 0 = lo4 where the optical depth 6, measured perpendicular to the radiating surface, is defined by the optical depth B. in the line center x = 0 through 0 = 6,/g, = &280. It is dominated by radiation processes, E = lOme, and a kinetic temperature 5 = 5 is assumed so that the formulas for the limiting case of low temperature apply. Figure 2 shows the source function s* and hence the density of excited atoms n2, the variable part of the excitation entropy o,,, - no In g,, and the excitation temperature T* as functions of the optical depth 0. The excitation entropy shows a similar behavior as the source function, and the excitation temperature is nearly constant and not essentially different from the kinetic temperature. The monotonic increase of the source function from the boundary (0 = 0) to the center (0 = 5 x 103) of the atmosphere is simply due to the fact that the radiation field is strongest at the center so that the absorption rate is higher there than near the surface. In Fig. 3, the mean intensity j;, the mean radiant entropy GZrand the mean radiant temperature TZ are plotted as functions of the frequency x for the optical depth B = 10. Strictly speaking, instead of ZZand T,, Orand pZ corresponding to the isotropic radiation field of equal spectral distribution of energy have been plotted, but the differences are for the example considered not very great (cf. Appendix D). The radiant entropy shows a similar behavior as the intensity whereas the radiant temperature is near the line center approximately constant and of the same order of magnitude as the kinetic temperature.
Radiative
transfer and irreversibility
81
The self-reversed core of the mean intensity is a typical feature of isothermal gases emitting a spectral line of large optical thickness under non-equilibrium conditions.(6~8*g*10*11’ The self-reversal steadily increases from the center to the surface of the atmosphere and can easily be understood from the monotonic increase of the source function from the boundary towards the interior (cf. Fig. 2).
FIG. 2. The source function s*, the variable part of the excitation entropy (I~=,,--no In gI, and the excitation temperature T*, all referred to their respective values in thermodynamic equilibrium, as functions of the optical depth 0. All quantities are symmetrical about the center of the atmosphere 0 = 5 x 103.
82
J. OXENIUS
lo-44 0
2
3
FIG. 3. The source function_ s*, the mean intensity ~1, the mean radiant entropy O,, and the mean radiant temperature T,, all referred to their respective values in thermodynamic equilibrium, as-functions of the frequency x at the optical depth 0 = 10. The plotted values of Ozand T, correspond to an isotropic radiation field. All quantities are symmetrical about the line center x = 0.
Radiative transfer and irreversibility
83
-0,25
I -o,sxIo-5 FIG. 4. The quantity 4, that is proportional to the local production of monochromatic radiant entropy, as a function of the frequency x at the optical depth 8 = 10. The plotted values of $. correspond to an isotropic radiation field. I/$ is symmetrical about the line center x = 0.
Finally, Fig. 4 shows for the optical depth 0 = 10 the quantity I,G~ which is proportional to the production of monochromatic radiant entropy. The radiation field has again been considered isotropic so that the plotted curve is, according to (4.49) a lower bound for the actual production of monochromatic radiant entropy; an estimate of the error thus introduced is given in Appendix D where it is also shown that negative I+$really occur within the framework of the model adopted in this paper. The order of magnitude of the total production of radiant entropy # can be estimated in the following way, assuming again isotropy. From (4.23) follows 1/,= s*/L+,
dx -j?&,$z
dx N”(5, >(s* -j._L+z dx)
(5.15)
if one takes into consideration that because of the exponential behavior of & = nM1j2e-5a only frequencies in the neighborhood of the line center x = 0 contribute to the integrals so that the temperature t,, which varies slowly there, can be replaced by an appropriate average value (&). On the other hand, eq. (4.4) yields
s* -
s
j-&, dx = ~(1 -s*)
(5.16)
J. OXEMUS
and hence for s* < 1 s*-
i
j;&dx
% E.
(5.17)
From (5.15) and (5.17) one gets (5.18)
* =cG,>. In the present example is 5 = 5 and E = 10A6; from Fig. 3 follows (&) = f(T/T,) so that 4 z 1.2 x 10e5 obtains. VI. CONCLUDING
M 12
REMARKS
The investigations of this paper show that the notions of entropy and temperature, originally introduced for describing thermodynamic equilibrium, can be supplied successfully to extreme non-equilibrium situations. Whereas the notion of a non-equilibrium entropy, due to Boltzmann and Planck, is fairly well known, at least in the form of the H-function which applies to the kinetic degrees of freedom, the notion of a non-equilibrium temperature, due to Planck, is practically not used in the literature. Non-equilibrium temperatures are, at best, introduced in a purely formal way, thus hiding their thermodynamic significance. For instance, if the excitation temperature is formally “defined” by writing the source function as the corresponding Planck function, it is overlooked that the excitation temperature follows in a unique way from Planck’s general definition such that the equation S* = B,,(T*) is a result and not a formal definition of T*; likewise, the relation I,(n) = B,,(T,(n)) is a result and not a formal definition of the radiant temperature T,(n) by means of the specific intensity I,(n). It is interesting to note that these two relations follow from two different forms of non-equilibrium entropies, namely from that of a Boltzmann and a Bose gas, respectively. Non-equilibrium temperatures may serve to facilitate the calculation of the entropy production. So is the transfer equation of radiant entropy most easily derived by making use of the notion of the temperature of a monochromatic light pencil. Non-equilibrium thermodynamics has had no need of generalized temperatures until now because it has dealt only with small deviations from thermodynamic equilibrium in which case one single temperature is locally defined. APPENDIX
A
At every point of the atmosphere there are uniquely given fluxes of radiant energy and radiant entropy, respectively. The amount of radiant energy of frequency v that flows at a given point in the direction n is given by lJ,(n’)n
and is hence the projection
. n’ da’
= n . sI,,(nr)n’ da’
of the vector F, = /d,,(n)
da
(A.1)
on this direction, i.e. F, is the flux of radiant energy of frequency v. The total flux of radiant energy is thus F = j-F’ dv.
(A-2)
Radiative transfer and irreversibility
85
For the divergence one gets div F, = div JnZJn) da = jn . grad Z,(n) dQ
(A-3)
or, by means of the transfer equation (2.8) n . grad Z,(n) = dZ,,(n)/ds = k0(+,/4rr)(S* -Z,(n))
(A-4)
where (cf. (2.9)) (A-5) div F, = &&,(S* -TV).
(A-6)
The divergence of the total flux of radiant energy is then div F = k,( S* -p&
dv)
(A-7)
which can be written div F = [Z&+/(1+E)][&,( T) -s&
dv]
(A.8)
taking (2.15) into account. In complete analogy, the flux of radiant entropy of frequency v is E, = j&(n)
dSZ
(A-9)
and the total flux of radiant entropy E =
s
E,dv.
(A. 10)
For the divergence one gets again div E, =
i
n . grad L,(n) da,
(A.ll)
or, using the transfer equation (3.35) n . grad L,(n) = dl,(n)/ds
= M+,/4nT,(n))(S*
-Z,,(n)),
div E, = k,+,/( S* - Z,,(n))(T,,(n))-l dS2/4rr
(A.12) (A.13)
and hence div E = k,//(S*
- Z,(n))(+,/T,(n)) dv dS2/47.
APPENDIX
(A.14)
B
In the framework of the model considered, the Planck function corresponding to the kinetic temperature T is an upper limit for both the source function and the specific intensity : S* I &o(T), UN
5 &o(T).
(B-1) (B-2)
J.
86
OXENIUS
Writing for the sake of brevity Z(s) = I&I, s) and the coordinate along the considered ray, the formal
K(S)
=
KJS)
solution
where s (2 0) denotes of the transfer equation
(2.8) dl(s)/ds
=
K(S)(
s*(S)
(B.3)
-I(s))
is given by I(s) = jS*(s’)
jK(s”) ds”)!c(.F’)
exp ( -
0
(B-4)
S,
where the boundary condition has been taken into account on the surface of the atmosphere (s = 0) from outside, variables s’ s y =
ds’
y’ =
ds”,
J-K@“)
that no radiation is impinging i.e. I(0) = 0.” With the new
ds”
IK(S”) 0
0
one gets Z(y)
= /S*(y’)ey
-Ydy’
0
Y
5
Y
hoax eylYdy’
1
= S&,
eetdt
s 0
0
s =
m
5
eetdt
%3x
S$,,,
0
i.e. the specific intensity inside the atmosnhere :
From
can never
exceed
(B.5) one gets for the source function
the maximum
in the left-hand
side the maximum SZ,,
and hence a fortiori relation from (B.5) and (B.6). *For simplicity, assumption.
(B.l).
an everywhere
of the source
SL,
+ cB,,( T) l+E
source function
’ S*max’ (B.6)
5 B,,(T)
On the other hand, the inequality
concave
function
(2.15) the inequality
s &L dv + E&S T) S*=P
value
surface is assumed.
(B.2) follows at once
The results do not depend
on this
Radiative
transfer and irreversibility
APPENDIX
87
C
In order to prove the inequality (cf. (4.48))
(C-1) under the assumption
(C.2) one has because of (4.9) to show that fi -
s
/.+I) dQ/4rr r 0
(C.3)
where p(n) = (1 + ~j(n)) In (1 + aj(n)) - orj(n) In aj(n),
(C.4)
r; = (l+ctj)In(l+aj)-~jlnorj,
(C.5)
cc = l/(er-1)
(C.6)
> 0.
One can write
(C.7) where A = ln(l+aj)-/ln(l+aj(n))dQ/4a,
(C-8)
B =jh(l+-i)-S/(n)In(l+&)z,
cc.9
so that (C.3) is proved if A > 0 and B 2 0 can be shown. To prove A > 0, one observes that for a non-negative functionflt)
holds where the bar denotes the average over a given t-interval; this follows from the fact that the arithmetical mean of non-negative quantities is always greater than their geometrical mean. Hence A = In [s(l +Mj(n)) dQ/4rr] -1 In (1 +aj(n)) dfi2/4rr = ln(l+orj(n))-ln(l+xj(n))
2 0.
(C. 10)
To prove B 2 0,one introduces a function T(n) through with
j(n) = j< 1 + T(n))
(C.11)
T(n) dQ/4rr = 0
(C.12)
I
88
J. OXENIUS
because of (C.2), and
T(n) 2
B =
-
1
because ofj(n) > 0. One can thus write
J_K#[ ln (1+-t) -l++-&j]$
s
=j
(1+5-(n)) In l+
with
q=
(C.13)
cxj> 0.
One now defines two non-negative functions T+(n)
=
(C.14) and
T’(n)
through
T-(n)
44
if
T(n)
0
if
7(n) < 0,
>
0
(C.15)
(C.16) with (C.17) because of (C.l2), and form
T-(n)
<
B =j
1 because of
(l+T+(n))h
[S +
s
T(n)
z
-
1.
Equation (C.13) now takes the
l+ T-(n)
l-l
+q_qT_(n)
1
-.1 d!A 4~
(C.18)
From In (1 +x) > x/(1 +x), valid for x b - 1, follows T+(n)
T+(n) 1 +q and from In (1 -x)
> - x/( 1 -x),
+qT+(n)
’
(1
+q)(l
(C.19)
+T’(n))
’
valid for x 6 1,
7-(n) 1 -t-q -47-(n)
7-(n)
(C.20)
--. ’ - (1 +q)(l -T-(n))
From (C.18), (C.19), and (C.20), one finally gets
B 2j
[S
(1 +T+W)(,
= [j/(1 +q)][JT+(n)
s
+q;;r;+(,,,; - (1-7-W) dn/4?r -ST-(n)
dQ/4a]
T-(n) (1 +q)(l
= 0.
using (C.17). This completes the proof of (C.3) and hence of (4.48).
-T-(n))
d!J 4?~
1 (C.21)
Radiative transfer and irreversibility
89
Because of (4.20) one has (A -6,)/A
= j%* -j,(n))&(n)
dQ/4n -(i* -&)iz
= s*(/&(n) dQ/4r -$A +(jz$, -j~,(n)E,(n)
dQ/4n)
(C.22)
taking (4.47) into account. The inequality (4.49) (C.23) is thus proved if jt -jj(n)S(n)
dLI2/4rr > 0
(C.24)
and 2 0
(C.25)
E(n) = ln (1 + l/c&t)),
(C.26)
s
t(n) dSI/4rr-g
can be shown where (cf. (4.16))
8 = ln(l+l/orj).
(C.27)
Relation (C.24) has already been proved since (cf. (C.9) and (C.21))
jt -jj(n)f(n)
dS2/4n = B 2 0.
To prove (C.25), one writes by means of (C.ll),
(C.28)
(C.14), (C.15), and (C.16)
+
(C.29) R+
where R",R-, and RC denote the ranges in which T(n) = 0, ~0, and >O, respectively. Because of In ( In (
l+ q(l_:-(n))) l+
4(1+:+(n)))
= ln(l+j
+ ln(l+(l+q;;(O:-(n)))~
=ln(l+3+ln(l-(1+~~1~~+(n)~~
(c.30) (c’31)
one has from (C.29)
(C.32)
J. OXENIUS
90
since{
= ln(l+l/q)and In
and from In (1 -x) In
(
' = 4x. From In(1 +x) > x/(1 +x) follows
R"+R-+R
r-(n)
7-(n)
l+
’ 1 +q-p-(n)
(1 +q)(l -7-(n))
T-(n) >- 1+q’
(C.33)
2 -x/( 1 -x) T+(n)
T+(Q) 2 -p. (1 +q)(l +T+(n)) 1 ’ -1 +q +47+(n) lfq
l(
T+(n)
(C.34)
By means of (C.33) and (C.34), one now obtains from (C.32) SE(n) ds2/4n -i
> (1 +q)-l[/T-(n)
ds2/4n -jT+(n)
dQ/4r]
= 0
(C.35)
taking (C.17) into account. This proves (C.25) and hence (4.50) and completes the proof of (C.23) and hence of (4.49). APPENDIX
D
The numerical values of the example treated in Section V.2 have been obtained by replacing everywhere the actual radiation field by an isotropic one. The errors introduced hereby will now be estimated in a rough way. For this purpose, one compares the semi-isotropic radiation field A(n) =
j,(l +S,) -
(
j,(l
-8,)
if
n.n,>O
if
n.n,>O
(D.1)
where no denotes the outer normal of the plane parallel atmosphere and IS21 < 1, with the isotropic radiation field of equal spectral energy density 3, = j,.
(D.2)
Using the formulas for the limiting case of low temperature (cf. Section V.l), one gets after some algebra for the relative errors of the mean radiant entropy, the mean radiant temperature, and the monochromatic entropy production Ox p=
-5s
(l+S,)ln(l+S,)-(I-S,)ln(l-S,) ,
2(1 + i,)
Ox
ln(1+S,)ln(1-S,)-~~zln(1-S2,)
i;-?;
--;_
^=ln(l+S,)ln(l-SJ-l,ln(l-Sz)+Iz’ TX
baa! 4,
1 =Fg
-25
[c--j;,
1+s, ln-----l-S,
where 6, = .$-lnjl.
1
ln(l -SYj) ,
(D.3)
(D.4)
(D-5)
(D-6)
Radiative transfer and irreversibility
91
The greatest errors occur at the boundary of the atmosphere where 6, = 1 since no radiation is incident from outside. In the example considered, the relative error of the mean radiant entropy turns out to be (6 per cent whereas that of the mean radiant temperature is about 50 per cent, i.e. TX z SF,. The errors for the optical depth 0 = 10 are correspondingly smaller. The error for the monochromatic entropy production at the optical depth 8 = 10 is much harder to estimate since the degree of anisotropy is expected to increase rapidly from the line center to the line wings. For the arbitrarily chosen values 6, = 4, +, 2, one finds relative errors of 9, 37, 95 per cent for the line center x = 0, and of - 3, - 11, - 30 per cent for the frequency x = 2, respectively. The actual I,&will therefore, at least in an interval 0 < x < 2, not too strongly deviate from the &corresponding to the isotropic case. On the other hand, the relative errors between #, and #, can be made as small as one pleases by choosing a sufficiently low temperature ([ B 1) as can be seen from (D.5) and (D.6). For the radiation field depends, apart from the total optical thickness, only on the parameter E which can be kept constant by choosing for any given temperature an appropriate electron density. This shows that negative #, really occur within the framework of the model adopted in this paper. Acknowledgement-The
author takes pleasure in acknowledging
discussions
with Dr. G. ECKER.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
9. 10. 11.
12. 13. 14. 15.
M. PLANCK, The Theory ofHeat Radiation. Dover, New York (1959). R. WILDT, Astrophys. J. 123, 107 (1956). R. N. THOMAS.Astroohvs. J. 125. 260 (1957). J. OXENIUSJQiRT 5; 7?1. ’ . ’ J. T. JEFFERIES and 0. R. WHITE, Astrophys. J. 132, 767 (1960). A. G. HEARN (a) Proc. Phys. Sot. 81, 648 (1963); (b) Proc. Phys. Sot. 84, 11 (1964). M. DOBROWOLNYand F. ENGELMANN,Nuovo Cim. 31, 965 (1965). J. T. JEFFEFUES and R. N. THOMAS,Astrophys. J. 127, 667 (1958); Astrophys. J. 129, 401 (1959); Astrophys. J. 131, 695 (1960). V. V. IVANOV, Sov. Astron. 6, 793 (1963); Sov. Astron. 7, 199 (1963). S. CUPERMAN,F. ENGELMANN,and J. OXENIUS,Phys. Fluids 6,108 (1963); Phys. Fluids. 7,428 (1964). E. H. AVRETTand D. G. HUMMER,Mon. Not. R. Astr. Sot. (in press). L. D. LANDAuand,E. M. LIFSHITZ,Statistical Physics, p. 116-l 18 and p. 155. PergamonPress, London, (1958). P. ROSEN,Phys. Rev. 96, 555 (1954). A. ORE, Phys. Rev. 98, 887 (1955.) S. R. DE GROOTand P. MAZUR, Non-equilibrium Thermodynamics, p. 22 North-Holland, Amsterdam (1962).
RADIATIVE
TRANSFER AND IRREVERSIBILITY* J.
Joint Institute
OXENIUSt
for Laboratory
$
Astrophysics,
Boulder, Colorado
(Received 12 July 1965) Abstract-The irreversibility of the interaction of radiation with matter is investigated by considering a stationary isothermal gas that emits the resonance line of atoms with only two discrete energy levels. In order to describe such a situation microscopically, the steady-state balance of the occupation numbers of the atomic levels and the equation of radiative transfer have to be taken into account simultaneously. The explicit expressions for the non-equilibrium entropies and temperatures of the radiation field and of the gas of the two-level atoms are derived in a systematic way. In the framework of the model considered, the kinetic temperature is an upper limit for both the radiant temperature and the excitation temperature. The local production of radiant entropy, integrated over the whole spectral line, is always positive, but, as a result of non-coherent scattering, it can be negative in certain frequency intervals. The radiant entropy produced by the entire atmosphere is, however, positive for every frequency emitted. Likewise, the total local entropy production is, in agreement with the Second Law, always positive. It is further shown that an anisotropic radiation field has a smaller entropy density and a greater production of radiant entropy than the isotropic radiation field of equal spectral distribution of energy. As an example, an atmosphere is treated numerically that emits a spectral line of large optical thickness under extreme non-equilibrium conditions. I.
INTRODUCTION
PLANCK(~) was the first to investigate the interaction of light and matter with respect to its irreversibility. Quite in contrast to the situation of thermodynamic equilibrium, the study of non-equilibrium phenomena requires that a model of the matter and its interaction with radiation is adopted. Planck chose a set of independent linear harmonic oscillators with parallel axes and could show that in the case of an interaction of these oscillators with a given radiation field the entropy of the total system, oscillators plus radiation, never decreases. As was pointed out already by Planck himself, the most severe limitation of this model lies in its restriction to a strictly monochromatic situation: Irreversibility due to changes of the spectral energy distribution of the radiation field remains necessarily outside of its scope. Thermodynamic aspects of radiative transfer processes were later studied by WILDT(~) who was mainly interested in atmospheres in radiative equilibrium, i.e. in situations in which the divergence of the radiant energy flux vanishes. The transfer equation used by him was formulated assuming “true” absorption and isotropic coherent scattering and hence excluding the effect of.non-coherent scattering. Wildt derived a number of inequalities for the radiant entropy that follow from macroscopic considerations alone, and was the first to formulate the transfer equation for the radiant entropy.
* Work supported in part by the Advanced Research Projects Agency (Project Defender) Contract DA-31-124-ARO-D-139. t On leave from Association Euratom-CEA, Fontenay-aux-Roses (Seine), France. $ Present address: Institut fiir Theoretische Physik der Universitlt Bonn, Germany. 65
under
66
J. OXENIUS
The present paper treats the irreversibility of the interaction of a radiation field with matter by means of a well-defined microscopic model. It considers the statistically steadystate of a self-excited gas that contains atoms with only two discrete energy levels. The irreversibleprocessto bestudied is the emission of the resonance line of the two-level atoms, In order to maintain a steady-state, the energy loss due to radiation has then to be counterbalanced by some energy-releasing reaction taking place inside the gas. The model chosen avoids two limitations of Planck’s oscillator model. First, the radiation field is not prescribed arbitrarily but follows in a unique way from the physical properties of the system and the boundary conditions. And second, the restriction to one single frequency is removed since line broadening effects are taken into account; i.e. non-coherent scattering is treated instead of the monochromatic coherent scattering studied by Planck. The main limitation of the present model consists in its restriction to only one atomic transition, excluding any coupling with other bound states and with the continuum. In general, the radiation of hot gases deviates strongly from that of a black body, i.e. from the radiation in thermodynamic equilibrium. In an investigation of irreversible radiation processes, the possibility of large deviations from thermal equilibrium has therefore to be taken into consideration from the outset in order to avoid a restriction to cases that are of only little interest. On the other hand, non-equilibrium thermodynamics has considered until now only processes that take place in the neighborhood of thermodynamic equilibrium. Section II of this paper describes the model chosen in greater detail and puts together some results of former investigations that will be used in the subsequent sections. Planck’s general definition of entropy, which is applicable also to non-equilibrium states, is used in Section III to derive in a short and systematic way the entropies, together with the corresponding temperatures, of the gas of the two-level atoms and of the radiation field. Some general theorems on temperatures, entropies, and entropy production, valid for the model treated, are listed in Section IV. It is shown that the kinetic temperature of the gas is an upper limit for both the radiant temperature and the excitation temperature of the two-level atoms. The production of radiant entropy has two sources: inelastic collisions and non-coherent scattering. While the local production of radiant entropy due to collisions is positive for every frequency, that due to non-coherent scattering can be negative within certain frequency intervals; the radiant entropy produced by the entire atmosphere is, however, positive for every frequency emitted. Likewise, the local production of radiant entropy, integrated over the whole spectral line, is positive. Furthermore, the total local entropy production that contains in addition to the production of radiant entropy a contribution due to the steady-state energy flux from an energy-releasing reaction, is, in agreement with the Second Law, positive. Finally, it is shown that an anisotropic radiation field has a smaller entropy density and a greater entropy production than the isotropic radiation field of equal spectral distribution of energy. The expressions derived in the former sections are then specialized in Section V to the practically important limiting case of low temperature by means of which a plane parallel atmosphere is treated numerically that emits a spectral line of large optical thickness under extreme non-equilibrium conditions. The radiation field used for this example has been taken from the literature.
Radiative transfer and irreversibility II.
THE
67
MODEL
In this paper, an isothermal gas in a statistically steady-state will be treated that contains atoms with only two discrete energy levels and no continuum. It will be assumed that the mass motion of each gas component vanishes, that no external electromagnetic fields are present, and that no radiation is incident on the gas from outside. The process to be investigated is the formation and emission of the resonance line of the two-level atoms when the corresponding radiant energy loss is counterbalanced by some energy-releasing reaction inside the gas such that a steady-state obtains. In order to determine the radiation field of the considered spectral line, the equation of radiative transfer and the stationarity condition for the occupation numbers of the two atomic energy levels have to be taken into account simultaneously. The microscopic form of the time-independent equation of radiative transferC3)
-Wn) = ds describes the change of the specific intensity Z,(n) of an unpolarized light pencil of frequency v over a distance ds in a direction that is specified by the unit vector n. The frequency vO refers to the center of the spectral line, nl and n2 are the number densities of two-level atoms in the ground state 1 and the excited state 2, and &a, B,,, and Aa, are the Einstein-Mime coefficients for absorption, induced emission, and spontaneous emission of the atomic transition 1 t) 2. Line broadening effects are taken into consideration by the absorption profile &(n) and the emission profile -q,(n); more precisely, 4,(n) dv dQ/4rr is that fraction of non-excited atoms that can absorb photons with a frequency in the range dv centered about v, and a direction inside the solid angle dQ centered about n, and T,(n) dv dCl2/4r is correspondingly that fraction of excited atoms that can emit these photons. Both profiles are therefore normalized in the following way: (2.2)
In (2.1), use has been made of the fact that the profiles for spontaneous emission and for induced emission are identical. (*) Furthermore, a constant energy hv,, has been ascribed to all photons of the spectral line. The simplifying assumption will now be made that the emission profile and the absorption profile are equal: TV(n) = +,(n). (2.3) It can be shown(*) that (2.3) holds exactly if the following two conditions are satisfied: First, the frequency distribution of scattered photons is in the atom’s rest frame given by the frequency dependence of the atomic absorption coefficient, and their angular distribution is isotropic; by scattering, the absorption of a photon followed by reemission of another photon is here to be understood. And second, the velocity distributions of the non-excited and excited atoms, respectively, are equal, i.e. Xl(v) = fi(v) where theflv)
denote normalized
distribution
functions.
(2.4)
68
J.
OXENIUS
A situation in which these two conditions are fulfilled obtains when the elastic collisions of the two-level atoms with one another or with other particles are so frequent that the corresponding relaxation time is smaller than the mean lifetime of the excited level.‘4’ In the opposite limiting case of vanishing elastic collisions, eq. (2.3) often holds approximately. (3.5.6b.7) It will be assumed that the gas of the two-level atoms has a kinetic temperature T, i.e. that the total velocity distribution function of all atoms, irrespective of whether they are excited or not, is Maxwellian. Furthermore, the kinetic temperature T is supposed to be equal to the electron temperature T,: T=
T,.
(2.5)
In view of (2.4), this means that the velocity distributions of the non-excited atoms, of the excited atoms, and of the free electrons are all Maxwell distributionsf,(v) corresponding to the same temperature T: fi(V)
= fi(v)
(2.6)
= L?(v) = fif(v)-
Since the velocity distributionf,(v) of the two-level atoms in the ground state is Maxwellian and hence isotropic, and since no electromagnetic fields are present, the absorption profile 4, is isotropicC4’ and satisfies because of (2.2) the normalization condition i
$,dv = 1.
(2.7)
Writing the transfer equation in its usual macroscopic
form
dl,,(n)/ds = K,(S* -IV(n)),
(2.8)
one gets by comparison with its microscopic form (2.1) and taking (2.3) into account an isotropic absorption coefficient Kv = h%(R&
(2.9)
--QJ-%l)(~V/4~)
and a source function S* = %&,l(R&, which is isotropic and independent relations Azl/B,,
(2.10)
-R&?&1)
of the frequency V. For because of the well-known
= 2hv;/c2,
(2.11)
g, BI, = g,Bn
where g,, g, are the statistical weights of the two energy levels, S* depends only on the frequency v0 that characterizes the energy difference of the two levels. The absorption coefficient (2.9) and the source function (2.10) depend for a given total density nl +n, of the two-level atoms only on the ratio n2/n, of the occupation numbers. This ratio follows from the stationarity equationC3) n,(&zj-%v
dv +
42)
=
nz(A21+
B,,/&+,
dv +
a,,)
(2.12)
where (2.3) has been taken into account. i,, denotes the mean intensity, 7, =
i
I,(n) dL2/4r,
(2.13)
69
Radiative transfer and irreversibility
and Q12,QR,,are the rate coefficients for excitation collisions and de-excitation collisions, respectively. For the sake of simplicity, it will be assumed that inelastic collisions only with free electrons are important. Both !& and !&, are then proportional to the electron density and are related to each other through %,/%
=
k&h)e-nY~‘kT
(2.14)
since the electron gas has a kinetic temperature T. Inserting the ratio nz/nl from (2.12) in the source function (2.10) and taking (2.11) and (2.14) into account now leads toC3’ 1 s* = &$, dv f &L(T) .E lfr s
(2.15)
BY&T) = (2hv~/c2)/(ehvofkT- 1)
(2.16)
where is the Planck function and l
= (QzJA,,)(l
-e-nvJkT)
= Q,,/B,,B,,(
T)
(2.17)
is a dimensionless quantity that depends on the electron density and the kinetic temperature. At low temperatures (e-h”oikT < l), the parameter E is simply the ratio of electronic collisional de-excitation to spontaneous radiative de-excitation of the upper atomic level and is therefore a measure for whether the atomic occupation numbers are determined by collisional processes (E B 1) or by radiative ones (E < 1). The steady-state source function (2.15) is composed of two parts : The first term is due to non-coherent scattering whereas the second term describes the “real” source of radiation due to collisional excitation. This is most easily seen from the divergence of the total flux of radiant energy (cf. Appendix A, eq. (A.8)) div F = [k,~/(l +E)][&,( T) -/7+$, dv] where k0 = hv,(n,B,, -nzBzl). that
(2.18)
It follows from Z,(n) < i&(T) (cf. Appendix B, eq. (B.2)) div F 2 0
(2.19)
which expresses the fact that one is dealing with a self-excited gas. In the approximation used that ascribes to all photons of the spectral line the same energy hvo, the energy flux F is proportional to the photon flux. For E = 0, div F vanishes so that no photons are created; the limiting case E = 0 thus corresponds to a purely scattering atmosphere where the source function is given by J~,c$,,dv. On the other hand, in a collisionally dominated gas (C = co) the source function is simply the Planck function B,.(T). In order to calculate the radiation field, the equation of radiative transfer (2.8) and the stationarity condition (2.15) have to be solved simultaneously, taking the boundary condition into account that no external radiation is incident on the gas. Supposing the absorption coefficient K, to be known, one can eliminate one of the two unknown quantities S* and I,(n) leading either to an integral equation for S* or to an integro-differential equation for I,(n). The solution of this problem has been discussed extensively in the literature.(6,8,9~10,11)
J. OXENIUS
70
III. ENTROPY
AND
TEMPERATURE
The entropy S of an arbitrary, equilibrium or non-equilibrium macroscopically, is according to PLANCK~) given by S=klnW
state that is defined (3.1)
where W denotes the number of independent microstates that are compatible with the macrostate under consideration. If the entropy can be written as a function of the energy U, the volume V, and the number of particles N, the temperature T of the macrostate is defined by l/T = (as/au),,,. (3 *I) For an ideal gas, to which the discussion will be limited, a macrostate can be characterized in the following way (12): One divides the quantum states of the gas into groups of neighboring states with almost identical physical properties. Be G(j) the number of quantum states in the group j and N(j) the occupation number of this group, one has for the mean occupation number of a quantum state of the group j j$j) = N(j)/G(f).
(3.3)
For a given division of the quantum state into groups, a macrostate can thus be characterized by specifying either the N(j) or the ti(j). Is W(j) the number of possibilities to distribute No’ particles over G(j) quantum states, one has
and hence for the entropy
W = RW’j’ i
(3.4)
S = k 2 In W(j).
(3.5)
1. Classical ideal gas The gas of the two-level atoms will be considered as an ideal gas in the classical limit. In the case of Boltzmann statistics one haso2) W(i) = G(i)N’t’/N(j)l.
(3.6)
Using Sterling’s formula, eq. (3.5) now takes the form S = kN_k
c G(j)j$j) ln j$j) j
(3.7)
where N = CNCi) is the total number of particles. i The total entropy So of the gas of two-level atoms is the sum of the entropies of the non-excited and of the excited particles: &=&fS2=
2 si. 1=1.2
(3.8)
Denoting again by h(v) the normalized velocity distribution function of two-level atoms in the excitation state i, one has for a division of the translational quantum states into groups corresponding to velocity ranges d3u N’:’ = N,(v)
d3v = N&(v) d3v
(3.9)
Radiative
71
transfer and irreversibility
and G’f = G,(v) d3u = g, VM3 d3v/h3
(3.10)
where M is the mass of a two-level atom and V an arbitrary, and (3.10), equation (3.7) leads to an entropy density
small volume. Using (3.9)
S,/ V = kni[ln (eg,M3/nrh3)
-JA(v)
In_&(v) d3u],
(3.11)
i.e. except for an additive constant, to the negative H-function. The local number densities n, = Ni/V have been introduced, and e is the base of the natural logarithms. In the model considered, the distribution functions A(v) are Maxwell distributions, h(v) = fM(v) = (M/27& T)3’2e- M”2’2kT, so that (3.11) transforms
(3.12)
into the formula of Sackur and Tetrode (3.13)
A!?,/V = kni[g + In (g,/Q3)] with the thermal de Broglie wavelength x = h(2aMkT)-1’2.
(3.14)
From (3.8) and (3.13) follows the total entropy density of the gas of two-level atoms S,/ V = k[n, ln (s&Q
+n2 ln
k2/n2>l-t~~o[~-3 ln Al
(3.15)
with the total number density of two-level atoms Izo = n,+ns.
(3.16)
The total entropy density (3.15) can now be written as the sum of the entropy density of the translational degrees of freedom S trans/ v = kn,[P - ln
(d3)1
(3.17)
and the entropy density of the excitation degrees of freedom S,,,/ v = k[fi, ln (gln&)
+n2 ln k2no/n2)l.
(3.18)
S trans corresponds to the entropy of a gas of N, = n, V non-localized mass-points of a thermal de Broglie wavelength A, and S,,, to the entropy of a system consisting of No = (nl +nJ Vlocalized particles that have two quantum states 1 and 2 with statistical weights gl and g2 and occupation probabilities nJn,, and n2/no. It should be stressed that such a decomposition of the total entropy is possible only when the respective velocity distributions of particles in the different excitation states are equal. Starting from (3.18) and the general temperature definition (3.2), the excitation temperature T* is given by l/T*
= (aS.Aa
Uexc)~,~o = (a(&,,/
V/%&J)n,
for U,,, = N2hvo is the excitation energy. Carrying out the differentiation or
(3.19) leads to
1/ T* = (k/&J ln (gZnl /gl%)
(3.20)
rt2/lz1 = (g2/g,)e-nYJkT* .
(3.21)
Expressing in (3.20) the ratio n2/n1 by the source function (2.10) and taking (2.11) into account, one gets T* = (hv,fk)/ln[l +(2hvE/c2P)] (3.22)
72
J. OXENIUS
so that the source function is the Planck function corresponding
to the temperature
T* :
S* = (2hv~/c2)/(e WkTX- 1) = B,& F). (3.23) 2. Radiation field The radiation field can be considered as an ideal photon gas. In the case of Bose statistics one haso’) J+‘(j)= (G’i’ + No’ _ 1) I /[( G’i’ -l)!W’!]. (3.24)
Using Sterling’s formula, equation (3.5) now takes the form s = k 2 G(j)[(l -+.j@) ln(1 +@)
-i(j) ln $)I.
Dividing the quantum states into groups corresponding elements of solid angle dQ, one has
(3.25)
to frequency
intervals dv and
No) = N,(n) dv da = VZ,(n) dv dQ/hv,c
(3.26)
because an unpolarized light pencil Z,(n) dv dQ of the considered spectral line has a photon density Z,,(n) dv dL?/hv,c; V denotes again an arbitrary, small volume. Since the number of quantum states of this group is G(j) E G,,(n) dv dQ = V2vz dv d!J/c3, the mean occupation
(3.27)
number of a quantum state of this group is given by z(j) = n,(n) = c2Z,(n)/2hv~.
(3.28)
A comparison with (3.25) now shows that an unpolarized light pencil of intensity Z,(n) has an entropy density per unit frequency and unit solid angleoJ3J4) y
=k:[(l+z)
ln[l+$)-sin%].
(3.29)
= c%OQ/ v
(3.30)
The specific entropy intensity is then(l) L(n)
and the local entropy density of the radiation field (3.31) The temperature T,(n) of a light pencil of specific intensity Z,(n) is now introduced means of the general relation (3.2)(l): l/T,(n)
= (aS/aQ
= aL,(n)/aZy(n)
by
(3.32)
since the energy U = VZJn) dv dQ/c corresponds to the entropy S = V&,(n) dv dQ/c and since the number of photons N has not to be held constant here because there is no conservation law for it. Carrying out the differentiation, one gets TV(n) =
(hv,/k)/ln
[l +(2hv~/c2ZV(n))]
so that the specific intensity is the Planck function corresponding
T,(n):
Z,(n) = (2hv~/c2)/(ehv~‘kT~cn) - 1) = BYo(T,(n)).
(3.33) to the temperature (3.34)
13
Radiative transfer and irreversibility
Finally, it follows from the equation intensity obeys the transfer equation(2) dL(n)lds
of radiative
= (‘G/T,(n))(S*
transfer
(2.8) that the entropy (3.35)
-Z,(n))
for the energy exchange of a light pencil with a material system is reversible if it takes place at the temperature T,,(n). Since the source function and the intensity can be written as Planck functions (cf. (3.23) and (3.34)), it follows from the transfer equation (2.8) that the energy exchange between a light pencil and the system of the excitation degrees of freedom satisfies the usual convention that the energy flows from the higher to the lower temperature: For instance, if T* > T,(n), then S* > Z,(n) and hence dZ,(n)/ds > 0, i.e. energy flows from the atoms to the light pencil; eq. (3.35) shows that then also dl,(n)/ds > 0 obtains. IV.
GENERAL
THEOREMS AND
ON
ENTROPY
TEMPERATURES,
ENTROPIES,
PRODUCTION
For the derivation of some theorems on temperatures, entropies, and entropy production, valid for the model considered, it is useful to introduce dimensionless quantities. Instead of the frequency v, one defines a dimensionless frequency x = (v-v,,)/AvD
(4.1)
with the Doppler width Av, = (~,/c)(2kT/M)~‘~.
(4.2)
The source function S* and the intensity Z,,(n) will be expressed by their values relative to the Planck function B,,(T): s* = S*/&,(T),
.L(n) = L(WG,(
T);
(4.3)
eq. (2.15) then takes the form s* = ( _,&dx++(l s
+E)
with the dimensionless mean intensity (4.5) and the dimensionless absorption
profile
A = &$v, Instead of the temperatures E = hvo/k T,
I
+,dx
= 1.
T, T*, and T,,(n), the dimensionless f* = hv,JkT*,
(4.6) quantities
E,(n) = hvdkT,(n)
(4.7)
will be introduced. Entropies are made dimensionless by dividing them by Boltzmann’s constant. For the entropy density tsexcof the excitation degrees of freedom one gets from (3.18) S,,,lk V = =exc = n1 ln (glno/nl) +n2 ln (g,n,/n,). (4.8)
74
J. OXENIUS
The entropy density per unit frequency and unit solid angle o,(n) and the entropy intensity A,(n) that correspond to the specific intensity j,(n) are according to (3.29) and (3.30) given by AvDS,(n)/k V = o,(n) = (2vgAvD/c3)pz(n) (4.9) and where the dimensionless
(4.10)
quantity
p,(n) has been introduced.
(2@vD/c2>4Q
= h,(n) =
Av,L,(n)/k
ln (I+$!$!)
= (1 +s,j
Finally,
the local entropy
-gins density
(4.11)
of the radiation
field o,,~ is (cf.
(3.31)) S&k 1. Inequalities
V s o,,~ =
SI
ox(n) dx da.
(4.12)
for the temperatures
Since the Planck function B,,,(T) is an upper limit for the source function and the specific intensity (cf. Appendix B, eqs. (B. 1) and (B.2)), one has the following two inequalities: s* i jz(n) On the other hand, be written
eqs. (3.23) and (3.24)
1,
(4.13)
I 1.
(4.14)
respectively,
show that the source function
s* = (eS - l)/(er* - 1)
can
(4.15)
and the intensity jz(n) = (er - l)/(erJ”’ The comparison
with (4.13) and (4.14) thus leads to the inequalities E* 2 6,
T” I
T,
E,(n) 2 E,
T,(n)
I
(4.17) (4.18)
T,
i.e. both the excitation temperature T* and the radiant exceed the value of the kinetic temperature T. 2. Production
(4.16)
- 1).
temperature
T,(n)
can never
of radiant entropy
The production of radiant entropy is given by the divergence of the flux of radiant entropy (cf. Appendix A). The local production of radiant entropy of frequency x is therefore (cf. (A.13)) div(Av,E,/k)
= s&=(n) dQ = x$,
(4.19)
where #, = +zs(s* -j&W,(n) x =
n142&,(T)(1
dQi4r
(4.20)
-ewr*>,
(4.21)
Radiative
and the total local production
transfer and irreversibility
15
of radiant entropy is given by (cf. (A.14)) div(E/k)
(4.22)
= Grad = x#
where (4.23) In the framework of the model considered, the following theorem holds: The local production of radiant entropy, integrated over the whole spectral line, is always positive, (4.24)
* 20.
In order to prove this theorem, one inserts the source function (cf. (4.4) and (4.5)) s* = (J‘J’j,(n’)d,dydn’/4n)/(l
+E)+E/(~ +E)
(4.25)
in (4.23) and writes the expression thus obtained in the form # = #‘O’/(l +E) +#“‘/(1
+E)
of a purely scattering where I+P) corresponds to the entropy production and zjCrn)to that of a collisionally dominated (E = co) gas. Explicitly one has
#Cm1= jj(l
-j&0)5,(n)+,
dx dV4n.
Eq. (4.24) is proved if it can be shown that both #(‘) and #(O”)are positive. The validity of the inequality $GQ 2 0
(4.26) (C = 0)
(4.28)
(4.29)
can immediately be seen from (4.14). Moreover, since the integrand of (4.28) is positive for every x, it follows that the radiant entropy locally produced by inelastic collisions is positive for every frequency. In order to prove the inequality $P) r 0, (4.30) one multiplies the second term of the right-hand side of (4.27) by jJ+, dy dCX/4v = 1 so that one can write
=
sss.i
=
liz(n)t,(n’) SSlS
[_@>E,(n’)-~l(n’>~z(n’)l~z+y dx dy dQ dQ’/(W2
--.AW4,(n’)WAv dx dy dQ dQ’/(W2
(4.3 1)
16
J. OXENIUS
where the last three integrals follow from the first by a simple exchange of the integration variables. Addition of the four integrals in (4.31) now yields
+ (jy(n) --_i,(n’)Ns(n’) -tyb))lMy
dx dy dQ dQ’l(4nY.
(4.32)
As can be seen from (4.16), both terms of the integrand are positive: (j, -j,)([, - tl) B 0. The inequality (4.30) and hence the theorem (4.24) are thus proved. An analogous theorem does not hold for the monochromatic entropy production #,, i.e. JI, can be negative. Since the radiant entropy produced by collisions is positive for every frequency, a negative local entropy production can only be due to non-coherent scattering. Physically, a negative & simply means that more photons of frequency x are scattered into other frequency ranges than are created by inelastic collisions and by scattering from other frequencies. However, the production of monochromatic radiant entropy, integrated over the whole volume of the gas, is positive for every frequency: s x$, d V 2 0.
(4.33)
By means of Gauss’ theorem, one can write Jx+,d V = Jdiv(Av,E,/k) =
=
$ $
n . (Av,E,/k)
dV dA
dA h,(n’)n .n’d!CI’ i
(4.34)
where (A.9) has been used and the integration dA is over the surface of the atmosphere whose outer normal is n. Since, by assumption, no radiation is incident on the surface of the gas from outside, one has*
Un’)
LO
if
n.n’>O
= 0
if
n.n’-cO
1
because h,(n) is never negative and vanishes only when the specific intensity vanishes (cf. (4.10) and (4.11)). From (4.34) and (4.35) follows (4.33) at once. 3. Total entropy production
In order to maintain a steady state, the radiant energy loss of the gas has to be compensated by a steady energy flux from some energy releasing reaction to the electrons (cf. Fig. 1). Quantitatively, the energy per unit volume and unit time hv,(n,!&, -nzC&) * For simplicity, an everywhere concave surface is assumed. The results do not depend on this assumption.
Radiative transfer and irreversibility
hv, ( n, a,,-
Electrons T
t
FIG. 1. Schematic
n2 a,,
1
Reaction T’
I
energy flow diagram for the model treated. The values indicated energies per unit volume and unit time.
has to flow from the reaction to the electron gas, giving rise to a production per unit time and unit volume S’jk V = 6’ = -(hv,,/kT’)(nlL2,,
are
of entropy (4.36)
-n2i22,)
if the reaction takes place at the temperature T’. Such energy flux is possible only when the reaction temperature is higher than the electron temperature: T’ 2 T
(4.37)
hvJkT’ s 6’ 5 6.
(4.38)
or Using (2.14), (2.17), (4.15), and (4.21), the entropy production algebra the form c? = -xc(l
(4.36) takes after some (4.39)
-s*)f’.
The total local entropy production is the sum of the production and the entropy production due to the energy flux just mentioned,
of radiant entropy
. . Qtot = %$,+K
(4.40)
since the entropy of the electron gas and the system of the excitation degrees of freedom, respectively, production vanishes, . Qe1
. =
The Second Law in its local formulation
Qexo
=
(4.41)
0.
now requires thaP5)
. dtot 2 0.
By means of (4.22), (4.26), and (4.40), the total entropy production
(4.42) can be written
itot= “[&O)“g,p”) -(l -s’)E’)]
(4.43)
18
J.
OXENIUS
where #(O) and #cm) are given by (4.27) and (4.28), respectively. Since I/(O) 3 0 has already been shown (cf. (4.30)), the inequality (4.42) will be proved if A = #‘“‘/(l can be shown.
Using
the inequality A=$
+E) -(l
t,(n)
-s*)[’
(4.44)
3 E 3 6’ (cf. (4.18) and (4.38)),
(1 -j,(n))E,(n)+,
Ess
2 0
dxg
-(I
one has
-s*>Y
1 (1 -j,(n))E’+,
2-
dxz
-(l
-s*)E’
l+E = 5’[(1 -si;$z because
of (4.4). The relations
dx)/(l
+E> -(l
-$*>I
(4.45)
= 0
(4.44) and hence (4.42) are therefore
proved.
4. Isotropy and anisotropy An arbitrary field j;, of equal
radiation fieldj,(n) spectral distribution
will now be compared with the isotropic radiation of energy, i.e. j,(n) is connected with J5 through
s
j,(n)
for every frequency are equal, too:
x so that according
dQ/4rr
= 3, = j^,
(4.46)
to (4.4) the two corresponding
source functions
s* = s”’
(4.47)
The following theorems hold for every frequency x: The mean local entropy density of an anisotropic radiation of the isotropic radiation field of equal energy density: J‘
field is smaller than that
o,(n) dQ2/4rr = 0, 5 ox.
The local entropy production of an anisotropic the corresponding isotropic radiation field :
radiation
(4.48) field is greater
than that of
(4.49)
VL 2 8,. The mean inverse temperature of an anisotropic of the corresponding isotropic radiation field: J These theorems
will be proved
Ss(n) dQ/4rr in Appendix
= i,
radiation
2 6,.
field is greater
than
that
(4.50)
C.
V. EXAMPLE
As an illustration, some numerical values of entropies, temperatures, and the production of radiant entropy will be calculated in this section. First, approximate formulas will
Radiative transfer and irreversibility
19
be derived for the limiting case of low temperature that almost always obtains in problems of line radiation. They will then be applied to an example of an atmosphere that emits a spectral line of large optical thickness under extreme non-equilibrium conditions. The radiation field used for the numerical calculations will be taken from the literature. 1. The limiting case of low temperature
Line radiation of atoms usually takes place at low temperatures, i.e. at temperatures that are small compared with the excitation energy of the spectral line, because at higher temperatures ionization is prevailing. The limiting case of low temperature means physically that the induced emissions are negligible compared with the spontaneous ones. Quantitatively, it is defined by erzco B 1,
et* 3 1,
erB 1,
(5.1)
where the last two inequalities are consequences of the first because of (4.17) and (4.18). Before the corresponding approximation can be carried out, all quantities have to be expressed by the temperatures. By means of (3.16) and (3.21) the densities of the non-excited and the excited atoms are given by rz2 = n,g,e-r*/(g, +gzemr*). nl = nogAg, +gze-r*), (5.2) Inserting these expressions in (4.8) leads to an entropy density of the excitation degrees of freedom %xc = no[gz%eer*/(g, +gzeer*) + ln(g, +gzeer*)l. (5.3) Using (4.16), the quantity p=(n) (cf. (4.11)) that determines the behavior of the radiant entropy takes the form p2(n) = .!J,(n)erJn)/(eTJn) - 1) - ln(e’J”) - 1).
(5.4)
Finally, the source function s* and the specific intensity j,(n) have already been written as functions of the temperatures in (4.15) and (4.16), respectively. The state of thermodynamic equilibrium is characterized by the equality of all temperatures: [ = [* = [Jn). The expressions for thermodynamic equilibrium can thus be obtained from the general expressions by replacing there all temperatures by the kinetic temperature. In particular, the source function (4.15) and the intensity (4.16) are then Planck functions, s* = j,(n) = 1, so that according to (4.20) the entropy production vanishes for all frequencies: #, = 0. The transition to the limiting case of low temperature, i.e. the application of the inequalities (5. l), now leads to the following approximate formulas: n1 =
no,
n2 =
~ok2/gl)e-t*~
%xc = no(g2/g1)5*e-z* am
= e-rJo(l se = eT-r* j,(n)
+no lng,,
+4,(n)), 9
= ee-rAn).
(5.5) (5.6) (5.7) (54 (5.9)
80
J. OXENIUS
One can now rewrite the occupation numbers, entropies, and temperatures in terms of the source function and the specific intensity because these latter quantities are usually calculated. It is useful to divide at the same time all quantities by their values in thermodynamic equilibrium. In this way, one gets for the density of excited particles n,jnih = s*,
(5.10)
i.e. the spatial distribution of excited atoms is directly given by the source function. Further, one obtains for the temperatures [*/t = 1 -(l/Q tZ(n)/E = 1 -(l/4)
Ins*,
(5.11)
lnjZ(n),
(5.12)
for the variable part of the excitation entropy (%XC--no
lngd/(~~%
-no Ing,)
= s*[l -(l/Q
Ins*],
(5.13)
In_idn)l
(5.14)
and for the radiant entropy s,(n)/3””
= pz(n)/pth
= j,(n)U
-(l/(1
+5))
where eth = 02~147~. 2. Numerical example The actual behavior of the entropies and temperatures will be illustrated by an example of an atmosphere of large optical thickness whose radiation field deviates strongly from that of thermal equilibrium. The numerical values of the source function and of the intensity have been taken from a paper by AVRETTand HUMMER~~). The atmosphere considered has the following characteristics: The line broadening is entirely due to the thermal Doppler effect; the normalized absorption profile is hence +I = ,-1/2e-~2. The atmosphere is homogeneous, plane parallel and of a total optical thickness 0 = lo4 where the optical depth 6, measured perpendicular to the radiating surface, is defined by the optical depth B. in the line center x = 0 through 0 = 6,/g, = &280. It is dominated by radiation processes, E = lOme, and a kinetic temperature 5 = 5 is assumed so that the formulas for the limiting case of low temperature apply. Figure 2 shows the source function s* and hence the density of excited atoms n2, the variable part of the excitation entropy o,,, - no In g,, and the excitation temperature T* as functions of the optical depth 0. The excitation entropy shows a similar behavior as the source function, and the excitation temperature is nearly constant and not essentially different from the kinetic temperature. The monotonic increase of the source function from the boundary (0 = 0) to the center (0 = 5 x 103) of the atmosphere is simply due to the fact that the radiation field is strongest at the center so that the absorption rate is higher there than near the surface. In Fig. 3, the mean intensity j;, the mean radiant entropy GZrand the mean radiant temperature TZ are plotted as functions of the frequency x for the optical depth B = 10. Strictly speaking, instead of ZZand T,, Orand pZ corresponding to the isotropic radiation field of equal spectral distribution of energy have been plotted, but the differences are for the example considered not very great (cf. Appendix D). The radiant entropy shows a similar behavior as the intensity whereas the radiant temperature is near the line center approximately constant and of the same order of magnitude as the kinetic temperature.
Radiative
transfer and irreversibility
81
The self-reversed core of the mean intensity is a typical feature of isothermal gases emitting a spectral line of large optical thickness under non-equilibrium conditions.(6~8*g*10*11’ The self-reversal steadily increases from the center to the surface of the atmosphere and can easily be understood from the monotonic increase of the source function from the boundary towards the interior (cf. Fig. 2).
FIG. 2. The source function s*, the variable part of the excitation entropy (I~=,,--no In gI, and the excitation temperature T*, all referred to their respective values in thermodynamic equilibrium, as functions of the optical depth 0. All quantities are symmetrical about the center of the atmosphere 0 = 5 x 103.
82
J. OXENIUS
lo-44 0
2
3
FIG. 3. The source function_ s*, the mean intensity ~1, the mean radiant entropy O,, and the mean radiant temperature T,, all referred to their respective values in thermodynamic equilibrium, as-functions of the frequency x at the optical depth 0 = 10. The plotted values of Ozand T, correspond to an isotropic radiation field. All quantities are symmetrical about the line center x = 0.
Radiative transfer and irreversibility
83
-0,25
I -o,sxIo-5 FIG. 4. The quantity 4, that is proportional to the local production of monochromatic radiant entropy, as a function of the frequency x at the optical depth 8 = 10. The plotted values of $. correspond to an isotropic radiation field. I/$ is symmetrical about the line center x = 0.
Finally, Fig. 4 shows for the optical depth 0 = 10 the quantity I,G~ which is proportional to the production of monochromatic radiant entropy. The radiation field has again been considered isotropic so that the plotted curve is, according to (4.49) a lower bound for the actual production of monochromatic radiant entropy; an estimate of the error thus introduced is given in Appendix D where it is also shown that negative I+$really occur within the framework of the model adopted in this paper. The order of magnitude of the total production of radiant entropy # can be estimated in the following way, assuming again isotropy. From (4.23) follows 1/,= s*/L+,
dx -j?&,$z
dx N”(5, >(s* -j._L+z dx)
(5.15)
if one takes into consideration that because of the exponential behavior of & = nM1j2e-5a only frequencies in the neighborhood of the line center x = 0 contribute to the integrals so that the temperature t,, which varies slowly there, can be replaced by an appropriate average value (&). On the other hand, eq. (4.4) yields
s* -
s
j-&, dx = ~(1 -s*)
(5.16)
J. OXEMUS
and hence for s* < 1 s*-
i
j;&dx
% E.
(5.17)
From (5.15) and (5.17) one gets (5.18)
* =cG,>. In the present example is 5 = 5 and E = 10A6; from Fig. 3 follows (&) = f(T/T,) so that 4 z 1.2 x 10e5 obtains. VI. CONCLUDING
M 12
REMARKS
The investigations of this paper show that the notions of entropy and temperature, originally introduced for describing thermodynamic equilibrium, can be supplied successfully to extreme non-equilibrium situations. Whereas the notion of a non-equilibrium entropy, due to Boltzmann and Planck, is fairly well known, at least in the form of the H-function which applies to the kinetic degrees of freedom, the notion of a non-equilibrium temperature, due to Planck, is practically not used in the literature. Non-equilibrium temperatures are, at best, introduced in a purely formal way, thus hiding their thermodynamic significance. For instance, if the excitation temperature is formally “defined” by writing the source function as the corresponding Planck function, it is overlooked that the excitation temperature follows in a unique way from Planck’s general definition such that the equation S* = B,,(T*) is a result and not a formal definition of T*; likewise, the relation I,(n) = B,,(T,(n)) is a result and not a formal definition of the radiant temperature T,(n) by means of the specific intensity I,(n). It is interesting to note that these two relations follow from two different forms of non-equilibrium entropies, namely from that of a Boltzmann and a Bose gas, respectively. Non-equilibrium temperatures may serve to facilitate the calculation of the entropy production. So is the transfer equation of radiant entropy most easily derived by making use of the notion of the temperature of a monochromatic light pencil. Non-equilibrium thermodynamics has had no need of generalized temperatures until now because it has dealt only with small deviations from thermodynamic equilibrium in which case one single temperature is locally defined. APPENDIX
A
At every point of the atmosphere there are uniquely given fluxes of radiant energy and radiant entropy, respectively. The amount of radiant energy of frequency v that flows at a given point in the direction n is given by lJ,(n’)n
and is hence the projection
. n’ da’
= n . sI,,(nr)n’ da’
of the vector F, = /d,,(n)
da
(A.1)
on this direction, i.e. F, is the flux of radiant energy of frequency v. The total flux of radiant energy is thus F = j-F’ dv.
(A-2)
Radiative transfer and irreversibility
85
For the divergence one gets div F, = div JnZJn) da = jn . grad Z,(n) dQ
(A-3)
or, by means of the transfer equation (2.8) n . grad Z,(n) = dZ,,(n)/ds = k0(+,/4rr)(S* -Z,(n))
(A-4)
where (cf. (2.9)) (A-5) div F, = &&,(S* -TV).
(A-6)
The divergence of the total flux of radiant energy is then div F = k,( S* -p&
dv)
(A-7)
which can be written div F = [Z&+/(1+E)][&,( T) -s&
dv]
(A.8)
taking (2.15) into account. In complete analogy, the flux of radiant entropy of frequency v is E, = j&(n)
dSZ
(A-9)
and the total flux of radiant entropy E =
s
E,dv.
(A. 10)
For the divergence one gets again div E, =
i
n . grad L,(n) da,
(A.ll)
or, using the transfer equation (3.35) n . grad L,(n) = dl,(n)/ds
= M+,/4nT,(n))(S*
-Z,,(n)),
div E, = k,+,/( S* - Z,,(n))(T,,(n))-l dS2/4rr
(A.12) (A.13)
and hence div E = k,//(S*
- Z,(n))(+,/T,(n)) dv dS2/47.
APPENDIX
(A.14)
B
In the framework of the model considered, the Planck function corresponding to the kinetic temperature T is an upper limit for both the source function and the specific intensity : S* I &o(T), UN
5 &o(T).
(B-1) (B-2)
J.
86
OXENIUS
Writing for the sake of brevity Z(s) = I&I, s) and the coordinate along the considered ray, the formal
K(S)
=
KJS)
solution
where s (2 0) denotes of the transfer equation
(2.8) dl(s)/ds
=
K(S)(
s*(S)
(B.3)
-I(s))
is given by I(s) = jS*(s’)
jK(s”) ds”)!c(.F’)
exp ( -
0
(B-4)
S,
where the boundary condition has been taken into account on the surface of the atmosphere (s = 0) from outside, variables s’ s y =
ds’
y’ =
ds”,
J-K@“)
that no radiation is impinging i.e. I(0) = 0.” With the new
ds”
IK(S”) 0
0
one gets Z(y)
= /S*(y’)ey
-Ydy’
0
Y
5
Y
hoax eylYdy’
1
= S&,
eetdt
s 0
0
s =
m
5
eetdt
%3x
S$,,,
0
i.e. the specific intensity inside the atmosnhere :
From
can never
exceed
(B.5) one gets for the source function
the maximum
in the left-hand
side the maximum SZ,,
and hence a fortiori relation from (B.5) and (B.6). *For simplicity, assumption.
(B.l).
an everywhere
of the source
SL,
+ cB,,( T) l+E
source function
’ S*max’ (B.6)
5 B,,(T)
On the other hand, the inequality
concave
function
(2.15) the inequality
s &L dv + E&S T) S*=P
value
surface is assumed.
(B.2) follows at once
The results do not depend
on this
Radiative
transfer and irreversibility
APPENDIX
87
C
In order to prove the inequality (cf. (4.48))
(C-1) under the assumption
(C.2) one has because of (4.9) to show that fi -
s
/.+I) dQ/4rr r 0
(C.3)
where p(n) = (1 + ~j(n)) In (1 + aj(n)) - orj(n) In aj(n),
(C.4)
r; = (l+ctj)In(l+aj)-~jlnorj,
(C.5)
cc = l/(er-1)
(C.6)
> 0.
One can write
(C.7) where A = ln(l+aj)-/ln(l+aj(n))dQ/4a,
(C-8)
B =jh(l+-i)-S/(n)In(l+&)z,
cc.9
so that (C.3) is proved if A > 0 and B 2 0 can be shown. To prove A > 0, one observes that for a non-negative functionflt)
holds where the bar denotes the average over a given t-interval; this follows from the fact that the arithmetical mean of non-negative quantities is always greater than their geometrical mean. Hence A = In [s(l +Mj(n)) dQ/4rr] -1 In (1 +aj(n)) dfi2/4rr = ln(l+orj(n))-ln(l+xj(n))
2 0.
(C. 10)
To prove B 2 0,one introduces a function T(n) through with
j(n) = j< 1 + T(n))
(C.11)
T(n) dQ/4rr = 0
(C.12)
I
88
J. OXENIUS
because of (C.2), and
T(n) 2
B =
-
1
because ofj(n) > 0. One can thus write
J_K#[ ln (1+-t) -l++-&j]$
s
=j
(1+5-(n)) In l+
with
q=
(C.13)
cxj> 0.
One now defines two non-negative functions T+(n)
=
(C.14) and
T’(n)
through
T-(n)
44
if
T(n)
0
if
7(n) < 0,
>
0
(C.15)
(C.16) with (C.17) because of (C.l2), and form
T-(n)
<
B =j
1 because of
(l+T+(n))h
[S +
s
T(n)
z
-
1.
Equation (C.13) now takes the
l+ T-(n)
l-l
+q_qT_(n)
1
-.1 d!A 4~
(C.18)
From In (1 +x) > x/(1 +x), valid for x b - 1, follows T+(n)
T+(n) 1 +q and from In (1 -x)
> - x/( 1 -x),
+qT+(n)
’
(1
+q)(l
(C.19)
+T’(n))
’
valid for x 6 1,
7-(n) 1 -t-q -47-(n)
7-(n)
(C.20)
--. ’ - (1 +q)(l -T-(n))
From (C.18), (C.19), and (C.20), one finally gets
B 2j
[S
(1 +T+W)(,
= [j/(1 +q)][JT+(n)
s
+q;;r;+(,,,; - (1-7-W) dn/4?r -ST-(n)
dQ/4a]
T-(n) (1 +q)(l
= 0.
using (C.17). This completes the proof of (C.3) and hence of (4.48).
-T-(n))
d!J 4?~
1 (C.21)
Radiative transfer and irreversibility
89
Because of (4.20) one has (A -6,)/A
= j%* -j,(n))&(n)
dQ/4n -(i* -&)iz
= s*(/&(n) dQ/4r -$A +(jz$, -j~,(n)E,(n)
dQ/4n)
(C.22)
taking (4.47) into account. The inequality (4.49) (C.23) is thus proved if jt -jj(n)S(n)
dLI2/4rr > 0
(C.24)
and 2 0
(C.25)
E(n) = ln (1 + l/c&t)),
(C.26)
s
t(n) dSI/4rr-g
can be shown where (cf. (4.16))
8 = ln(l+l/orj).
(C.27)
Relation (C.24) has already been proved since (cf. (C.9) and (C.21))
jt -jj(n)f(n)
dS2/4n = B 2 0.
To prove (C.25), one writes by means of (C.ll),
(C.28)
(C.14), (C.15), and (C.16)
+
(C.29) R+
where R",R-, and RC denote the ranges in which T(n) = 0, ~0, and >O, respectively. Because of In ( In (
l+ q(l_:-(n))) l+
4(1+:+(n)))
= ln(l+j
+ ln(l+(l+q;;(O:-(n)))~
=ln(l+3+ln(l-(1+~~1~~+(n)~~
(c.30) (c’31)
one has from (C.29)
(C.32)
J. OXENIUS
90
since{
= ln(l+l/q)and In
and from In (1 -x) In
(
' = 4x. From In(1 +x) > x/(1 +x) follows
R"+R-+R
r-(n)
7-(n)
l+
’ 1 +q-p-(n)
(1 +q)(l -7-(n))
T-(n) >- 1+q’
(C.33)
2 -x/( 1 -x) T+(n)
T+(Q) 2 -p. (1 +q)(l +T+(n)) 1 ’ -1 +q +47+(n) lfq
l(
T+(n)
(C.34)
By means of (C.33) and (C.34), one now obtains from (C.32) SE(n) ds2/4n -i
> (1 +q)-l[/T-(n)
ds2/4n -jT+(n)
dQ/4r]
= 0
(C.35)
taking (C.17) into account. This proves (C.25) and hence (4.50) and completes the proof of (C.23) and hence of (4.49). APPENDIX
D
The numerical values of the example treated in Section V.2 have been obtained by replacing everywhere the actual radiation field by an isotropic one. The errors introduced hereby will now be estimated in a rough way. For this purpose, one compares the semi-isotropic radiation field A(n) =
j,(l +S,) -
(
j,(l
-8,)
if
n.n,>O
if
n.n,>O
(D.1)
where no denotes the outer normal of the plane parallel atmosphere and IS21 < 1, with the isotropic radiation field of equal spectral energy density 3, = j,.
(D.2)
Using the formulas for the limiting case of low temperature (cf. Section V.l), one gets after some algebra for the relative errors of the mean radiant entropy, the mean radiant temperature, and the monochromatic entropy production Ox p=
-5s
(l+S,)ln(l+S,)-(I-S,)ln(l-S,) ,
2(1 + i,)
Ox
ln(1+S,)ln(1-S,)-~~zln(1-S2,)
i;-?;
--;_
^=ln(l+S,)ln(l-SJ-l,ln(l-Sz)+Iz’ TX
baa! 4,
1 =Fg
-25
[c--j;,
1+s, ln-----l-S,
where 6, = .$-lnjl.
1
ln(l -SYj) ,
(D.3)
(D.4)
(D-5)
(D-6)
Radiative transfer and irreversibility
91
The greatest errors occur at the boundary of the atmosphere where 6, = 1 since no radiation is incident from outside. In the example considered, the relative error of the mean radiant entropy turns out to be (6 per cent whereas that of the mean radiant temperature is about 50 per cent, i.e. TX z SF,. The errors for the optical depth 0 = 10 are correspondingly smaller. The error for the monochromatic entropy production at the optical depth 8 = 10 is much harder to estimate since the degree of anisotropy is expected to increase rapidly from the line center to the line wings. For the arbitrarily chosen values 6, = 4, +, 2, one finds relative errors of 9, 37, 95 per cent for the line center x = 0, and of - 3, - 11, - 30 per cent for the frequency x = 2, respectively. The actual I,&will therefore, at least in an interval 0 < x < 2, not too strongly deviate from the &corresponding to the isotropic case. On the other hand, the relative errors between #, and #, can be made as small as one pleases by choosing a sufficiently low temperature ([ B 1) as can be seen from (D.5) and (D.6). For the radiation field depends, apart from the total optical thickness, only on the parameter E which can be kept constant by choosing for any given temperature an appropriate electron density. This shows that negative #, really occur within the framework of the model adopted in this paper. Acknowledgement-The
author takes pleasure in acknowledging
discussions
with Dr. G. ECKER.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
9. 10. 11.
12. 13. 14. 15.
M. PLANCK, The Theory ofHeat Radiation. Dover, New York (1959). R. WILDT, Astrophys. J. 123, 107 (1956). R. N. THOMAS.Astroohvs. J. 125. 260 (1957). J. OXENIUSJQiRT 5; 7?1. ’ . ’ J. T. JEFFERIES and 0. R. WHITE, Astrophys. J. 132, 767 (1960). A. G. HEARN (a) Proc. Phys. Sot. 81, 648 (1963); (b) Proc. Phys. Sot. 84, 11 (1964). M. DOBROWOLNYand F. ENGELMANN,Nuovo Cim. 31, 965 (1965). J. T. JEFFEFUES and R. N. THOMAS,Astrophys. J. 127, 667 (1958); Astrophys. J. 129, 401 (1959); Astrophys. J. 131, 695 (1960). V. V. IVANOV, Sov. Astron. 6, 793 (1963); Sov. Astron. 7, 199 (1963). S. CUPERMAN,F. ENGELMANN,and J. OXENIUS,Phys. Fluids 6,108 (1963); Phys. Fluids. 7,428 (1964). E. H. AVRETTand D. G. HUMMER,Mon. Not. R. Astr. Sot. (in press). L. D. LANDAuand,E. M. LIFSHITZ,Statistical Physics, p. 116-l 18 and p. 155. PergamonPress, London, (1958). P. ROSEN,Phys. Rev. 96, 555 (1954). A. ORE, Phys. Rev. 98, 887 (1955.) S. R. DE GROOTand P. MAZUR, Non-equilibrium Thermodynamics, p. 22 North-Holland, Amsterdam (1962).