### Title

### Relativistic statistical thermodynamics of dense photon gas

### Author(s)

### Tsintsadze, LN; Kishimoto, Y; Callebaut, DK; Tsintsadze, NL

### Citation

### PHYSICAL REVIEW E (2007), 76(1)

### Issue Date

### 2007-07

### URL

### http://hdl.handle.net/2433/50247

### Right

### Copyright 2007 American Physical Society

### Type

### Journal Article

### Textversion

### publisher

**Relativistic statistical thermodynamics of dense photon gas**

Levan N. Tsintsadze*and Yasuaki Kishimoto

*Department of Fundamental Energy, Graduate School of Energy Science, Kyoto University, Japan*

Dirk K. Callebaut

*Physics Department, University of Antwerp, Antwerpen, Belgium*

Nodar L. Tsintsadze*

*Department of Plasma Physics, Tbilisi State University, Tbilisi, Georgia*

共Received 1 February 2007; published 30 July 2007兲

We discuss some aspects of interactions of high-frequency electromagnetic waves with plasmas, assuming that the intensity of radiation is sufficiently large, so that the photon-photon interaction is more likely than the photon-plasma particle interaction. In the stationary limit, solving the kinetic equation of the photon gas, we derive a distribution function. With this distribution function at hand, we investigate the adiabatic photon self-capture and obtain the number density of the trapped photons. We employ the distribution function to calculate the thermodynamic quantities for the photon gas. Having expressions of the entropy and the pressure of the photon gas, we define the heat capacities and exhibit the existence of the ratio of the specific heats⌫, which equals 7 / 6 for nonrelativistic temperatures. In addition, we disclose the magnitude of the mean square fluctuation of the number of photons. Finally, we discuss the uniform expansion of the photon gas.

DOI:10.1103/PhysRevE.76.016406 PACS number共s兲: 52.27.Ny, 52.38.⫺r, 52.40.Db

**I. INTRODUCTION**

The recent development of astronomical observations has revealed that our universe is full of enigmatic explosive phe-nomena, such as jets, bursts, and flares. It is possible now to study extremely complex phenomena共supernova explosion, Gamma-ray bursts, etc.兲 of astrophysics in laboratories using intense and ultraintense lasers. Intense lasers have been used to investigate hydrodynamics, radiation flow, opacities, etc., related to supernova explosions, giant planets, and other as-trophysical systems 关1兴. Thus the study of the properties of

such radiation 共strong and superstrong laser pulse, non-thermal equilibrium cosmic field radiation, etc.兲 is of vital importance. The development of compact, high-power, short pulse, efficient lasers is a fast moving technology. In the field of superstrong femtosecond pulses, it is expected that the character of the nonlinear response of the medium will radi-cally change. Currently, lasers produce pulses whose inten-sity approaches 1022W / cm2 关2兴. With a further increase of

intensity 关3兴, we may encounter novel physical processes,

where the quantum electrodynamic description may be needed. Recently, the nonlinear collective effects in quantum electrodynamics has been reviewed in Ref.关4兴.

We have shown in Ref. 关5兴, and later Medvedev in 关6兴,

where thermodynamic properties of a photon gas in electron-positron plasmas were studied 共results of 关6兴 were

recalcu-lated recently in 关7兴兲, that the behavior of photons in a

plasma is radically different from that in a vacuum. Namely, plasma particles perform oscillatory motion in the field of electromagnetic 共EM兲 waves affecting the radiation field. The oscillation of electrons in an isotropic homogeneous

plasma leads to the index of refraction, which depends on the frequency of the radiation, and is not close to unity for a dense plasma, i.e.,

*R*2=*k*
2* _{c}*2
2 = 1 −

*p*2 2, 共1兲 where

*p*=

### 共

4*e*2

_{n}*m0e*␥

### 兲

1/2for an electron-ion plasma 共neglecting
the ion contribution兲 and *p*=

### 共

24*e*2_{n}

*m0e*␥

### 兲

1/2

for an
electron-positron plasma *共−e, m _{0e}, n, and* ␥ are electron charge, the
rest mass, density, and the relativistic gamma factor of the
electrons, respectively兲.

Rewriting Eq.共1兲 in terms of an energy =ប and
*mo-mentum p =បk 共where ប is the Planck constant divided by*
2*兲 and introducing m*_{␥}=ប*p/ c*2, we obtain the expression

for the energy of a single photon
␥*= c共p*␥2*+ m*␥2*c*2兲1/2*= m*␥*c*2

### 冉

1 −*u*_{␥}2
*c*2

### 冊

−1/2

, 共2兲

which is expressed through the standard formula for the ve-locity of energy transport

*u*_{␥}*= c*

### 冉

1 −*p*2 2

### 冊

1/2 = *k*. 共3兲

For the momentum of a photon we can write

*p*ជ_{␥}=*បk*ជ*= m*_{␥}␥_{␥}*u*ជ_{␥}*= m*_{␥}

### 冉

1 −*u*␥

2

*c*2

### 冊

−1/2

*u*ជ_{␥}. 共4兲
The form of Eq. 共2兲 coincides with the expression for the

total relativistic energy of massive particles, so that a rest
*mass m*_{␥} is associated with the photon in a plasma关5,6,8兴.

We note here that two important features of photons follow
from Eq. 共2*兲. Namely, first at p*_{␥}= 0, _{␥}*= m*_{␥}*c*2 _{is not zero.}

Second, the rest mass of photons depends on the plasma *Also at Department of Plasma Physics, Institute of Physics,

*density, or the volume as m*_{␥}=* _{c}*ប2

### 共

4*e*2

*m0e*␥

*Ne*

*V*

### 兲

1/2 . In view of this analogy between a photon in a plasma and a free material particle, we can treat the photon gas in the plasma just as a subsystem of particles that have nonzero rest mass.In the approximation of geometric optics we may work in
terms of rays共photons兲 instead of waves. In a homogeneous
isotropic medium the direction of the ray coincides with that
of the normal of the wave surface. In practice, however, we
often have to deal with pulses formed by a group of waves.
For the wave packet, i.e., a quasimonochromatic group of
waves, the Fourier component of wave energy is by
defini-tion very “sharp” and appreciably different from zero only in
a narrow range of frequencies and wave vectors near the
carrier*and k of the pulse. In plasmas, as follows from Eq.*
共1*兲, the group velocity u=*_{k}⬍c. Thus the wave packets of

light are propagated with a group velocity which is less than the speed of light, in accordance with the theory of relativity 关9兴. It is also well-known that the introduction of group

ve-locity is valid in the case of weak field共for the linear waves兲. However, for strong nonlinear waves in plasmas a concept of group velocity is meaningless. In this case we should define the mean velocity of the group of photons taking into ac-count their interaction with plasma particles. We note that the wide range of applicability of the approximation of geo-metrical optics is due to the fact that the properties of plasma usually vary slowly in space and time, i.e., the properties of the medium change very little over distances of the order of the wavelength共or of some characteristic length兲.

Let us recall some purely quantum mechanical features of
a macroscopic system. It is well-known that there is an
ex-tremely high density of levels in the energy eigenvalue
spec-trum of a macroscopic system. We know also that the
num-ber of levels in a given finite range of the energy spectrum of
a macroscopic system increases exponentially with the
*num-ber of particles N in the system, and separations between*
levels are given by numbers of the 10*−N*_{. Therefore we can}

conclude that in such a case the spectrum is almost continu-ous and a quasiclassical approximation is applicable. To sup-port this statement, we will discuss some conditions which will allow us to use a quasiclassical approximation. We start from the uncertainty principle in the relativistic case关10兴 for

photons. In the relativistic theory a coordinate uncertainty in a frame of reference in which the particle is moving with energy is

*⌬q ⬃c*ប
=

*c*

. 共5兲

Estimating this quantity for the isotropic plasma, we obtain
for the underdense plasma, *䉭q⬃ 共 is the wavelength兲,*
and for the overdense plasma *䉭q⬃c/**p*. This means that

the coordinates of the photon are meaningful only in those cases where the characteristic dimensions of the problem are large in comparison with the wavelength or the anomalous skin depth.

We now consider the quantization of an EM field. In the
quantum field theory, the Hamiltonian has the same form as
*in classical field theory, the only difference is that now E and*

*B are operators, i.e.,*

*H*ជ*f*=

1 8

### 冕

*共E*

2* _{+ B}*2

_{兲dr}_{ជ}

_{共6兲}

and the eigenvalues of this Hamiltonian are

*H =*

### 兺

**k,**

### 冉

*n***k,**+

1

2

### 冊

ប*共k兲,*共7兲

*where the occupation numbers n*

**k,**are integers, andstands

for the polarization.

The eigenvalues of the momentum operator are

*p*ជ=

### 兺

**k,**

### 冉

*n*_{k,}_{}+1

2

### 冊

*បk*ជ. 共8兲 The expressions 共7兲 and 共8兲 enable one to introduce the

concept of photons, i.e., the EM field as an ensemble of
particles each with energyបand momentum*បk*ជ. The
*occu-pation numbers n***k,** now represent the numbers of photons

*with given k*ជand polarization .

The properties of a photon gas are known to be similar to
*the classical properties when the photon numbers n*_{k,}_{} are
large. This statement allows us to define the condition for a
value of an amplitude of the electric field, which indicates
the validity of the classical approach of the photon gas. To
this end, we shall estimate the total field energy per unit
volume, which is proportional to *兩E兩*2_{. In the quasiclassical}

limit, the total number of proper oscillations with the
*mag-nitude of the wave vector in the interval dk is*

*Vk*2*dk*
2 =
*V*2
2* _{c}*3

*R*2

*d*

*d*共

*R兲d*. 共9兲

Noting Eq.共1兲 for an isotropic plasma expression 共9兲 reduces

to

*V*2

2* _{c}*3

*Rd*. 共10兲

For the energy density of the field we have
*兩E兩*2_{=}

### 冕

_{ប}

_{}

_{n}_{共}

_{}

_{兲} 2

_{Rd}_{}2

*3 ⬃*

_{c}*R*ប 2

*3 4*

_{c}

_{n}_{共}

_{}

_{兲.}

_{共11兲}

As we have mentioned above there is a similarity between
*the quantum and the classical system, provided n*_{}Ⰷ1, i.e.,
when

*兩E兩 Ⰷ 共បcR兲*1/2

### 冉

*c*

### 冊

2

. 共12兲

From this it is clear that for the static field, i.e.,= 0,*兩E兩 is*
always classical. The same situation occurs for the overdense
*plasma, as R→0. In general, a high-frequency EM field, if*
sufficiently weak, can never be quasiclassical. Thus the
in-equality共12兲 is the required condition, which allows the EM

field to be treated as quasiclassical.

**II. FIRST LAW OF RELATIVISTIC THERMODYNAMICS**
We now consider a system which is a dilute gas composed
of electrons, ions, and photons*共e−i−*␥兲, or electrons,

*posi-TSINTSADZE et al.* **PHYSICAL REVIEW E 76, 016406**共2007兲

trons, and photons*共e−p−*␥兲, and describe this compressible
and continuous medium in terms of its macroscopic
proper-ties such as entropy, pressure, density, temperature, etc.

First, we calculate the thermodynamical quantities devel-oping the statistical mechanics in the presence of a strong EM field. It was shown in Ref. 关11兴 that in the case of the

relativistically intense 共circularly polarized兲 EM waves
*propagation into a plasma, the momentum e*_{␣}*A*ជ_{⬜}*/ c共A*ជ_{⬜}is the
perpendicular component of the vector potential of the EM
waves,␣stands for the particle species兲 can be much larger
than the perpendicular components of the thermal
momen-tum of the particles. Hence the perpendicular momenmomen-tum of
*particles is just p*ជ_{⬜␣}= −*e*␣* _{c}A*ជ⬜, whereas the momentum of
par-ticles along the propagation of EM waves remains thermal.
In the following, we study a closed system for a period of
time that is long compared with its relaxation time. This
implies that the system is in complete statistical equilibrium.

*Introducing E as the internal energy in a volume V of the*three component gas, the first law of thermodynamics reads

*共index t stands for total兲*

*dEt= dQt− PtdV,* 共13兲

*where Pt*is the total pressure, or

*Pt= Pe+ Pi共p兲+ P*␥. 共14兲

In the case when a plasma is in a superstrong EM field, the
pressure becomes anisotropic. For instance, in the case of a
relativistically intense circularly polarized EM field the total
pressure is written as
*Pt*=

### 兺

␣*共P*⬜␣

*+ P*储␣

*兲 + P*␥, 共15兲 where

*P*

_{⬜␣}=2 3

*n*

_{␣}

*m*

_{0␣}

*c*2

*a*

_{␣}2

### 冑

*1 + a*

_{␣}2

*K*0共␣

### 冑

*1 + a*␣2兲

*K*1共␣

### 冑

*1 + a*␣2兲 , 共16兲

*P*储␣= 1 3

*n*␣

*T*␣. 共17兲

*Here n*

_{␣}=

*n*0␣冑

*1+a*␣ 2

*K*1共␣兲

*K*1共␣

### 冑

*1 + a*␣

2_{兲 is the density of particles,}

*a*_{␣}=* _{m}e*␣

*A*ជ⬜

0␣*c*2,␣=

*m*0␣*c*2

*T*_{␣} *, and K*ᐉ*共X兲 is the McDonald function of*

ᐉ order.

Deriving expressions共16兲 and 共17兲 use was made of the

distribution function
*f*_{␣}*= B*␦

### 冉

*p*ជ

_{⬜␣}+

*e*␣

*cA*ជ⬜

### 冊

exp### 再

−*c*

### 冑

*m*

_{0␣}2

*c*2

*+ p*

_{⬜␣}2

*+ p*

_{储}2

_{␣}

*T*

_{␣}

### 冎

, 共18兲*where B is the normalization constant and*␦

*共x兲 is Dirac’s*function. If we integrate expression共18

*兲 over p*ជ

_{⬜}, we obtain the distribution function, which was derived in关11兴, i.e.,

*f*_{␣}*共p*储␣*,a*␣2兲 =

### 冕

*dp*ជ⬜

*f*␣

*共p*储

*, p*ជ⬜

*,a*␣2兲 =

*n*0␣

*m*

_{0␣}

*c*1

*K*1共␣兲 exp兵−␣

### 冑

*1 + a*␣2

*+ p*2储␣/共m0␣2

*c*2兲其. 共19兲 We note here that distribution functions共18兲 and 共19兲 give a

complete description of the microscopic properties of the gas in the presence of superstrong radiation.

In Eq.共13*兲 the dQt*is the amount of heat that is gained or

lost by the system, which has the form

*dQt= TedSe+ Ti _{共p兲}dSi_{共p兲}+ T*␥

*dS*␥, 共20兲

where

*S*_{␣}*= − V*

### 冕

*dp*储

### 冕

*dp*ជ⬜

*f*␣

*ln f*␣ 共21兲

is the entropy of the particles.

Introducing the entropy per particle and using expression
共19兲, we obtain
*S*_{␣}
*N* = −
1
*n*_{␣}

### 冕

*dp*储

*f*␣

*共p*储␣

*,a*␣ 2

*␣*

_{兲ln f}*共p*储␣

*,a*␣2兲. 共22兲

*After substitution of f*_{␣}*共p*储␣*, a*␣2兲 into Eq. 共22兲, a simple

*inte-gration over p*储gives

*S*_{␣}
*N* = −

### 冋

ln*n*

_{␣}

*m*

_{0␣}

*cK*1共␣兲 + 1 −

_{␣}

### 冑

*1 + a*

_{␣}2

*K*2共␣

### 冑

*1 + a*␣ 2

_{兲}

*K*1共␣

### 冑

*1 + a*␣2兲

### 册

. 共23兲 In order to calculate the pressure and the entropy of the pho-ton gas, we use the Bose distribution function关5兴. The resultis
*P*_{␥}= *T*␥
4_{}
␥
2
2* _{共បc兲}*3

### 兺

ᐉ=1 ⬁*e*ᐉ␥ ᐉ2

*K*2共ᐉ␥兲 共24兲 and

*S*

_{␥}=

*VT*␥ 3

_{}␥ 2 2

*3*

_{共បc兲}### 兺

ᐉ=1 ⬁*e*ᐉ␥ ᐉ2

### 冋

ᐉ␥### 冉

1 − ᐉ␥ 4### 冊

*K*3共ᐉ␥兲 +ᐉ 2

_{}␥ 2 4

*K*1共ᐉ␥兲

### 册

, 共25兲 where_{␥}=

*m*␥

*c*2

*T*

_{␥}= ប

*T*

_{␥}

### 共

4*e*2

*m0e*␥

*Ne*

*V*

### 兲

1/2 .We now suppose that in each subsystem the entropy is
*conserved, i.e., Se, Si共p兲, and S*␥ are constant. We note here

that the relaxation in a photon-plasma system is a two-stage
process. First, the statistical equilibrium is established in
each subsystem independently, at first in a plasma, since
pho-tons usually have much longer mean free paths than charged
particles, and then in a photon gas. Slower processes of the
equalization of the photon and the plasma temperatures will
*take place afterwards. Since for an adiabatic process dS⬅0,*
we obtain the adiabatic equation for material particles from
Eq.共23兲,

*n*_{␣}
*K*1共␣兲
exp兵−␣

### 冑

*1 + a*␣2

*G*其 = const, 共26兲 where

*G =K*2共␣

### 冑

*1 + a*␣ 2

_{兲}

*K*1共␣

### 冑

*1 + a*␣2兲 .

For clarity, we consider three cases. First, for the relativistic
temperatures_{␣}

### 冑

*1 + a*

_{␣}2Ⰶ1, we get

*n*_{␣}
*T*_{␣}

### 冉

1 +*e*_{␣}2*A*_{⬜}2

*T*_{␣}2

### 冊

= const. 共27兲This expression shows that the thermal kinetic energy due to
the thermal motion of particles along the propagation of EM
*waves, T*_{␣}*= T*储␣, dominates the energy of the waves, and the

second term in the bracket in Eq. 共27兲 is less than unity.

Hence we can neglect the second term in the bracket to ob-tain

*VT*_{␣}*= V*⌫−1*T*_{␣}= const,

from which follows the expression for the ratio of the specific heats

⌫ =*CP*

*CV*

= 2.

In the opposite limit, that is for the nonrelativistic
tempera-tures_{␣}

### 冑

*1 + a*

_{␣}2Ⰷ1, we obtain

*n*_{␣}

*T*_{␣}1/2exp兵␣共1 −

### 冑

*1 + a*␣

2_{兲其 = const.} _{共28兲}

Finally, in the case when the temperature is ultrarelativistic,

*T*_{␣}*Ⰷm*_{0␣}*c*2*, and also the radiation, i.e., a*_{␣}2Ⰷ1, then
␣

### 冑

*1 + a*␣2⬇

*e*_{␣}*兩A*_{⬜}兩

*T*_{␣} and can be of the order of unity. The

adia-batic equation now reads

*n*_{␣}
*T*_{␣}1/2exp

### 再

−*e*_{␣}*兩A*_{⬜}兩

*T*_{␣}

### 冎

= const. 共29兲For the subsystem of photons, the asymptotic behavior of
Eq.共25兲 for_{␥}Ⰶ1 leads to

*S*_{␥}*= S*_{0␥}共1 + 0.83_{␥}兲, 共30兲
where the second term is due to the mass of the photon, and

*S*_{0␥}=4_{45}2

### 共

*T*␥

_{បc}### 兲

3*V is the entropy of the photon gas in vacuum.*

For the case_{␥}Ⰷ1 Eq. 共25兲 becomes

*S*_{␥}*= S*_{0␥}0.48_{␥}3/2. 共31兲
In this case the entropy depends on the temperature and the
volume as follows:

*S*_{␥}*⬃ T*_{␥}3/2*V*1/4. 共32兲
Thus, for the adiabatic process, we obtain

*T*_{␥}*V*1/6*= T*_{␥}*V*⌫−1= const. 共33兲
We specifically emphasize that in contrast to the vacuum
case, we can here define the ratio of the specific heats for the

photon gas, and in the case of nonrelativistic temperatures
the ratio of the specific heats for the photon gas is⌫=7_{6}.

As we have indicated in the Introduction, the nature of photons in plasmas is quit different from the one in vacuum. In plasma the photon has a rest mass that depends on the volume, and hence we can write for the mean square fluc-tuation of the number of photons

*具共⌬N*␥兲2典 = −*T*␥*N*␥
3
*V*2

### 冉

*V*

*P*

_{␥}

### 冊

*T*

_{␥}. 共34兲

The derivation of this equation is well-known关12兴. The

limi-tations on its validity were pointed out, and a discussion on the mean square relative fluctuation in number of particles for an ideal relativistic Bose gas was reported by Dunning-Davies关13兴.

We now examine fluctuations in the distribution of
*pho-tons over the various “quantum” states. Let nK* be their

*oc-cupation numbers in the Kth quantum state. The mean values*
*具nK典=n*␥of these numbers are

*n*_{␥}=
1
exp

## 再

*共K兲 −*␥

*T*

_{␥}

## 冎

− 1 . 共35兲Recalling Eq.共34兲, we get

*具共⌬nK*兲2*典 = T*␥

*n*_{␥}

共36兲

or

*具共⌬nK*兲2*典 = n*␥*共1 + n*␥兲. 共37兲

It is important to emphasize that in Eq. 共37兲 the first term

reflects the corpuscular behavior of the photons, whereas the
second term is of wave origin. More precisely, it is the result
of the irregular interference of EM waves. One can see from
Eq.共37*兲 that in the case when 兩共K兲−*_{␥}*兩ⰇT*_{␥}, the first term
is larger than the second one. This implies that photons are
neutral particles. In the opposite case *兩共K兲−*_{␥}*兩ⰆT*_{␥}, i.e.,
for the classical approach of fluctuation of EM waves, Eq.
共37兲 exhibits that the relative fluctuations of the number of

photons does not decrease, when the mean number of pho-tons increases, so that

*具共⌬nK*兲2典

*n*_{␥}2 ⬃ 1.

Thus we may conclude that in the range*兩共K兲−*_{␥}*兩ⰇT*_{␥}, i.e.,

*n*_{␥}Ⰶ1, the radiation resembles the ideal gas of the

particles-photons, and in the range *兩共K兲−*_{␥}*兩ⰆT*_{␥}*, i.e., n*_{␥}Ⰷ1, the
radiation represents the system of classical electromagnetic
waves.

**III. BOLTZMANN H-THEOREM FOR PHOTON GAS**
Recently in Ref.关14兴 a new version of the Pauli equation

for the photon gas was derived from a general kinetic

*equa-TSINTSADZE et al.* **PHYSICAL REVIEW E 76, 016406**共2007兲

tion 共which is of the type of the Wigner-Moyal equation 关15兴兲 for the EM spectral intensity 关16–18兴. In the limit of

the spatial homogeneity for the distribution function the Pauli equation reads

*N共k*ជ*,t兲*
*t* =

### 兺

±### 冕

*d*3

*k*

### ⬘

共2兲3*W*±

*共k*ជ

### ⬘

*,k*ជ兲 ⫻

### 冋

*共k*ជ兲

*共k*ជ

### ⬘

兲*N共k*ជ

### ⬘

*,t兲 − N共k*ជ

*,t兲*

### 册

. 共38兲*Here W*

_{±}

*共k*ជ

### ⬘

*, k*ជ兲 is the scattering rate

*W*±*共k*ជ

### ⬘

*,k*ជ兲 = 4

*p*4兩␦

*共q兲兩*2

*共k*ជ

*± q*ជ/2兲

*共k*ជ兲␦

*共⍀ − q*ជ

*u*ជ±兲, 共39兲

*where k*ជ

### ⬘

*= k*ជ

*+ q*ជ

*, u*ជ

_{±}=

*共kជ±qជ/2兲c*2

*共kជ±qជ/2兲*, *共k*ជ*兲, k*ជ and *⍀, q*ជ are the

fre-quencies, wave vectors of the transverse and longitudinal
photons *共photonikos兲, respectively, and N共k*ជ*, t*兲 is the
distri-bution function of photons.

We now discuss some implications of Eq.共38兲. Namely,

this equation exhibits the irreversible processes, and is the mathematical basis for a H-theorem. The relaxation process is accompanied by an increase in the entropy of the photon gas. Note that the equation type of Eq. 共38兲 has been

ob-tained for the first time by Pauli for a quantum system and applied to study of irreversible processes 关19兴. Later Van

Hove 关20兴, Prigogine 关21兴, and Chester 关22兴 developed a

general theory of irreversible processes. Namely, it was
shown by them that the statistical equilibrium of the system
is triggered by a small perturbation in potential energy, and
the probability of the transition*共k*ជ*, k*ជ

### ⬘

兲 can be calculated by the first order approximation of the nonstationary theory of perturbation.Equation 共38兲, derived for a dense photon gas, is pure

classical and describes the three wave interaction. Namely, the photon passing through the photon bunch absorbs and emits photonikos, with frequencies⍀= ⫿共−

_{⬘}

兲 and wave
*vectors q*ជ=

*⫿共k*ជ

*− k*ជ

### ⬘

兲. The integral in Eq. 共38兲 is the elasticcollision integral and describes the photon scattering process
on the variation of shape of the photon bunch. This equation
indicates that the equilibrium of the photon gas is triggered
by the perturbation␦=␦*共n*/*no*␥兲.

In the limit of spatial homogeneity and quasiclassical ap-proximation we can define the entropy of a photon gas as

*S = − KBV*

### 冕

*d*3*k*

43*兵N共k*ជ*,t兲ln N共k*ជ*,t兲 − 关N共k*ជ*,t兲 + 1兴ln共1 + N兲其,*
共40兲
*where KB* is Boltzmann’s constant.

Differentiating this expression with respect to time, we
obtain
*dS*
*dt* *= VKB*

### 冕

*d*3

*k*43 ln

### 冉

*1 + N共k*ជ

*,t兲*

*N共k*ជ

*,t兲*

### 冊

*N共k*ជ

*,t兲*

*t*. 共41兲

From Eq.共38兲, where we take* _{共kជ}*共kជ兲

_{⬘}

兲⬇1 since we consider

*the case, when the wave number q of the photoniko is much*
*less than the wave number k of the photons, we substitute*

*N共k,t兲*

*t* into Eq.共41兲 to obtain

*dS*
*dt* *= VKB*

### 兺

±### 冕

*d*3

*k*共2兲3

### 冕

*d*3

*k*

### ⬘

共2兲3ln### 冉

*1 + N共k*ជ

*,t兲*

*N共k*ជ

*,t兲*

### 冊

*W*±

*共k*ជ

### ⬘

*,k*ជ兲

*⫻关N共k*ជ

### ⬘

*,t兲 − N共k*ជ

*,t兲兴.*共42兲 Bearing in mind that the expression under the integrals in Eq.共42

*兲 is invariant under the transformations k*ជ

*→k*ជ

### ⬘

*and k*ជ

*←k*ជ

### ⬘

, we can rewrite this equation in the form*dS*
*dt* =
*VKB*
2

### 兺

_{±}

### 冕

*d*3

*k*共2兲3

### 冕

*d*3

*k*

### ⬘

共2兲3*W*±

*共k*ជ

### ⬘

*,k*ជ兲 ⫻ln

### 冉

*关1 + N共k*ជ

*,t兲兴N共k*ជ

### ⬘

*,t*兲

*关1 + N共k*ជ

### ⬘

*,t兲兴N共k*ជ

*,t兲*

### 冊

*关N共k*ជ

### ⬘

*,t兲 − N共k*ជ

*,t*兲兴. 共43兲

*By the definition W*±

*and N共k*ជ

*, t兲 in the integrand are*

posi-tive, and the function

*F = ln*

### 冉

*1 + N共k*ជ

*,t兲*

*1 + N共k*ជ

### ⬘

*,t兲*

*N共k*ជ

### ⬘

*,t兲*

*N共k*ជ*,t兲*

### 冊

*关N共k*ជ

### ⬘

*,t兲 − N共k*ជ

*,t兲兴*共44兲

*is non-negative in any case, i.e., N共k*ជ

### ⬘

*, t兲⬎N共k*ជ

*, t兲 or reverse.*We thus obtain the required result

*dS*

*dt* 艌 0, 共45兲

expressing the law of increase of the entropy of the photon gas. Note that equality occurs at equilibrium.

**IV. ADIABATIC PHOTON SELF-CAPTURE**

In this section, we discuss the phenomenon of photon capture by some potential well. To this end, we consider the distribution of photons in a slowly applied field, which is a function of the density and the relativistic factor of particles,

*U = g关n*_{␣}*共r*ជ*, t兲,*␥_{␣}*共r*ជ*, t兲兴.*

*Let l and*be the characteristic length and time of
varia-tion of the potential. We suppose that

Ⰷ *l*

*u*. 共46兲

With this condition in mind, we employ the equation derived
in Ref.关25兴,
*k*ជ·ⵜជ*rN共r*ជ*,t,k*ជ兲 −
*p*2
*2c*2ⵜជ*r*
*ne*
␥*e*
·ⵜជ*kN共r*ជ*,t,k*ជ兲 = 0. 共47兲

In the following, we consider the case when the density and
*the relativistic factor are functions only of the distance r*
from a fixed point. Then the solution of Eq.共47兲 is

*N共r,k兲 = n*_{0␥}*f*_{␥}*关共k*ជ*,r*ជ*兲兴 = n*_{0␥} 1
共20
2_{兲}3/2exp

### 再

−*k*2

*+ kp*2

_{␦}2

_{0}2

### 冎

, 共48兲 where0is the spectral width, and␦= ␦*n*
*n*0␥
+

### 冉

1 ␥− 1 ␥0### 冊

,*kp*=

*p*

*c*. 共49兲

We specifically note here that␦can be positive as well as
negative. Namely, in the case when the density of particles
has a cavity, i.e., _{n}n

0= 1 −

兩␦*n*_{兩}

*n*0, and ␥=␥0, ␦ is negative

### 共

␦= −_{␥}1

0

### 兩

␦*n*

### 兩

*/n*0

### 兲

. Next, in the case when the density does not*change, i.e., n = n*0, but␥⬎␥0, i.e., there is a focusing of EM

waves, then␦ is again negative. Whereas, in the case when
*both n and* ␥ change, then ␦ can be positive as well as
negative.

If␦⬍0 in some region, and in the rest of the space␦ ⬎0, then we have two sorts of photons. First, photons with ␦⬎0 have a Gaussian-Boltzmann distribution throughout the space, and the density of photons is given as

*n*_{␥}*共r兲 =*

### 冕

*dk*ជ

*N共r,k兲 = n*

_{0␥}exp

### 再

−*kp*

2_{␦}

2_{0}2

### 冎

; 共50兲 but in the case, when there are some photons in the cavity, then the motion of the photons takes place in a finite region of space, i.e., they are trapped in the potential well*U = −kp*2兩␦兩. In other words, for the trapped photons we have

*k*_{0}2*+ kp*

2_{␦}_{= 0, N共r,k兲=N共0兲, and for them the wave number}

varies between 0艋k艋k*p*兩␦兩1/2, whereas for the untrapped

*photons, k⬎kp*兩␦兩1/2*. Therefore we can now represent n*␥as

*n*_{␥}*共r兲 = n*_{␥}*trap共r兲 + n*_{␥}*untr共r兲,* 共51兲
where
*n*_{␥}*trap*
*n*_{0␥} =
4
3

### 冑

### 冉

*kp*

### 冑

2_{0}兩␦兩 1/2

### 冊

3 共52兲 and for the untrapped photons we have*n*_{␥}*untr*
*n*_{0␥} =
4

### 冑

*e*

*k*2

_{p}_{兩}␦兩/2

_{0}2

### 冕

0 ⬁*d*2

*e*−2 =

### 冉

1 −_{冑}

4
### 冕

0 0*d*2

*e*−2

### 冊

*e*0 2 , 共53兲 where=

_{冑}

_{2}

*k*0 and 0=

*kp*兩␦兩1/2

冑_{2}_{0} . Equations 共52兲 and 共53兲

*ex-hibit that, when kp*兩␦兩1/2Ⰷ

### 冑

20*, then n*␥

*untr→0, whereas n*␥

*trap*

increases as a third power, i.e., almost all photons are trapped. In the opposite limit,0Ⰶ1, for the density of

pho-tons, we obtain

*n*_{␥}*= n*_{0␥}

### 再

1 +_{0}2− 8 15

### 冑

05/2

### 冎

.

**V. UNIFORM EXPANSION OF PHOTON GAS**
We next consider the uniform expansion of the photon
gas. To this end, we employ the equation of continuity of the

photon gas derived in Ref.关14兴. In the past Kompaneets 关23兴

has shown that the establishment of equilibrium between the photons and the electrons is possible through the Compton effect. In his consideration, since the free electron does not absorb and emit, but only scatters the photon, the total num-ber of photons is conserved. Using the kinetic equation of Kompaneets, Zel’dovich and Levich关24兴 have shown that in

the absence of absorption the photons undergo Bose-Einstein condensation. Recently it was shown that another mecha-nism exists 共“Compton” scattering type兲 of the creation of equilibrium state and Bose-Einstein condensation in a non-ideal dense photon gas关25,14兴. Hereafter, we assume that the

total number of photons is conserved.

In the following the dynamics of the photon gas is deter-mined by the constancy of the entropy. Equations 共30兲 and

共32兲 yield the following expressions, first for the

ultrarelativ-istic photon gas, i.e.,_{␥}*⬇cp*_{␥},

*T共t兲 = T*0

### 冉

*V*0

*V共t兲*

### 冊

1/3_{1}1 +

### 冉

*V共t兲*

*V*0

### 冊

1/3, 共54兲and second for the nonrelativistic photon gas, i.e., _{␥}
*⬇m*␥*c*2+
*p*_{␥}2
*2m*␥,
*T共t兲 = T*0

### 冉

*V*0

*V共t兲*

### 冊

1/6 , 共55兲 where= 0.29*m*␥

*共V*0

*兲c*2

*T*_{0} *, T*0 *and V*0 are initial temperature and

volume.

*In order to determine the explicit dependence T共t兲 and*

*V共t兲, we study the spherically symmetric case. In this case*

the equation of continuity takes the form
*n*_{␥}
*t* +
1
*r*2
*rr*
2* _{n}*
␥

*ur*= 0. 共56兲

The solution of which we represent as

*n*_{␥}*共t兲 = n*_{0␥}

### 冉

*R*0

*R共t兲*

### 冊

3 ,*ur= u*0

*r*

*R共t兲*, 共57兲

where the suffix 0 denotes the constant initial value. Substituting Eq.共57兲 into Eq. 共56兲, we obtain

*dR共t兲*

*dt* *= u*0 *or R共t兲 = R*0*+ u*0*t.* 共58兲

Substituting Eq. 共58兲 into Eqs. 共54兲 and 共55兲, we can now

explicitly express also the time dependence of the tempera-ture. The result is for the ultrarelativistic photon gas

*T共t兲 = T*0
*R*0
*R共t兲*
1
1 +*R共t兲*
*R*0
共59兲

and for the nonrelativistic photon gas

*TSINTSADZE et al.* **PHYSICAL REVIEW E 76, 016406**共2007兲

*T共t兲 = T*0

### 冉

*R*0

*R共t兲*

### 冊

1/2

. 共60兲

Thus we may conclude that the cooling of the photon gas is slower in the nonrelativistic case than in the ultrarelativistic case, as is evident from Eqs.共59兲 and 共60兲.

**VI. SUMMARY**

We have investigated the interaction of spectrally broad and relativistically intense EM radiation with a plasma. We have obtained the condition which allows the EM field to be treated as quasiclassical. We have studied the system of a dilute gas composed of electrons, ions, and photons共or elec-trons, posielec-trons, and photons兲, and described it in terms of macroscopic properties. We have calculated all thermody-namic quantities developing the statistical mechanics in the presence of a strong EM field. We have demonstrated the existence of the ratio of the specific heats,⌫, which equals 7 / 6. We have also disclosed the magnitude of the mean square fluctuation of the number of photons, and shown that the relative fluctuation of the number of photons does not

decrease, when the mean number of photons increases. We have discussed the Boltzmann H-theorem in a photon gas. In addition, we have studied the adiabatic photon self-capture and defined the number of trapped photons. Finally, we have considered the uniform expansion of the photon gas and ex-plicitly expressed the time dependence of temperature and volume. EM radiation has played a crucial role in opening up new frontiers in physics. The distribution law discovered by Planck accurately describes the equilibrium properties of an assembly of photons over a vast range of temperatures and scales, from terrestrial cavity radiation to hot stellar atmo-spheres, and, of course, including the cosmic background radiation. However, there are changes in Planck’s law and photon thermodynamics, as discussed in this and previous 关5,6兴 papers, which may play a role in an as yet undiscovered

phenomenon.

**ACKNOWLEDGMENT**

This work was partially supported by ISTC Grant, Project G-1366.

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