Table Of ContentA Kinetics Of The Non-Equilibrium Universe.
II. A Kinetics Of The Local Thermodynamical
Equilibrium’s Recovery.
Yu.G.Ignatyev, D.Yu.Ignatyev
Kazan State PedagogicalUniversity,
Mezhlauk str., 1, Kazan 420021,Russia
1 Abstract
1
Ithasbeenresearchedthekineticsofthethermalequilibrium’sestab-
0
lishment in an early Universe under the assumption of the recovery of
2
interaction scaling of elementary particles in rangeof superhigh energies.
n Thecaseofthethermalequilibrium’sweakinitialviolationandbasiccos-
a
mological consequences of the thermal equilibrium’s violation have been
J
researched.
1
] 1 Introduction
c
q
- In the previous paper of one of the authors [1] it was shown, that in the case
r
g of the scaling behavior of the particles’ cross-section of interaction in range of
[ superhigh energies:
1
Const
v σ , (1)
tot
6 ∼ s
6
where s is a kinematic invariant of the four-particle reaction (details see in
3
[1]), the initial particles’ distribution in the expanding Universe is not to be
0
. equilibrium,butcanberandom. Inthispaperweinvestigatethe kineticsofthe
1
processeswithelementaryparticlesintheearlyUniverseundertheconditionsof
0
scalingofinteractionswiththe purposetoclarifytheboundariesofrandomness
1
1 oftheinitialparticles’distribution. Asthecross-sectionofelementaryparticles’
: interactionatthatwe willuse anasymptoticcross-sectionofscattering,UACS,
v
incorporated in papers [2], [3]1:
i
X
2π 2π
r σ (s)= = , (2)
a 0 s 1+ln2 s sΛ(s)
s0
(cid:16) (cid:17)
where s =4 - the square of the total energy of two colliding Planck masses,
0
s
Λ(s)=1+ln2 Const. (3)
s ≈
0
1AsinthepreviouspaperwewilluseasystemofunitsG=~=c=1.
1
2 Kinetic equations for superthermal particles
2.1 A simplification of the relativistic integral of collisions
Therelativistickineticequationsforhomogenousisotropicdistributionsf (t,p)
a
have the form (see the previous paper [1], details see in [4], [5]):
∂f a˙ ∂f 1
a a
p = J (t,p), (4)
ab⇆cd
∂t − a ∂p m2+p2
a b,c,d
X
p
where a(t) is a scale factor of the Friedmann’s world:
ds2 =dt2 a2(t)[dχ2+ρ2(χ)(sin2θdϕ2+dθ2)]; (5)
−
p2 = g pαpβ, (α,β =1..3); (6)
αβ
−
J (t,p) is an integral of four-particle reactions [6], [7]:
ab⇆cd
J (t,p)=π4 dπ dπ dπ δ(4)(P +P P P )
ab b c d a b c d
− − ×
Z
[(1 fa)(1 fb)fcfdMcd→ab 2 (7)
× ± ± | | −
(1 fc)(1 fd)fafb Mab→cd 2],
− ± ± | |
characters correspond to bosons (+) and fermions ( ), Mi→f are invariant
± −
amplitudes of scattering (line means the average by particles’ states of polar-
ization), dπ is a normed differential of volume of the a particle’s momentum
a
space:
ρ dp1dp2dp3
dπ =√ g a , (8)
a − (2π)3p
4
ρ is a factor of degeneration.
a
Let us simplify an integral of four-particle interactions (7), using the prop-
erties of the distributions’ isotropy f (t,p). For the fulfilment of two inner
a
integrations we will proceed to the local c.m. system, integration in which is
carried out by the elementary way. After the inverse Lorentz transformation
and conversion to the spherical system of coordinates in the momentum space
we obtain ([8]):
∞
2S +1 qdq
b
J (p)=
ab −8(2π)4p m2+q2×
Z b
0
s+ 0 p 2π
sds 1
dtM(s,t)2 dϕ
× s+m2 m216πλ2 | | ×
sZ− a− b −λZ2/s Z0
f (p )f (q )[1 f (p ∆)][1 f (q +∆)]
a 4 b 4 c 4 d 4
{ ± − ± −
f (p ∆)f (q +∆)[1 f (p )][1 f (q )] ,
c 4 d 4 a 4 b 4
− − ± ± }
2
where
ts m2 m2
∆= p q (p +q ) a− b
−λ2 4− 4− 4 4 s −
(cid:20) (cid:21)
ts ts
cosϕ 1+
− s−λ2 λ2 ×
(cid:18) (cid:19)
m2+m2 λ2+4m2p2+4m2q2 1/2
4p q 1 a b b 4 a 4 ,
4 4
× − s − s
(cid:20) (cid:18) (cid:19) (cid:21)
λ2 =s2 2s(m2 +m2)+(m2 m2)2,
− a b a− b −
a function of a triangle, s,t are the kinematic invariants (see [1]),
s± =m2a+m2b +2(p4q4±pq).
It is necessary at that to bear in mind the definition of the total cross-section
of an interaction [9], (see also the previous paper [1]):
0
1
σ = dtF(s,t), (9)
tot 16πλ2
−λZ2/s
where we denoted as is generally accepted:
F(t,s)= M(s,t)2. (10)
| |
In the ultrarelativistic limit:
p s
i ; λ s2, (11)
m →∞⇒ m2 →∞ →
i i
aforecited expressions are greatly simplified:
∞ 4pq 1 2π
(2S +1 ds
b
J (p)= dq dxF(x,s) dϕ
ab −32(2π)4p s ×
Z Z Z Z
0 0 0 0
f (p)f (q)[1 f (p ∆)][1 f (q+∆)]
a b c d
{ ± − ± −
f (p ∆)f (q+∆)[1 f (p)][1 f (q)] , (12)
c d a b
− − ± ± }
where a new variable was incorporated (see [1]):
t
x= (13)
−s
and
∆=x(p q) cosϕ x(1 x)(4pq s). (14)
− − − −
p
3
2.2 An integral of collisions for a distributions’ weak de-
viation from the equilibrium
Let us investigate at first a weak violation of the thermodynamical equilibrium
in the hot model, when the main part of particles, n (t), lies in the thermal
0
equilibrium state, and only for the minor part of particles, n (t) the thermal
1
equilibrium is violated (see Fig. 1):
n (t) n (t). (15)
1 0
≪
Figure 1: The schematic representation of the distribution function’s deviation
from the equilibrium.
Henceforth in this paper we will suppose that distribution functions different
not greatly from the equilibrium ones in the range of small values of energy,
smaller than certain unitary limit, p = p (or T = T ), below which scaling is
0 0
absent, and can be violated at energies, above the limit:
1
f0 = , p<p ;
a µ +E (p) 0
a a
fa(p)≈ exp(− T )±1 (16)
∆f (p); f0(p) ∆f (p) 1, p>p ,
a a ≪ a ≪ 0
whereµ (t)arethechemicalpotentials,T(t)isatemperatureoftheequilibrium
a
component of plasma. Thus, in range p > p it can be observed the anomaly
0
greatnumber ofparticlesin comparisonwith the equilibriumone, butminor at
that (. (15)) in comparison with the total number of equilibrium particles.
Let us investigate the process of relaxation of the distribution f (p) to the
a
equilibrium f0(p). The problem in such a setting for the special case of the
a
4
initial distribution f(t = 0,p) was solved earlier in [2], [3]. Here we will give
the general solution of this problem. At that, as it will be obvious from the
further arguments, the cosmological plasma can be formally considered as a
two-componentsystem-equilibriumwiththedistributionf0(t,p),andnonequi-
a
librium superthermal, with the distribution δf (t,p) = Ψ(t,p), where number
a
of particles in the nonequilibrium component is small, but energy density, as a
matter of fact, is random.
Let us investigate an integral of collisions (12) in range
p>p T. (17)
0
≫
In consequence of the inequality (16) in this range we can neglect collisions of
superthermalparticlesbetweenthemselves,notgoingbeyondtheaccountofthe
superthermal particles’scattering on equilibrium particles. Therefore the value
of one of momentums in the integral of collisions, P′ = p ∆, or q′ = q+∆
−
must lie in a thermal range, another’s value - in a superthermal one, outside
the unitary limit. A subintegral value of the integral of collisions is extremely
small outside this range. In consequence of this circumstance we can neglect
the secondmember incurly brackets(12), since itcancompete with the firstin
asymptoticallysmallvariationrangesofvariablesxandϕ: x(1 x).T/p 0.
Thestatisticalfactorsoftype[1 f (p′)]inthe firstmember of−anintegral→(12)
a
±
cannoticeablydifferfromonebesidesonlyintherangeofmomentums’thermal
values. As a result the integral of collisions (12) in the investigated range of
momentums’ values can be written down in a form:
(2S +1)∆f (p)
b a
Jab↔cd(p)|p>p0 = (2π)3p ×
∞ 2p(q4+q) 1
qf0(q)dq ds
b dxF(x,s). (18)
× m2+q2 16π
Z0 b 2p(qZ4−q) Z0
p
Using the definition of the totalcross-sectionof scattering(12), we obtainfrom
(18):
Jab↔cd(p)|p>p0 =
∞ 2p(q4+q)
(2S +1)∆f (p) qf0(q)dq
= b a b σ s(s)ds. (19)
(2π)3p m2+q2 tot
Z0 b 2p(qZ4−q)
p
Substituting eventually an expression for σ in the inner integral in form of
tot
UACS, (2), carrying out an integration with a logarithmic accuracy and sum-
ming up the obtained expression by all channels of reactions, we find finally:
J (p) =
a |p>p0
∞
(2S +1)ν q2f0(q) dq
= ∆f (p) b ab b , (20)
a
− π m2+q2Λ(s¯)
Xb Z0 b
p
5
where
1
s¯= pq4,
2
ν isanumberofchannelsofreactions,inwhichasortparticlescanparticipate
ab
a.
Let us calculate values of the integral (20) in extreme cases.
A scattering on nonrelativistic particles. If b sort equilibrium particles
are nonrelativistic, i.e., q m , the integral (20) is reduced to the expression:
b
≪
J (p) =
a |p>p0
n0(t) ν
= 8π2∆f (p) b ab , (m >T). (21)
a pm b
− mb 1+ln2 b
Xb 2
A scattering on ultrarelativistic particles. If b sort equilibrium particles
are ultrarelativistic, i.e., m T, moreover their chemical potential is small,
b
≪
- µ T, then calculating the integral (20) with respect to the equilibrium
b
≪
distribution (16), we find:
J (p) =
a |p>p0
π N˜T2(t)
= ∆f (p), (m T, µ T), (22)
−31+ln2Tp/2 a b ≪ b ≪
where
1 1 1
˜ = (2S+1)+ (2S+1) =N + N ;
B F
N 2" 2 # 2
B F
X X
N is a number of sorts of equilibrium bosons, F - fermions.
B
To estimate contributions of nonrelativstic and relativistic particles to the
integral of collisions in an equilibrium component, we first will calculate their
concentrations. Aconcentrationofultrarelativisticparticlesinthehotmodelis
resultedfromthe expression(16)forthe distributionfunctionofanequilibrium
component by means of the substitution
E(p)=p; µ =0 (23)
a
into the formula for the determination of particles’ number density (see [1]):
∞
2S +1
n (t)= a f (t,p)p2dp. (24)
a 2π2 a
Z
0
Thus we find (see [1]):
(2S +1)T3
a
n (t)= g ζ(3) (25)
a π2 n
Aconcentrationofnon-relativisticequilibriumparticlesatundertheassumption
of conservation of their numberisviolatedinproportiontoa−3(t). Thereforein
6
conditions of equilibrium’s weak violation the ratio of nonrelativistic particles’
densitytodensityofrelictphotonsisapproximatelyconstant(sinceT a(t)−1:
∼
n (t)
0 Const=δ 10−10 10−9. (26)
n (t) ≈ ∼ ÷
γ
Calculating the ratio of contributions to the integral of nonrelativistic and ul-
trarelativistic particles’ collisions, we obtain:
24πn0 64T(t) T
J /J b =ζ(3)δ 10−9 , (27)
non ultra ∼ m T2 πm ∼ m
b b b
-ratioofcontributionsissmallatT 109m anddiminisheswithtime. There-
b
≪
fore in future we will neglect the contribution of nonrelativistic particles in the
integral of collisions.
3 A Relaxation Of A Superthermal Component
On Equilibrium Particles
Substituting the received expression (22) for the integral of collisions into the
kineticequations(4),weobtainakineticequation,whichdescribesanevolution
of an ultrarelativistic superthermal component in the equilibrium cosmological
plasma:
∂∆f a˙ ∂∆f πN˜ T2(t)
a a
p p = ∆f . (28)
∂t − a ∂p − 3 1+ln2pT/2 a
(cid:18) (cid:19)
Taking into account the fact, that the variable:
=a(t)p, (29)
P
is an integral of motion [4], we proceed to variables t, in the equation (28);
P
for any function Ψ(t,p) at that the following relation takes place:
∂Ψ(t,p) a˙ ∂Ψ(t,p) ∂Ψ(t, )
p = P . (30)
∂t − a ∂p ∂t
At such a substitution the kinetic equation (28) can be easily integrated in
quadratures. For convenience in future we will define more exactly the normal-
ization of the variable . At that there appears a necessity to compare values,
P
which are used in the nonequlibrium model, with corresponding values of the
standard cosmological scenario, since all observed cosmological parameters are
interpretedinSCSterms. Wealsoshouldkeepinmindtwosynchronousmodels
of Universe: the real - nonequilibrium model with macroscopic parameters
M
P(t) and the ideal - equilibrium model , which in given point of time t
0
M
possesses certain macroscopic parameters P (t).
0
7
An Ultrarelativistic Universe
Let us consider the Universe with the ultrarelativistic equation of state2 (a
characteristic of a barotropic line is ρ=1/3):
ε=3p. (31)
Then according to Einstein’s equations an energy density of the Universe is
changed by law:
1
εa4 =Const; ε= , (32)
32πt2
and a scale factor is changed by law:
a(t) t1/2. (33)
∼
Fromthe otherhand, anenergydensityofanequilibriumplasmais determined
via its temperature by the relation (see [1]):
π2T4
ε = . (34)
0
N 15
Therefore,if the Universe was filled up only by equilibrium plasma, its temper-
ature T (t) would change by law (see [1]):
0
1/4
45
1/4T (t)= t−1/2 ( a−1), (35)
N 0 32π3 ∼
(cid:18) (cid:19)
- here we take into account a possible weak dependence of an effective number
ofequilibriumtypes ofparticles fromtime, (t). So,let usdefine moreexactly
N
the formula (29) by following way:
p= 1/4T (t). (36)
0
PN
Such is the meaning of the momentum variable according to this formula:
P
to within the numerical factor of order of one is the relation of particles’
P
energytotheiraverageenergyinthesamepointoftimeinthelocallyequilibrium
ultrarelativistic Universe.
Thus, solving the kinetic equation (28) with the account of relations (30)
and (36), we find its solution:
t
ξ(t, ) y2(t′)dt′
∆f (t, )=∆f0( )exp P , (37)
a P a P − √t′
P Z
0
where:
∆f0( )=∆f (0, ),
a P a P
2Itshouldbenoted, thatpressureandmomentum havethesamedenotations.
8
isaninitialdeviationfromtheequilibriumandthereisincorporatedthedimen-
sionless function:
T(t)
y(t)= (38)
T (t)
0
and the parameter, weakly dependent from variables t, :
P
π ˜ 45 1/4 1
ξ(t, )= N ; (39)
P 3√N (cid:18)32π3(cid:19) Λ(PTT0/2)
Λ(x)=1+ln2x. (40)
Values 1 correspond to the approximation p p T(t).
T
P ≫ ≫ ≈
Since T(t) is a temperature of the equilibrium component of plasma, and
T (t) is a temperature of the completely equilibrium in a given point of time
0
Universe, the following condition is always fulfilled:
y(t)61. (41)
To be the correct solution of the kinetic equations, the function ∆f (t, )
a
P
has to satisfy in all times to the integral condition (15). Since according to
the solution (36) the distribution function’s ∆f (t, ) deviation from the equi-
a
P
librium strictly diminishes with time, for the validity of the solution (36) it is
sufficient to the function ∆f (t, ) to satisfy the condition (15) in the initial
a
P
point of time. This gives:
∞
2
∆f0( ) 2d y3, (42)
a P P P ≫ 3/4 0
Z0 N
where y =y(0)61.
0
Asanexamplewewillconsiderarelaxationofasuperthermalcomponentat
theinitialdistributioninformofastaircasefunctionforthedensityofparticles’
number:
π2∆N˜
, 6 ;
∆f0( )= P0P2 P P0 , (43)
P 0, ;
0
P P
so that:
∞
1
∆N˜ = ∆f0( ) 2d (44)
π2 P P P
Z
0
isaninitialconformaldensityofthenon-equilibriumparticles’number. OnFig.
2 an evolution of the superthermal component for such a distribution, at that
we laid y(t) 1 is shown .
≡
9
1
0.8
0.6
0.4
0.2
0 20 40 60 80 100
P
Figure 2: A relaxation of a superthermal component for the distribution (43)
with the assumption y(t)=1 at =100, ˜/√ =10. A relative magnitude
0
P N N
of the distribution function of particles’ number density by conformal energies
is shown. Top-down (along the figure’s left border) there are firm lines:
P
t=0, t=0,01, t=0,1, t=1, t=10 and t=100000;a dotted line is t=3.
Time is measured in seconds.
Let us remind that cosmologicaltime t is measured in Planck units. There-
fore a native question, if methods of classical (nonquantized) kinetics can be
used in times of order of severalPlanck times, appears. A condition of applica-
bility of a particles’ semi-classical description in the cosmologicalsituation is a
relation, resulting from the Heisenberg’s uncertainty relation:
Et 1. (45)
≫
According to (35) and (36):
1/4
45
E =p= t−1/2. (46)
P 32π3
(cid:18) (cid:19)
Therefore a condition of applicability of a particles’ semi-classical description
(45) takes form:
32π2
t 2 2,65. (47)
P ≫ 45 ≈
r
Such consideration at t 1 is justified for sufficiently great values of the con-
∼
formal momentum 1, which exactly correspond superthermal particles.
P ≫
Thus, a semi-classical description of particles is applicable in Planck times of
an evolution of the Universe with the more validity the lesser is a ratio of a
10
