A Kinetics of non-equilibrium Universe. III. Stability of non-equilibrium scenario

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by  Yu. G. Ignatyev

A Kinetics of non-equilibrium Universe. III.
Stability of non-equilibrium scenario.
Yu.G.Ignatyev, D.Yu.Ignatyev
Kazan State PedagogicalUniversity,
Mezhlauk str., 1, Kazan 420021,Russia
Abstract
1 Aninfluenceofinitialdistributionnonequilibriumparametersonther-
1 modynamical equilibrium recovery processes in early Universe under the
0
assumption ofelementary particles’ scaling of interactions in rangeof su-
2
perhigh energies is researched.

n
a
J 1 Introduction

1
In previous papers of the Authors the model of superheat particles’ cosmolog-

]
c ical evolution in conditions of scaling of interactions was built up [1, 2]. For
q elementary particles’ cut set of interactions at that the Universal Asymptotic
- Cut Set of Scattering, introduced in papers [3, 4]1 was used:
r
g
2π 2π
[
Σ0(s)= = , (1)
1 s 1+ln2 ss0 sΛ(s)
v
(cid:16) (cid:17)
8 wheres isakinematic invariantoffour-piecereaction(detailsseein[1]), s0 =4
6 is a square of two colliding Planck masses’ total energy,
3
0 2 s
Λ(s)=1+ln Const, (2)
1. s0 ≈
0
In [2] a relaxation of cosmological plasma’s superheat component on equilib-
1
rium component was researching under the assumption that particles number
1
: in superheat component is far less than the equilibrium particles number. In
v
particular,itwasshownthatthesolutionofkineticequationdescribinganevolu-
i
X tionoftheultrarelativisticsuperheatcomponentintheequilibriumcosmological
r plasma has a form:
a
t
ξ(t, ) y2(t′)dt′
0
∆f (t, )=∆f ( )exp P , (3)
a P a P − √t′ 
P Z0
 
where ∆f0( ) = ∆f (0, ) is an initial deviation from equilibrium (detail see
a P a P
in [2]) and the dimensionless function is introduced:
T(t)
4 4
y(t)= ; σ y ; σ0 y (0). (4)
T0(t) ≡ ≡
1systemofunitsG=~=c=1.
1
whereT(t)is atemperature ofplasma’sequilibriumcomponent, T0(t) isa tem-
perature in the same point of time in completely equilibrium Universe and the
dimensionless momentum variable (conformal momentum) is introduced:
P
p= 1/4T0(t). (5)
PN
Farther is an efficient number of particles’ equilibrium types;
N
π ˜ 45 1/4 1
ξ(t, )= N ; (6)
P 3√ 32π3 Λ( TT0/2)
N (cid:18) (cid:19) P
is a parameter, weakly dependent on variables t, ,
P
1 1 1
˜ = (2S+1)+ (2S+1) =N + N ;
B F
N 2" 2 # 2
B F
X X
N is a number ofsorts ofequilibriumbosons, N is a number ofsortsofequi-
B F
libriumfermions. Relativetemperaturey(t)ofplasma’sequilibriumcomponent,
presentingaparameterofthenonequilibriumdistribution(3),isdeterminedvia
the integral equation of energy-balance:
∞
15
4 3 0
y + (2S +1) ∆f ( )
π4 a P a P ×
Xa Z0
t
ξ(t, ) y2(t′)dt′
exp P d =1, (7)
× − √t′  P
P Z0
 
where(2S +1)isastatisticalfactor. From(7)inzeropointoftimefollowsthe
a
relation [2]:
∞
15
3 0
π4 (2Sa+1) P ∆fa(P)dP =1−σ0. (8)
Xa Z0
Let us introduce corresponding to [2] a new dimensionless time variable τ :
ξ
τ = √t (9)
0
P
where ξ =ξ( 0);
P
∞
3 0
(2S +1) d ∆f ( )
a a 0 PP a P
P0 = P R∞ (10)
(2S +1) d 2∆f0( )
a a 0 PP a P
P R
2
is an averagevalue of momentum variable in point of time t=0, Z(τ):
P
τ
2 ′ ′
Z(τ)=2 y (τ )dτ . (11)
Z0
Then subject to (8) equation (7) is reduced to the form([2]):
y =[1 (1 σ0)Φ(Z)]1/4, (12)
− −
where
∞
(2S +1) d 3∆f0( )e−ZP0/P
(t) a a 0 PP a P
Φ(Z)= P = . (13)
P(0) P a (2Sa+R1)∞0 dPP3∆fa0(P)
P R
2 Numerical Model
In paper [2] distribution function’s initial deviation from the equilibrium was
represented in the most plain form:
A
0
∆f (x)= χ(1 x), k 0, (14)
3(k2+x2)3/2 − →
0
P
where χ(z) is a Heaviside function (a staircasefunction), x= / 0 is a dimen-
P P
sionless momentum variable, A, 0 and k are certain parameters; k parameter
P
is introducedto providea convergenceofdistributionfunction’s all moments in
range of momentum’s small values. A distribution function of superheat parti-
cles’ energy density at that has a form, close to the spectrum of the so-called
white noise, when all energy’s values are equiprobable up to the certain crit-
ical value 0, after which distribution breaks. From the qualitative point of
P
view an evolution of the initial distribution (14) is reduced to the “corrosion”
ofnonequlibriumdistributionspectrum’slow-energypartwhileenergydensity’s
constancyinhigh-energypartconserves,whatdidnotallowtomakepredictions
abouttheformofdistribution’stailarea. Ingivenpaperweselectadistribution
function’s deviation in more realistic form, allowing to carry out a research of
therelaxationprocessdependenceonparametersoftheinitialdistribution. The
paper’s main goal at that is a stability’s research of the nonequlibrium cosmo-
logical scenario [2] with respect to the parameters of the superheat particles’
initial distribution. So, let us specify the initial distribution in form:
∆f0( )=Ae−αP, (15)
P
so that:
∞
1 2A
∆N˜ = ∆f0( ) 2d = (16)
π2 P P P π2α3N −
Xa Z0
3
is an initial conformal density of nonequlibrium particles’ number. Let us cal-
culate an initial conformal energy density of nonequilibrium particles:
∞
1 3!A
3
ε˜1 = π2 ∆f0PP dP = π2α4N. (17)
Xa Z0
Thus, we obtain for the initial averageconformal energy:
∆ε˜ 3
0 = = . (18)
P ∆N˜ α
Letuscalculatetheparameterσ0 =ε0/(ε0+ε1),whereε0isanenergydensityof
plasma’sequilibriumcomponentandε1 is anonequlibriumcomponent’senergy
density. Plasma’s equilibrium component is described via the distribution:
p 1/4T0 1/4
f0 =Bexp =Bexp PN =Bexp PN . (19)
−T − T − y
(cid:16) (cid:17) (cid:18) (cid:19) (cid:18) (cid:19)
Calculating conformalparticles number densities and theirs energies relative to
this distribution, we obtain:
y3B2! y4B3!
N˜0 = π2 N1/4; ε˜0 = π2 . (20)
Comparing (20) with an energy density of equilibrium plasma, which is deter-
mined via its temperature by means of the relation (see [1]):
2 4
π T
ε0 = , (21)
N 15
we find:
π4
B = . (22)
90N
Then:
π2
0 = y3 5/4. (23)
N 45 N
Using (17), (20) and (22) we obtain:
π4α4
A=(1 σ0) . (24)
− 90
Then according to the referred above nonequilibrium particles’ condition of
smallness:
n1(t) n0(t), (25)
≪
it has to be:
α(1 σ0) y03N1/4. (26)
− ≪
On Fig.7-9 the graphs of the initial staircase and exponential distributions at
equalenergydensities andparticles’averageenergiesforvariousnonequlibrium
parameters are shown in comparison.
4
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0 2 4 6 8 10 12
x
Figure 1: Comparisonofstaircaseandexponentialinitial distributions at equal
in both cases energy densities ε˜1 and average energies of particles 0; α=0.9.
P
Staircase distribution is a thin line, exponential distribution is a heavy line.
10
8
6
4
2
0 2 4 6 8 10 12 14 16 18 20
x
Figure 2: Comparisonof staircaseandexponential initialdistributions at equal
in both cases energy densities ε˜1 and average energies of particles 0; α=0.5.
P
Staircase distribution is a thin line, exponential distribution is a heavy line.
5
1200
1000
800
600
400
200
0 20 40 60 80 100
x
Figure 3: Comparisonof staircaseandexponential initialdistributions at equal
in both cases energy densities ε˜1 and average energies of particles 0; α=0.1.
P
Staircase distribution is a thin line, exponential distribution is a heavy line.
3 Research of nonequilibrium distribution’s re-
laxation
Calculating function Φ(Z) relative to the distribution (15) according to the
formula (13), we obtain:
9Z3 2(3Z+6)K0(2√3Z)
Φ(Z)= +
2 9Z2

4√3(3+6Z)K1(2√3Z)
+ , (27)
27Z5/2 !
where K (z) is a Bessel function of the second sort (McDonald dunction) [5]:
ν
∞
K (z)= √nzy e−zcoshtsinh2νtdt,
ν 2yΓ(ν+ 1)
2 Z0
1
(z)>0, (ν)> . (28)
ℜ ℜ −2
6
Relation of τ and Z is determined via the following formula 2
Z
1 dU
=τ, (29)
2 1 (1 σ0)Φ(U)
Z0 − −
p
in which an obtained value of Φ(Z) (27)
is to be substituted. On Fig. 4-7 the graphs showing an influence of initial
distribution’s parameters on plasma’s temperature relaxation process are rep-
resented.
1
0.9
0.8
0.7
y
0.6
0.5
0.4
0 1 2 3 4 5
T
Figure 4: Relaxation of plasma’s temperature to the equilibrium:
y =T(τ)/T0(τ) subject to the parameter σ0: bottom-up
σ0 =0,01; 0,1; 0,3; 0,5; 0,9. Values of dimensionless time variable τ are put
on the abscissa axis.
2detailsseein[2].
7
1
0.995
0.99
y
0.985
0.98
0.975
0 1 2 3 4 5
T
Figure 5: Relaxation of plasma’s temperature to the equilibrium:
y =T(τ)/T0(τ) for the parameter σ0 =0.9 for staircase function (dotted line)
and Ae−αp function (firm line). Values of dimensionless time variable τ are
put on the abscissa axis.
1
0.95
0.9
y
0.85
0.8
0.75
0 1 2 3 4 5
T
Figure 6: Relaxation of plasma’s temperature to the equilibrium:
y =T(τ)/T0(τ) for the parameter σ0 =0.3 for staircase function (dotted line)
and Ae−αp function (firm line). Values of dimensionless time variable τ are
put on the abscissa axis.
8
1
0.9
0.8
0.7
y
0.6
0.5
0.4
0 1 2 3 4 5
T
Figure 7: Relaxation of plasma’s temperature to the equilibrium:
y =T(τ)/T0(τ) for the parameter σ0 =0.01 for staircasefunction (dotted line)
and Ae−αp function (firm line). Values of dimensionless time variable τ are
put on the abscissa axis.
On Fig. 4 the results of numerical modelling of equilibrium component’s
heating process by superheat particles for the initial distribution (15) in terms
ofequation(29) subject to the initialdistribution’s parametersareshown. It is
obviousfromthispicturethattemperatureofplasma’sequilibriumcomponentis
saturateduptothevalueT0(τ)atτ 3 4.OnFig.5-7temperature’srelaxation
∼ −
processofthestaircaseandexponentialdistributionsforvariousvaluesofinitial
distribution’s nonequilibrium parameter σ0 is shown. It is clear from these
pictures,that inthe caseofthe exponentialdistribution temperaturerelaxesto
theequilibriumratherslowerthaninthecaseofthe staircasefunction,however
qualitative behavior of y(τ) functions’s graphs coincides with each other.
Using the results of equation (29) numerical integration in formula for the
relaxationofthe nonequilibriumdistribution(3),weobtaintime dependence of
distribution function’s deviation from the equilibrium. On Fig.8 the results of
equations’(29) and(3) simultaneousnumericalintegrationsubject to the value
of initial distribution’s nonequilibrium parameter are shown.
Thus,anevolutionofdistributionintimevariableZ doesnotdependexplic-
itly from the nonequilibrium parameter σ0, but the relation of real time τ and
time variable Z do. This relationis givenby formula (29) and is representedin
graphical form on Fig.8.
9
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0 0.5 1 1.5 2 2.5 3
x
Figure 8: Relaxation of superheat component for distribution (15) (A=1)
subject to time parameter Z: top-down - Z =0.5, Z =1, Z =2, Z =4.Value
of dimensionless time variable x is put on the abscissa axis, value lg(1+f) is
put on the ordinate axis.
2.5
2
Max
Z_ 1.5
1
0 2 4 6 8
Tau
Figure 9: Dependence of distribution’s maximum on dimensionless time τ(Z),
calculated by formula (3) relative to the initial distribution (15) subject to
initial distribution’s nonequilibrium parameter σ0. Top-down: σ0 =0.01,
σ0 =0.1, σ0 =0.3, σ0 =5, σ0 =0.9,
10