Abstract:A formulation of nonequilibrium thermo field dynamics has been performed using the

Unification of Thermo Field Kinetic and

Hydrodynamics Approaches in Theory of Dense

Quantum–Field Systems

Mykhailo Tokarchuk1,2,∗and Petro Hlushak1

1

2

3

4

5

1 _{InstituteforCondensedMatterPhysicsoftheNationalAcademyofSciencesofUkraine,1,SvientsitskiiStr.,}

79011Lviv,Ukraine;[email protected],[email protected]

2 _{Nationaluniversity"Lvivskapolitechnika",12,S.BanderaStr.,79013Lviv,Ukraine}

* Correspondence:[email protected]

Abstract: Aformulationofnonequilibriumthermofielddynamicshasbeenperformedusingthe nonequilibriumstatisticaloperatormethodbyD.N.Zubarev. Generalizedtransferequationsfor aconsistentdescriptionofkineticsandhydrodynamicsofthedensequantum-fieldsystemwith stronglycoupledstatesarederived.

Keywords: nonequilibrium thermofieldd ynamics,k inetics,h ydrodynamics,k ineticequations, transportcoefficients,coupledstates,quark-gluonplasma

6

1. Introduction 7

A problem of accounting for the bound states (clusters) [15,16] formed by particles is 8

particularly important in the development of the theories of nonequilibrium processes of temperature 9

quantum-field systems, such as nuclear matter [1–14]. Kinetic and hydrodynamic processes in 10

a hot, compressed nuclear matter, which appears after ultrarelativistic collisions of heavy nuclei 11

[5,12,14,17–21] are mutually connected, and, therefore, the coupled states between nucleons should be 12

consider. This is of great importance for the analysis and correlation of final reaction products. 13

Obviously, a nucleon interaction investigation based on a quark-gluon plasma is a sequential 14

microscopic approach to the dynamical description of reactions in a nuclear matter. The problems of a 15

dense quark-gluon matter were discussed in detail in [2,3,10,11,23–27]. 16

In his recent works [15,16,19] G. Röpke noted the importance of constructing a nonequilibrium 17

theory in which along with hydrodynamic parameters a cluster distribution function are taken into 18

account, similarly to the case of the classical theory of non-equilibrium processes of dense gases and 19

liquids [28–31]. 20

In modern theoretical studies of the nonequilibrium properties of quark-gluon plasma [10,11,23, 21

24], which is one of the states of nuclear matter, one of the most widely used statistical concept is the 22

entropy of Tsallis and Renyi [32–41]. At the same time, the important problem of the construction 23

of kinetic and hydrodynamic equations for nuclear matter of high density and high temperature is 24

not sufficiently addressed for these systems. However, within the framework of the Gibbs statistics, 25

the equations of hydrodynamics and thermodynamics were already considered in many papers 26

using the method of the Zubarev’s nonequilibrium statistical operator [42–51], the projection operator 27

method [52,53] and kinetic equations [54–57]. Thus, we propose an approach to solve these problems 28

based on the the nonequilibrium thermofield dynamics [58–60] in the formulation of the method of 29

nonequilibrium statistical operator [61–63]. Below, in the second section of this paper, we consider the 30

nonequilibrium thermo field dynamics in the formulation of the nonequilibrium statistical operator 31

method [64–66] in Renyi statistics. Next, in the third section, generalized equations for the consistent 32

description of kinetic and hydrodynamic processes which take into account the bound states that 33

emerge in the temperature quantum-field system will be presented. 34

We use the nonequilibrium statistical operator method in the thermofield formulation [61,62], 36

where the mean values corresponding to the observables can be found using the nonequilibrium 37

thermovacuum state vector|$(t)ii: 38

hAit_{=hh1|A$(t)ii=hh1|Aˆ|}

$(t)ii, (1)

where ˆAis a superoperator acting on the state|$(t)ii. The nonequilibrium thermovacuum state vector 39

|$(t)iisatisfies the Schrödinger equation [61]: 40

∂

∂t|$(t)ii − 1

ih¯[H,$(t)] EE

=0, (2)

or 41

∂

∂t|$(t)ii −

1

i¯hHˆ|$(t)ii=0. (3)

Here, the total Hamiltonian ˆHtakes the form: 42

ˆ

H=H−H˜, (4)

wherehh1|_Hˆ =0, andH=H(aˆ+, ˆa), ˜H=H(∗)(a˜+, ˜a)are superoperators constructed from the creation 43

and annihilation of superoperators of the thermal Liouville space [61,68,69]. The superoperatorsH 44

and ˜Hare defined in [61]. 45

In the nonequilibrium statistical operator method in the thermofield formulation [61,62], the 46

nonequilibrium thermovacuum state vector as a solution of the Schrödinger equation (3) with a source 47

−ε

|$(t)ii − |$rel(t)ii

, with the projection taken into account can be found in the form 48

|$(t)ii=|$_rel(t)ii+ t Z

−_∞

dt0eε(t0−t)_T(t,t0₎

1− P_rel(t0) 1

i¯hHˆ|$rel(t

0_{)ii. (5)}

HereT(t,t0) =exp₊nRt t0dt0

1− P_rel(t0)₁ i¯hHˆ

o

is the evolution operator with the projection taken into 49

account, where exp₊is the ordered exponential,ε→+0 after the thermodynamic limit transition. 50

Prel(t) |. . .ii

=|$rel(t)ii +

∑

n

δ|$rel(t)ii

δhh1|pˆn|$(t)iihh1|pˆn|. . .ii (6)

−

∑

n

δ|$rel(t)ii

δhh1|pˆn|$(t)iihh1|pˆn|. . .iihh1|. . .ii

is the Kawasaki-Ganton projection operator, which acts only on the state vectors|. . .iiand has the 51

operator propertiesP_rel(t)|$(t0)ii = |$rel(t)ii,Prel(t)|$rel(t0)ii = |$rel(t)ii, Prel(t)Prel(t0) = Prel(t). 52

The relevant thermovacuum state vector |$rel(t)ii = $ˆrel(t)|1ii, is normalized in accordance with 53

the relationhh1|$rel(t)ii = hh1|$ˆrel(t)|1ii = 1, where ˆ$rel(t)is the relevant statistical superoperator. 54

The relevant thermovacuum state vector of the system can be defined as follows. We assume that 55

hpnit = hh1|pˆn|$(t)iiis the set of observed variables describing the nonequilibrium system state, 56

wherepnare the operators constructed on the respective creation and annihilation operatorsa+_l and 57

al. The relevant statistical operator$rel(t)is determined from the extremum of the Renyi entropy 58

functional 59

LR(t) = 1

1−qlnhh1|(|$

0_(t)ii)q₋

αhh1|$0(t)ii −

∑

under the additional condition that the mean valueshpnitare given with the normalization condition 60

hh1|$ˆ(t)|1ii=1 preserved. The Lagrange parametersαandF_n∗(t)are determined from the respective 61

normalization condition and self-consistency conditions: 62

h. . .it_rel=hh1|. . .|$rel(t)ii, hpnit=hpnirelt =hh1|pˆn|$rel(t)ii. (7) The relevant statistical operator$rel(t)then becomes

63

$rel(t) = 1

ZR(t) h

1−q−1 q

∑

∗

n(t)δpˆn(t) i_q−1₁

, (8)

whereqis the Renyi parameter,δpˆn(t) =pˆ− hh1|pˆn|$(t)ii, and 64

ZR(t) = DD

1 h

1− q−1 q

∑

∗

n(t)δpˆn(t) i_q−1₁EE

, (9)

is the partition function. The sum overncan denote the summation over the wave vectork, the kind 65

of particles and a whole series of quantum numbers, such as spin. From (8) atq=1, we obtain the 66

relevant statistical operator corresponding to Gibbs statistics [61]: 67

$rel(t) =exp

n

−Φ(t)−

∑

F_n∗(t)pn o

, (10)

whereΦ(t) =ln Sp exp{−_∑nFn∗(t)pn}is the Massieu-Planck functional. Substituting (8) in (5), we 68

now obtain the nonequilibrium thermovacuum vector 69

|$(t)ii=|$rel(t)ii+

∑

Z t

−_∞dt 0

eε(t0−t)_T(t,t0₎ 1

R

0

dτ$τ

rel(t

0

)Jn(t0)$rel(t)1−τ(t0) EE

Fn∗(t0), (11)

where Jn(t) = [1− P(t)]1_qψ−1(t)p˙ˆn are the operators of the generalized flows describing the 70

dissipative processes ˙ˆpn =−_i¯1_hHˆpˆnin the system. The projection operatorP(t)acts on operators and 71

has the structure 72

P(t)(. . .) =hh1|. . .|$rel(t)ii+

∑

δ

Z 1

0 dτ$

τ

rel(t)ψ

−1_(t)F

m(t) (12)

+

∑

n

f_mn−1(t)δpˆn

$−_relτ

_DD

. . .

Z 1

0 dτ$

τ

rel(t)δpˆn$

−τ

rel(t)$rel(t) EE

,

where δ[. . .] = [. . .]− hh1|[. . .]|$rel(t)ii and fmn(t) = δhh1|_δpˆ_Fm_n(|_t$₎(t)ii. The operator ψ(t)has the form 73

ψ(t) =1−q−_q1∑_nF_n∗(t)δpˆn(t). Using the nonequilibrium thermovacuum state vector|$(t)iigiven by 74

(11), we obtain the transport equations for the nonequilibrium meanshh1|pˆn|$(t)iiin the thermofield 75

representation. For this, we use the identity 76

∂

∂thh1|pˆn|$(t)ii=hh1|p˙ˆn|$(t)ii=hh1|p˙ˆn|$rel(t)ii+hhJn(t)|$(t)ii. (13)

Averaging the last term in the right-hand side with |$(t)ii given by (11), we obtain the transport 77

∂

∂thh1|pˆn|$(t)ii = hh1|p˙ˆn|$rel(t)ii (14)

+

∑

n0 Z t

−∞dt

0_eε(t0−t)DD_p˙ˆ

nT(t,t0)

Z 1

0 dτ $

τ

rel(t0)Jn0(t0)$1_rel−τ(t0) EE

F_n∗0(t0).

Transport equations (14) take the memory effects into account and can be used to describe 79

nonequilibrium processes in quantum Bose and Fermi systems in concrete cases in the framework 80

of the nonequilibrium thermofield dynamics of nonextensive statistics. In particular, a system of 81

relativistic transport equations for a consistent description of the kinetic and hydrodynamic processes 82

in a quark-gluon system was derived in [62] using the nonequilibrium statistical operator method in 83

the thermofield representation in Gibbs statistics. The advanced approach in terms of Renyi statistics 84

can be generalized to the case of relativistic systems, and this observation is important [32,34–41]. This 85

subject will be described in forthcoming works. 86

3. Thermo field transport equation with taking into account coupled states 87

We will consider a quantum field system in which coupled states can appear between the particles. 88

Let us introduce annihilation and creation operators of a coupled state(Aα)withA-particle: 89

a_Aα(p) =

∑

1,...,A

ΨAαp(1, . . . ,A)a(1). . .a(A), a+_Aα(p) =

∑

1,...,A

Ψ_A∗_αp(1, . . . ,A)a+(1). . .a+(A), (15)

whereΨAαp(1, . . . ,A)is a self-function of the A-particle coupled state,αdenotes internal quantum 90

numbers (spin, etc.),pis a particle momentum, the sum covers the particles. Annihilation and creation 91

operatorsa(j)anda+_{(j)satisfy the following commutation relations:}

92

[a(l),a+(j)]σ=δl,j, [a(l),a(j)]σ= [a+(l),a+(j)]σ =0, (16)

whereσ-commutator is determined by[a,b]σ = ab−σbawithσ = ±1: +1 for bosons and−1 for 93

fermions. 94

The Hamiltonian of such a system can be written in the form: 95

H=

∑

A,α

Z _dpdq

(2π¯h)6 p2

2mAa

+

Aα

p−q 2

a_Aαp+ q

2

(17)

+1

2

∑

∑

Z _dpdp0_dq

(2π¯h)9 VAB(q)a

+

Aα

p+q−p 0

2

ˆ

n_Bβ(q)a_Aα

p−q−p

0

2

,

where VAB(q) is interaction energy between A- and B-particle coupled states, qis a wavevector. 96

Annihilation and creation operatorsa_Aα(p)anda+_Aα(p)satisfy the following commutation relations: 97

[a_Aα(p),a+_Bβ(p0)]_σ=δA,Bδ_α,βδ(p−p

0₎

,

[a_Aα(p),a_Bβ(p0)]_σ= [a+_Aα(p),a+_Bβ(p0)]_σ =0. (18)

ˆ

nBβ(q)in (17) is a Fourier transform of theB-particle density operator: 98

ˆ

nBβ(q) =

Z _dp

(2πh¯)3a

+

p−q₂ap+q₂.

As parameters of a reduced description for the consistent description of the kinetics and hydrodynamics 99

of a system, where coupled states between the particles can appear, let us choose nonequilibrium 100

hh1|nˆ_Aα(r,p)|$(t)ii= f_Aα(r,p;t) = f_Aα(x;t), x={r,p}, (19)

heref_Aα(x;t)is a Wigner function of theA-particle coupled state where 102

ˆ

n_Aα(r,p)≡nˆ_Aα(x) =

Z _dq

(2π¯h)3e

−1 i¯hq·raˆ+

Aα

p−q 2

ˆ

a_Aαp+q

2

(20)

is the Klimontovich density operator; and the average value of the total energy density operator 103

hh1|Hˆ(r)|$(t)ii=hh1|H(r)$(t)ii. (21)

By thisR

dr H(r) = H, ˆH(r)is a superoperator of the total energy density which is constructed on 104

annihilation and creation superoperators ˆa_Aα(p)and ˆa+_Aα(p). The latter satisfy commutation relations 105

(18). Following [61], one can rewrite relevant statistical operator ˆ$rel(t),|$rel(t)ii = $ˆrel(t)|1iiand 106

with (8) fromq=1 for the mentioned parameters of a reduced description in the form: 107

ˆ

$_rel(t) =exp

(

−Φ∗(t)− Z

drβ(r;t) Hˆ(r)−

∑

Z _dp

(2πh¯)3µAα(x;t)nˆAα(x) !)

, (22)

where Lagrange multipliersβ(r;t)andµ_Aα(x;t)can be found from the self-consistency conditions, 108

correspondingly: 109

hh1|_Hˆ(r)|$(t)ii = hh1|Hˆ(r)|$_rel(t)ii, (23) hh1|nˆAα(x)|$(t)ii = hh1|nˆAα(x)|$rel(t)ii, (24) Φ∗_{(t)is the Massieu-Planck functional and it can be defined from the normalization condition :}

110

Φ∗_{(t) =}

ln

**

1

exp (

− Z

drβ(r;t) Hˆ(r)−

∑

Z _dp

(2π¯h)3µAα

(x;t)nˆ_Aα(x)

!) ++

. (25)

Using now the general structure of nonequilibrium thermo field dynamics (14), one can obtain a 111

set of generalized transport equations forA-particle Wigner distribution functions and the average 112

interaction energy: 113

∂

∂thh1|nˆAα(x)|$(t)ii=hh1|n˙ˆAα(x)|$q(t)ii (26)

+

Z dr0

t Z

−∞

dt0 eε(t0−t)

ϕ_nHAα(x,r0;t,t0)β(r0;t0)

+

∑

B,β

Z dx0

t Z

−∞

dt0 eε(t0−t)_ϕAαBβ

∂

∂thh1|Hˆ(r)|$(t)ii=hh1|H˙ˆ(r)|$q(t)ii (27)

+

Z dr0

t Z

−∞

dt0eε(t0−t)

ϕ_HH(r,r0;t,t0)β(r0;t0)

+

∑

B,β

Z dx0

t Z

−∞

dt0eε(t0−t)_ϕBβ

Hn(r,x0;t,t0)β(r0;t0)µBβ(x

0_;t0_),

wherex0 ={r0,p0},dx0= (2π¯h)−3dr0dp0. Here 114

ϕ

Aα

Bβ

nn(x,x0;t,t0) =

** 1 ˆ J_n Aα

(x,t)T(t,t0)

1 R 0

dτ $τ

rel(t0)Jn_Bβ(x

0_;t0₎

$1_rel−τ(t0)

++

, (28)

ϕA_nHα(x,r0;t,t0) =

** 1 ˆ J_n Aα

(x,t)T(t,t0)

1 R 0

dτ $τ_rel(t0)JH(r0;t0)$1_rel−τ(t0)

++

, (29)

ϕB_Hnβ(r0,x0;t,t0) =

** 1 ˆ

JH(r,t)T(t,t0) 1 R 0

dτ $τ_rel(t0)J_n

Bβ

(x0;t0)$1_rel−τ(t0)

++

, (30)

ϕ_HH(r,r0;t,t0) =

** 1 ˆ

JH(r,t)T(t,t0) 1 R 0

dτ $τ_rel(t0)JH(r0;t0)$1_rel−τ(t0)

++

(31)

are generalized transport cores which describe dissipative processes. In these formulae 115

JH(r;t) =

1−P(t0)H˙(r),

J_n

Aα

(r,p;t) = 1−P(t0)n˙_Aα(x) (32)

are generalized flows, ˙H(r) =−1

i¯h[H,H(r)], ˙nAα(r,p) = −

1

i¯h[H,nAα(x)], P(t)is a generalized Mori 116

projection operator in thermo field representation. It acts on operators 117

P(t)P=hh|_Pˆ|$rel(t)ii+ Z

dr δhh1|Pˆ|$rel(t)ii δhh1|Hˆ(r)|$(t)ii

H(r)− hh1|_Hˆ(r)|$(t)ii

(33)

+

∑

A,α

Z _drdp

(2π¯h)3

δhh1|Pˆ|$rel(t)ii δhh1|nˆ_Aα(x)|$(t)ii

n_Aα(x)− hh1|nˆ_Aα(x)|$(t)ii

and has all the properties of a projection operator: 118

P(t)H(r) =H(r), P(t)P(t0) =P(t), P(t)n_Aα(r,p) =n_Aα(r,p), 1−P(t)P(t) =0.

The obtained transport equations have the general meaning and can describe both weakly and 119

strongly nonequilibrium processes of a quantum system with taking into consideration coupled states. 120

In the next step we will construct such annihilation and creation superoperators, for which 121

the relevant thermo vacuum state vector is a vacuum state. Analysing the structure of relevant 122

statistical superoperator (22), one can mark out some part which would correspond to the system of 123

noninteracting quantumA-particles. Let us write ˆ$rel(t)in an evident form and separate terms which 124

ˆ

$rel(t) =exp

−Φ∗_(t)−Z _dr

β(r;t) (34)

×

∑

A,α

Z _drdp

(2π¯h)3

p2 2mA

ˆ

n_Aα(x)−µ_Aα(x;t)nˆ_Aα(x)

− Z

drβ(r;t)Hˆint(r) )

.

Using operator equality (AandBare some operators) 126

eA+B=



1+

1 Z

0

dτeτ(A+B)Be−τ(A+B) 

eA,

the relation for ˆ$rel(t)can be rewritten in the following form: 127

ˆ

$rel(t) = 

1−

Z

drβ(r;t)

1 Z

0

dτ$ˆτq(t)Hˆint(r) $ˆrel(t)

−τ



$ˆ0_rel(t), (35)

where 128

ˆ

$0_rel(t) =exp (

Φ(t)− Z

drβ(r;t)

∑

Z _drdp

(2π¯h)3

_p2

2mA ˆ

n_Aα(x)−µ_Aα(x;t)nˆ_Aα(x)

)

, (36)

or 129

ˆ

$0_rel(t) =exp (

Φ(t)− Z

drβ(r;t)

∑

Z _dp

(2π¯h)3bAα(x;t)nˆAα(x) )

, (37)

where b_Aα(x;t) =

p2 2mAnˆAα

(x)−µ_Aα(x;t)nˆ_Aα(x)

. Relevant statistical superoperator ˆ$0_rel(t) is 130

bilinear on annihilation and creation superoperators ˆa_Aα(P) and ˆa+_Aα(P), as well as on the 131

non-perturbed part of Hamiltonian ¯H0. One can write the total relevant superoperator as some 132

non-perturbed part of ˆ$0_rel(t)and the part which describes interaction of quantum particles in the 133

relevant state. Further, we introduce the following designation: 134

ˆ

$_rel(t) =$ˆ0_rel(t) +$ˆ0_rel(t), (38)

where 135

ˆ

$0_rel(t) =−

Z

drβ(r;t)

1 Z

0

dτ$ˆτ_rel(t)Hˆint(r) $ˆrel(t)

−τ

ˆ

$0_rel(t). (39)

Relevant (relevant) thermo vacuum states|$ˆrel(t)iiand|$ˆ0_rel(t)iiare not vacuum states for annihilation 136

and creation superoperators ˆa_Aα(P), ˆa+_Aα(P), ˜a_Aα(P), ˜a+_Aα(P). But for|$ˆ0_q(t)iione can construct new 137

superoperators ˆγ_Aα(P), ˜a+_Aα(P), ˜γ_Aα(P), ˜γ+_Aα(P)as a linear combination of superoperators ˆa_Aα(P), 138

ˆ

a+_Aα(P)and ˜a_Aα(P), ˜a+_Aα(P)in order to satisfy the conditions: 139

ˆ

γ_Aα(P;t)|$0_rel(t)ii=0, hh1|a˜+_Aα(P;t) =0,

˜

γ_Aα(P;t)|$0_rel(t)ii=0, hh1|γ˜+_Aα(P;t) =0.

(40)

To achieve this let us consider an action of annihilation superoperators ˆa_Aα(P;t), ˜a_Aα(P;t)on relevant 140

ˆ

a_Aα(P;t)|$0_rel(t0)ii = fAα(P;t−t0)a˜+Aα(P;t)|$

0

rel(t0)ii, (41) ˜

a_Aα(P;t)|$0_rel(t0)ii = σfAα(P;t−t0)aˆ

+

Aα(P;t)|$

0 rel(t0)ii,

where superoperators ˆa_Aα(p;t), ˆa+_Aα(p;t), ˜a_Aα(p;t), ˆa+_Aα(p;t)are in the Heisenberg representation 142

ˆ

a_Aα(P;t) =e−i¯1hH¯0t_aˆ

Aα(P)e

1

i¯hH¯0t_{, a˜}

Aα(P;t) =e

−1 i¯hH¯0t_a˜

Aα(P)e

1 i¯hH¯0t_, ˆ

a+_Aα(P;t) =e−i¯1hH¯0t_aˆ+

Aα(P)e

1

i¯hH¯0t_{, a˜}+

Aα(P;t) =e

−1 i¯hH¯0t_a˜+

Aα(P)e

1 i¯hH¯0t_,

and satisfy commutation relations: 143

h ˆ

a_Aα(P;t), ˆa+_Bβ(P0;t)i

σ

=δA,Bδα,βδ(P−P

0_),

h ˜

a_Aα(P;t), ˜a+_Bβ(P0;t)i

σ

=δA,Bδ_α,βδ(P−P

0 ),

h ˆ

a_Aα(P;t), ˜a_Bβ(P0;t)i

σ

=haˆ+_Aα(P;t), ˜a+_Bβ(P0;t)i

σ

=0.

It is necessary to note that superoperators ˆH(r), ˆn_Aα(x) are built on superoperators ˆa_Aα(p+ q₂), 144

ˆ

a+_Aα(p−q₂), ˜a_Aα(p+q₂), ˜a+_Aα(p− q₂). Therefore, for convenience here a unit denotion was introduced 145

for arguments likeP=p±q₂. This should be taken into account in further calculations where obvious 146

expressions are needed. 147

According to general relations of [61,62], we can introduce new operators ˆγ_Aα(P;t), ˜a+_Aα(P;t), 148

˜

γ_Aα(P;t), ˜γ+_Aα(P;t)via superoperators ˆa_Aα(P;t), ˆa+_Aα(P;t), ˜a_Aα(P;t), ˜a+_Aα(P;t): 149

ˆ

γ_Aα(P;t) =

q

1+σn_Aα(P;t,t0) "

ˆ

a_Aα(P;t)− nAα(P;t,t0)

1+σn_Aα(P;t,t0) ˜

a+_Aα(P;t)

# ,

˜

γ+_Aα(P;t) = q

1+σn_Aα(P;t,t0)

˜

a_A+_α(P;t)−σaˆ_Aα(P;t). (42)

Relations (42) satisfy conditions (40). Here 150

n_Aα(p,q;t,t0) =nAα(P;t,t0) =hh1|a˜

+

Aα(P;t)a˜Aα(P;t)|$

0 rel(t0)ii

=hh1|a˜+_Aα(p−q

2;t)a˜Aα(p+ q 2;t)|$

0 rel(t0)ii,

is a relevant distribution function ofA-particle coupled states in momentum spacep,q, which is 151

calculated with the help of relevant thermo vacuum state vector$0_rel(t0)ii(37). Function fAα(P;t−t0) 152

in formulae (41) is connected withnAα(P;t,t0)by the relation 153

fAα(P;t−t0) =

nAα(P;t,t0)

1+σnAα(P;t,t0) .

Superoperators ˆγ_Aα(P;t)and ˜γ_Aα(P;t), ˜a+_Aα(P;t)and ˜γ+_Aα(P;t)satisfy the “canonical” commutation 154

relations: 155

h ˆ

γ_Aα(P;t), ˜a+_Bβ(P0;t) i

σ

=δA,Bδα,βδ(P−P

0_),

h ˜

γ_Aα(P;t), ˜γ+_Bβ(P0;t) i

σ

=δA,Bδ_α,βδ(P−P

0 ), h

ˆ

γ_Aα(P;t), ˜γ_Bβ(P0;t) i

σ

=ha˜+_Aα(P;t), ˜γ+_Bβ(P0;t) i

σ

=0.

Inversed transformations to superoperators ˆa_Aα(P;t), ˜a+_Aα(P;t)are easily obtained from (42): 156

ˆ

a_Aα(P;t) =

q

1+σn_Aα(P;t,t0) "

ˆ

γ_Aα(P;t) + nAα(P;t,t0)

1+σn_Aα(P;t,t0) ˜

γ+_Aα(P;t) #

,

˜

a+_Aα(P;t) =

q

1+σn_Aα(P;t,t0)γ˜+_Aα(P;t) +σγˆ_Aα(P;t). (44)

ˆ

γ_Aα(P;t), ˜a+_Aα(P;t), ˜γ_Aα(P;t), ˜γ+_Aα(P;t)could be defined as some operators of annihilation and creation 157

ofA-quasiparticle coupled states, for which relevant thermo vacuum state|$0_rel(t0)ii(37) is a vacuum 158

state. In such a way, we obtained relations of dynamical reflection of superoperators ˆa_Aα(P;t), ˆa+_Aα(P;t), 159

˜

a_Aα(P;t), ˜a+_Aα(P;t)to new superoperators of “quasiparticles” ˆγ_Aα(P;t), ˜a+_Aα(P;t), ˜γ_Aα(P;t), ˜γ+_Aα(P;t). 160

A set of transport equations (26), (27) together with dynamical reflections (42), (44) of 161

superoperators in the thermo field space constitute the basis for a consistent description of the kinetics 162

and hydrodynamics of a dense quantum system with strongly coupled states. Both strongly and 163

weakly nonequilibrium processes of a nuclear matter can be investigated using this approach, in which 164

the particle interaction is characterized by strongly coupled states, taking into account theirs nuclear 165

nature [1–4]. 166

Weakly nonequilibrium processes can be described when the fluctuations of the parameters 167

δβ(~r;t) =β(~r;t)−β,δµAα(x;t) =µAα(x;t)−µAαare small, whereβandµAαare equilibrium values 168

for temperature and chemical potential respectively In this case, the system of equations (26), (27) 169

will have a similar structure, but is closed with respect tohh1|δnˆ_Aα(x)|$(t)ii,hh1|δHˆ(r)|$(t)ii, where 170

δnˆ_Aα(x) =nˆ_Aα(x)− hh1|nˆ_Aα(x)|$0ii,δHˆ(r) =Hˆ(r)− hh1|Hˆ(r)|$0ii,|$0iiis the equilibrium thermo 171

vacuum state vector of the systems. 172

In addition, by designing a system of equations on moments 1,Pof distribution function we 173

obtain, respectively, the equation of the thermo field hydrodynamic for the dense quantum - field 174

systems. 175

These questions require separate consideration and will be investigated in future work. 176

4. Conclusions 177

We generalized the nonequilibrium thermo field dynamics in the frames of Zubarev’s 178

nonequilibrium statistical operator method [61] within the framework of Reniy statistics. Based 179

on this approach and Gibbs statistics the generalized equations of consistent description of kinetics 180

and hydrodynamics for dense quantum field systems with strongly bound states were obtained. Using 181

this approach, one can investigate both strongly and slightly nonequilibrium processes of nuclear 182

matter, when the interaction between particles of the latter is characterized by strongly bound states of 183

internucleon nature [2,3]. 184

185

1. Negele, J.W. The mean field theory of nuclear structure and dynamics.Rev. Mod. Phys.1982, 54, 913-1015. 186

2. Toneev, V.D.; Shultz, H.; Gudima,K.K.; Röpke G. Towards study of hot and dense nuclear matter in heavy ion 187

collisions.Physics of Elementary Particles and Nuclei (Particles & Nuclei)1986, 17, 1093 (in Russian). 188

3. Röpke, G.; H.Shultz, H.; Gudima, K.K.; Toneev V.D. Dynamical approaches to heavy-ion collision in 189

intermediate energy region. Physics of Elementary Particles and Nuclei (Particles & Nuclei)1990, 21, 364 (in 190

Russian). 191

4. McLerran, L. The physics of the quark-gluon plasma.Rev. Mod. Phys.1986, 58, 1021-1064. 192

5. Blättel, B.; Koch, V.; Mosel, U. Transport-theoretical analysis of relativistic heavy-ion collisions. Rep. on Progr. 193

in Phys.1993, 56, 1-62. 194

6. Mrówczy ´nsky, S.In: Quark-qluon plasma.World Scientific, Singapore,1990. 195

7. Cleymans, J.; Gavai, R.V.; Suhonen, E. Quarks and gluons at high temperatures and densities.Phys. Rep.1986, 196

9. McLerran, L. Probes of the quark-gluon plasma as it might be produced in ultra-relativistic nuclear collisions. 199

Acta Phys. Polon. B1985, 16, 669-682. 200

10. Shuryak, E.V. What RHIC experiments and theory tell us about properties of quark-gluon plasma?Nucl. Phys. 201

A2005, 750, 64. https://doi.org/10.1016/j.nuclphysa.2004.10.022. 202

11. Schuryak, E. Strongly coupled quark-gluon plasma in heavy ion collisions. Rev. Modern. Phys. 2017, 89, 203

035001(61). 204

12. Jaiswal, A.; Roy, V. Relativistic hydrodynamics in heavy-ion collisions: general aspects and recent 205

developments.Advan. High. Ener. Phys.2016, ID9623034, 39. 206

13. Florkowski, W.; Heller, M.P.; and Spalinski, M. New theories of relativistic hydrodynamics in the LHC era. 207

Report Progr. Phys.2018, 81, 046001. 208

14. Li, Y. A flow paradigm in heavy-ion collisions. arXiv: 1712.04580,2018, 45p. 209

15. R ¨opke, G. Nuclear matter EoS including few-nucleon correlations.Nuovo Cim. C2016, 39, 392. 210

16. R ¨opke, G. Correlations and clustering in dilute matter. Chap. 2.Nuclear Particle Correlations and Cluster Physics, 211

(Edit. Schroder Wolf-Udo, University of Rochester, USA), World Scientific,2017, 31. 212

17. Hirino, T.; van der Kolk, N.; and Bilandzic, A. Hydrodynamics and Flow.Lect. Notes Phys.2010, 785, 139. 213

DOI: 10.1007/978-3-642-02286-9_4. 214

18. Florkowski, W. Hydrodynamic description of ultrarelativistic heavy-ion collisions. arXiv: 1712.05162,2017, 215

25p. 216

19. R ¨opke, G.; Shlomo, S.; at all. Density determinations in heavy ion collisions.Phys. Rev. C2013, 88, 024609. 217

20. Hempel, M.; Hagel, K.; Natowitz, J.; R ¨opke, G.; and Typel, S. Constraining supernova equations of state with 218

equilibrium constants from heavy-ion collisions.Phys. Rev. C2015, 91, 045805. 219

21. Bastian, N-U.F.; Blaschke, D.; Fischer, T.; and R ¨opke, G. Nowards a Unified Quark-Hadron-Matter Equation 220

of State for Applications in Astrophysics and Heavy-Ion Collisions.Universe2018, 4, 67. 221

22. Elze,H.-Th.; Heinz, U. Quark-gluon transport theory.Phys. Rep.1989, 183 82. 222

23. Lee, T.D. The strongly interacting quark-gluon plasma and future physics. Nucl. Phys. A2005, 750, 1. 223

https://doi.org/10.1016/j.nuclphysa.2004.11.003. 224

24. Gyulassy, M.; and I. McLerran, I. New forms of QCD matter discovered at RHIC.Nucl. Phys. A2005, 750, 30. 225

https://doi.org/10.1016/j.nuclphysa.2004.10.034. 226

25. Romatschke, P. Do nuclear collisions create a locally equilibrated quark-gluon plasma?Eur. Phys. J.2017, 77. 227

21. 228

26. Romatschke, P. Relativistic Fluid Dynamics Far From Local Equilibrium.Phys. Rev. Lett.,2018120, 012301(5). 229

27. Baier, R.; Romatschke, P.; and Son, D.T.; Starinets, A.O.; Stephanov, M.A. Relativistic viscous hydrodynamics, 230

conformal invariance, and holography.J. High Energy Phys.2008, 4, 100. 231

28. Zubarev, D.N.; Morozov, V.G. Formulation of boundary conditions for the BBGRY hierarchy with allowance 232

for local conservation laws.Theor. Math. Phys.1984, 60, 814. DOI:10.1007/BF01018982. 233

29. Zubarev, D.N.; Morozov, V.G.; Omelyan, I.P.; Tokarchuk, M.V. Kinetic equations for dense gases and liquids. 234

Theor. Math. Phys.1991, 87, 412. DOI:10.1007/BF01018982. 235

DOI:10.1007/BF01016582 236

30. Zubarev, D.N.; Morozov, V.G.; Omelyan, I.P.; Tokarchuk, M.V. Unification of the kinetic and hydrodynamics 237

approaches in theory of dense gases and liquids.Theor. Math. Phys.1993,96, 997. 238

DOI:10.1007/BF01019063. 239

31. Markiv, B.; Omelyan, I.; Tokarchuk, M. Consistent Description of Kinetics and Hydrodynamics of Weakly 240

Nonequilibrium Processes in Simple Liquids. J. Stat. Phys.2014, 155, 843. 241

DOI:10.1007/s10955-014-0980-4. 242

32. Biro, T.S.; and Molnar, E. Fluids dynamical equations and transport coefficients of relativistic gases with 243

non-extensive statistics.Phys. Rev. C2012, 85, 024905. 244

33. Shen, k-M.; Biro, T.S.; Wang, En-Ke. Different non-extensive models for heavy-ion collisions.Physica A2018, 245

492, 2353. 246

34. Osada, T.; and Wilk,G. Nonextensive/Dissipative Correspondence in Relativistic Hydrodynamics. Progr. 247

Theor. Phys. Suppl.2008, 174, 168-172. 248

35. Osada, T.; and Wilk,G. Erratum: Nonextensive hydrodynamics for relativistic heavy-ion collisions.Phys. Rev. 250

C2008, 77, 044903. https://doi.org/10.1103/PhysRevC.78.069903 251

36. Osada, T.; and Wilk,G. Nonextensive perfect hydrodynamics – a model of dissipative relativistic 252

hydrodynamics?Cent. Eur. J. Phys.2009, 7, 432-443. DOI:10.2478/s11534-008-0163-5. 253

37. Osada, T. Relativistic hydrodynamical model in the presence of long-range correlations.Phys. Rev. C2010, 81, 254

024907. https://doi.org/10.1103/PhysRevC.81.024907. 255

38. Lavagno, A. Relativistic nonextensive thermodynamics. Phys. Lett. A 2002, 301, 13-18. 256

http://doi.org/10.1016/S0375-9601(02)00964-7. 257

39. Gianpiero,G.; Lavagno, A.; and Pigato, D. Nonextensive statistical effects in the quark-gluon plasma formation 258

at relativistic heavy-ion collisions energies.Cent. Eur. J. Phys.2012, 10, 594-601. 259

https://doi.org/10.2478/s11534-011-0123-3. 260

40. Lavagno, A.; Pigato, D. Nonextensive statistical effects and strangeness production in hot and dense nuclear 261

matter.J. Phys. G: Nucl. Part.2012, 39, 125106. 262

DOI: 10.1088/0954-3899/39/12/125106. 263

41. Lavagno, A.; Pigato, D. Nonextensive nuclear liquid-gas phase transition.Physica. A2013, 392, 5164-5171. 264

http://doi.org/10.1016/j.physa.2013.06.048. 265

42. Zubarev, D.N.; Prozorkevich, A.V.; Smolyanskii, S.A. Derivation of nonlinear generalized equations of 266

quantum relativistic hydrodynamics.Theor. Math. Phys.1979, 40, 821-831. 267

43. Prozorkevich, A.V.; Samorodov, V. L.; Smolyanskii, S. A. Quantum relativistic hydrodynamics of systems 268

with broken symmetry. I. Local-equilibrium state.Theor. Math. Phys.1982, 52, 920-926. 269

44. Smolyanskii, S. A. Quantum relativistic hydrodynamics of systems with broken symmetry II. Nonequilibrium 270

state.Theor. Math. Phys.1982, 52, 809-814. 271

45. Hosoya, A.; Sakagami,M.A.; Takao,M. Nonequilibrium thermodynamics in field theory: Transport coefficients. 272

Ann. Phys.1984, 154, 229-252. 273

46. Becattini, F.; and Tinti, L. Nonequilibrium thermodynamical inequivalence of quantum stress-energy and 274

spin tensors.Phys. Rev. D2013, 87, 025029. 275

https://doi.org/10.1103/PhysRevD.87.025029. 276

47. Tinti, L.Progress in Mathematical Relativity, Gravitation and Cosmology. Springer Berlin Heidelberg,2014. 277

48. Becattini, F.; Bucciantini, L.; Grossi, E.; Tinti, L. Local thermodynamical equilibrium and theβframe for a 278

quantum relativistic fluid.Eur. Phys. J. C2015, 75, 191. 279

DOI:10.1140/epjc/s10052-015-3384-y. 280

49. Hayata, T.; Hidaka, Y.; and Nomi, T.; Hongo, M. Relativistic hydrodynamics from quantum field theory on 281

the basis of the generalized Gibbs ensemble method.Phys. Rev. D2015, 92, 065008(1-14). 282

50. Harutyunyan, A.; Sedrakian, A.; and Rischke, D.H. Relativistic Dissipative Fluid Dynamics from the 283

Non-Equilibrium Statistical Operator.Particles2018, 1, 155-165. 284

51. Harutyunyan, A.; Sedrakian, A.Bulk Viscosity of the Hot Quark Plasma from from the Non-Equilibrium 285

Statistical Operator.Particles2018, 1, 212-229. 286

52. Koide, T.; and Kodama, T. Transport coefficient of non-Newtonian fluid and causal dissipative hydrodynamis. 287

Phys. Rev. E2008, 78, 051107(1-11). 288

53. Huang, Xu-Gu; Koide, T. Shear viscosity, bulk viscosity, and relaxation times of causal dissipative relativistic 289

fluid-dynamics at finite temperature and chemical potential.Nucl. Phys. A2013, 889, 73-92. 290

54. Morozov, V.G.; Ropke, G.; Holl, A. Kinetic Theory of Quantum Electrodynamic Plasma in a Strong 291

Electromagnetic Field: I. The Covariant Formalism.Theor. Math. Phys.2002, 131, 812-831. 292

55. Morozov, V.G.; Ropke, G.; Holl, A. Kinetic Theory of Quantum Electrodynamic Plasma in a Strong 293

Electromagnetic Field: II. The Covariant Mean-Field Approximation.Theor. Math. Phys.2002, 132, 1029-1042. 294

56. Blaizot, J.-P.; Li, Yan. Onset of hydrodynamics for a quark-gluon plasma from the evolution of momentsof 295

distribution functions.J. High Energy Phys.2017, 11, 1-26. 296

57. Tinti, L.; Jaiswal, A.; Ryblewski, R. Quasiparticle second-order viscous hydrodynamics from kinetic theory. 297

Phys. Rev. D2017, 95, 054007(1-11). 298

58. Arimitsu, T.; Umezawa, H. A General Formulation of Nonequilibrium Thermo Field Dynamics. Progr. Theor. 299

Phys.1985, 74, 429-432. 300

https://doi.org/10.1143/PTP.77.32. 303

60. Arimitsu, T. A canonical formalism of dissipative quantum systems. Non-equilibrium thermofield dynamics. 304

Condens. Matter Phys.1994, 4, 26-88. DOI:10.5488/CMP.4.26. 305

61. Zubarev, D.N.; Tokarchuk, M.V. Nonequilibrium thermofield dynamics and the nonequilibrium statistical 306

operator method I. Basic relations.Theor. Math. Phys.1991, 88, 876-893. 307

62. Tokarchuk, M. V.; Arimitsu, T.; Kobryn, A.E. Thermo field hydrodynamic and kinetic equations of dense 308

quantum nuclear systems.Condens. Matter Phys.1998, 1(15), 605-642. 309

DOI:10.5488/CMP.1.3.605. 310

63. Glushak, P.A.; Markiv, B.B.; Tokarchuk, M.V. Zubarev’s Nonequilibrium Statistical Operator Method in the 311

Generalized Statistics of Multiparticle Systems.Theor. Math. Phys.2018, 194, 57-73. 312

64. Zubarev, D.N.Nonequilibrium Statistical Thermodynamics, Consultant Bureau, New-York,1974. 313

65. Zubarev, D.N.; Morozov, V.G.; and Ropke, G.Statistical mechanics of nonequilibrium processes, Akademie Verlag, 314

Berlin,1996, 1. 315

66. Zubarev, D.N.; Morozov, V.G.; and Ropke, G.Statistical mechanics of nonequilibrium processes, Vol. 2, Akademie 316

Verlag, Berlin,1997, 2. 317

67. Markiv, B.B.; Tokarchuk, R.M.; Kostrobij, P.P.; Tokarchuk, M.V. Nonequilibrium statistical operator method in 318

Renyi statistics.Physica A2011, 390, 785-791. 319

http://doi.org/10.1016/j.physa.2010.11.009. 320

68. Takahashi, Y.; and Umezawa, H. Thermo field dynamics.Collective Phenomena1975, 2, 55-80;Int. J. Mod. Phys. 321

B1976, 10, 1755-1805. http://dx.doi.org/10.1142/S0217979296000817. 322

69. Umezawa, H.; Matsumoto, H.; Tachiki, M.Thermo field dynamics and condensed states. North-Holland; 323