# Peer-Review Record

# Unification of Thermo Field Kinetic and Hydrodynamics Approaches in the Theory of Dense Quantum–Field Systems

*Reviewer 1:*Anonymous

*Reviewer 2:*Anonymous

*Reviewer 3:*Anonymous

*Reviewer 4:*Anonymous

**Round 1**

*Reviewer 1 Report*

The paper is interesting and gives some new insight on the whole problem of the description of bound states in the framework of Zubarev's non-equilibrium statistical operator approach. It seems rather fruitful to combine Zubarev's approach with the quantum thermofield approach due to Umezawa et al. Thus the formulation of the problem and its solution are quite reasonable.

Further, it seems also promisable to plug all this machinery in the surrounding of non-extensive Tsallis - Renyi statistics because the clusters of nucleon as well as quark-gluon are evidently not

extensive. Nevertheless, it is known that the Renyi statistics preserves the additivity property (whereas Tsallis does not). So it would be rather **advisory **(**but not obligatory!**) to point in the Conclusion the real merits of using this more complex statistics (comparing to Gibbs one). May be, it would be useful for the authors to consult with the review article Rudoi Yu.G. Generalized

Information Entropy and Non-Canonical Distribution in Statistical Mechanics, *Theor. and Math. Phys. *135: 1 (2003) 451-496 (but it is not necessary to cite it directly in the considered paper).

*Author Response*

Reply to Reviewer 1 is in file Response_Ref1.pdf.

Author Response File: Author Response.pdf

*Reviewer 2 Report*

The entropy definitions proposed by Tsallis and Renyi to describe non-equilibrium steady states introduce a parameter q that determines a degree of the system state deviation from thermodynamic equilibrium. In the general case its value is unknown. In the process of establishing this steady state the parameter* q* should evolve with time. It means that similar to the integral kernels in nonlocal thermodynamic relationships obtained by Zubarev, the value of q-parameter is a nonlocal functional of the process history and should depend on the imposed constraints and initial conditions. The more information was introduced into the system via imposed constraints and initial and boundary conditions, the farther the system state deviates from equilibrium. So, by using Renyi statistics without this information it is impossible to describe the system states far from equilibrium.

Comments for author File: Comments.pdf

*Author Response*

Reply is in file Response_Ref2.pdf

Author Response File: Author Response.pdf

*Reviewer 3 Report*

The paper is devoted to a consistent description of kinetic and

hydrodynamic processes in dense quantum systems with bound states,

within Zubarev’s method of the nonequilibrium statistical operator

(NSOM). Although the paper contains only a formal derivation

of rather involved transport equations, it can, in principle, serve as a

first step toward simpler and more tractable equations of motion

for treating specific problems. Therefore I will recommend the

publication of the manuscript once the following points are clarified:

1. The authors use the formalism of the so-called ``nonequilibrium

thermofield dynamics’’ instead of the more traditional

formulation of NSOM in terms of the density matrix. The authors

should explain (say, in Introduction) why it is advantageous to use the

``thermofield dynamics’’ formalism for the problems considered in the

paper.

2. A similar equation arises about the Renyi definition of the

nonequilibrium entropy used in the paper. Why this definition is

more adequate than the Gibbs definition in the context of the problems

under consideration?

3. The authors should add some comments in the last section about

calculation of the ``cores’’ (28) – (31) appearing in the resulting

transport equations.

Finally, the English style in the paper must be improved.

*Author Response*

Reply is in file Response_Ref3.pdf

Author Response File: Author Response.pdf

*Reviewer 4 Report*

The authors have applied Zubarev’s method of the nonequilibrium statistical operator in the thermofield formalism to derive the generalized transport equations for many-body quantum systems with arbitrarily strong interactions. A novel feature of the work is the derivation of transport equations with consideration of strongly bound states (clusters), which can be important for kinetic and hydrodynamic studies of the strongly interacting matter formed in the heavy-ion collision experiments. I recommend the manuscript to be published in “Particles”.

However, I would like to raise several questions and suggest some minor changes which could improve the paper.

a) Firstly, I would like the authors to explain briefly the meaning of some basic terms and conventions so that a reader who is not very familiar to the thermofield formalism would not have difficulties. In particular, I would like the following questions to be addressed:

1. What is the meaning of the double brackets denoting the states such as |1>>

or |\rho (t) >> ? Does this notation imply anything different from the common Dirac notation |1>?

2. What in general the terms “thermofield, thermovacuum” imply? What is the definition of the thermovacuum state? (why not just vacuum?)

3. Why A is supposed to be a superoperator in Eq. (1) if it acts on the states? According to the common definition, a superoperator is a transformation which acts on operators rather than states.

4. Why the local chemical potentials in Eq. (22) are considered to be also momentum-dependent rather than only space-dependent? In other words, why the matching conditions given by Eq. (24) are posed on the particle densities with given momentum and not on the integral densities?

b) Some technical remarks

1. I guess the creation and annihilation operators in the equation following the line 98 should carry index $B\beta$ as well, shouldn’t they? Also there is a problem in obtaining Eq. (34) from Eq. (17), it seems that the integral in Eq. (17) should be over “q” variable, consequently n(q) will become n(p).

2. The integration over the 3-space coordinate r in Eqs. (34) and (36) is duplicated.

3․ There is a mistake in the formula after the line 126: the second exponent in the square brackets should not contain the operator B, i.e., one should replace $\exp{-\tau(A+B)}$ → $\exp{-\tau A}$. One can easily check this, for instance, by considering a particular case when B = -A.

4. The operatores $n_A\alpha(x)$ are duplicated in Eq. (37), thus, they should be removed either from Eq. (37), or from the definition of $b_A\alpha(x,t)$ introduced after the equation.

5. I wonder why the new set of creation and annihilation operators defined in the line 138 again contains $\tilde{a}^+_{A\alpha}$ instead of $\hat{\gamma}^+_{A\alpha}$. Please check this also in the next two pages.

6. I think some phrases and descriptions do not sound properly, such as

“coupled state” → maybe “bound state” ?

“temperature quantum-field system” → maybe “thermal quantum-field system” ?

“nucleon interaction” → maybe “nuclear interaction” or “nucleon-nucleon interaction” ?

c) I noticed a few linguistic or typing errors as well, which are needed to be corrected as follows:

line 13, consider → considered

line 18, function are taken → function is taken

line 29, based on the the → based on the

line 109, correspondingly → respectively

line 136, Relevant (relevant) → Relevant (???)

line 168, \beta(\vec r, t) → \beta(\bm r, t) (please use the same notation for vectors everywhere)

line 169, … respectively In this case … → … respectively. In this case … (add “.”)

line 178, in the frames → in the framework

line 182, slightly → weakly

d) Finally, the references are not listed in the right order as they appear in the text (e.g., the first references which appear in the text are numbered as [15,16]).

Comments for author File: Comments.pdf

*Author Response*

Reply is in file Response_Ref4.pdf

Author Response File: Author Response.pdf

**Round 2**

*Reviewer 4 Report*

The revised version of the manuscript provides the necessary background on the methods of the thermo-field dynamics and the nonequilibrium statistical operator used in the paper. This work can be of special importance for kinetic/hydrodynamic description of strongly correlated nuclear matter or quark-gluon plasma which appear in the heavy-ion collisions or in the cores of compact stars. I find the manuscript appropriate for publication in Particles.

P.s. Note that there is still an inaccuracy in the lines 169, 179, 185, 188, 191, 193, 221, where

$ \tilde{\gamma}^+_{A\alpha} $ should be replaced by $ \hat{\gamma}^+_{A\alpha} $, and in Eqs. (46) and (49), where $ \tilde{a}^+_{A\alpha} $ should be replaced by $ \hat{\gamma}^+_{A\alpha} $.

I also noticed that the reference [22] is still missing from the text.

*Author Response*

Reply is in file Response_Ref4_v2.pdf

Author Response File: Author Response.pdf

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