The functional generalization of the Boltzmann-Vlasov equation and its Gauge-like symmetry

I recommend for publication of this paper in SciPost Physics Journal after the questions / comments are addressed.

-The premise of the paper it interesting.

-It brings some alternative concepts in statistical mechanics that can be helpful in addressing the proposed problem

- The author is honest about the degree of speculation

-The paper is very speculative

- The regime of validity of the ideas that the author proposes is fully relegated to the Appendix, without a sufficient discussion in the main text;

-Multiple typos, incomplete phrases and not well defined notation, which when accumulated with the previous points, impair the understanding of the paper;

Dear Editorial Board and Author,

The present paper attempts to address the problem that hydrodynamics seems to work in systems with a small number of particles using a generalization of the Boltzmann equation, which considers the evolution of the single particle distribution function, that is now considered as a stochastic variable.

Among the expected acceptance criteria, I think this paper meets only the criterion of providing a link to new link between different research areas. Namely, some notions in non-equilibrium quantum field theory and hydrodynamics. It is indeed a possible way to regard the problem, but this not uncontroversial and to this point, extremely speculative.

I would not recommend the publication of this manuscript without major editing, since the accumulation of the various points below combined with the admitted high degree of speculation make it difficult to do otherwise. I apologize if the report is too long, but I think it is necessary, given that many unorthodox ideas are employed. I explain the points below. I hope they are clear enough.

1.1 Since the difference between the Gibbsian and the Boltzmannian notions of entropy is not usually discussed in this field and the papers [1,18] are by the same author, a few sentences about it could make the discussion more self-contained.

1.2 The current introduction has more than the context and the summary of achievements. It also has content that is part of the bulk of the paper. This contradicts general acceptance criteria of this publication.

2. I did not understand the content of the phrase (second paragraph of the introduction): "Yet no indication exists that if we somehow tightly selected for initial geometry, we would not have extra uncertainties due to dynamics, the sign of 'perfect' hydrodynamics ". I don't see the link between the "the sign of 'perfect' hydrodynamics " and the remainder of the phrase.

3. If I understand correctly, in Eq. (1) $\Lambda$ is a scale beyond which the equation would not be valid, but Eq. (4) is a completely different equation and yet it possesses the same parameter there. Are they really the same, or a different cut-off?

4. On the last phrase of the paragraph below eq. (3), was $f(x_1, x_2,...,f_{n})$ really intended or a typo? If not a typo what does it mean? Are there momenta here? This is not defined neither in the introduction nor in the Appendix.

5.0 It would be interesting to discuss the difference between this approach and the Wigner function approach. Wouldn't the 'function' BBGKY hierarchy of ref. [13] encompass these fluctuations? Is the functional formalism a way to take into accounts higher moments in the 'function' BBGKY?
5.1 Subscript missing in Eq. (5)? $\langle O \rangle_{f_{2}}$ instead of $\langle O \rangle$ ?
5.2 Would not there be a minus sign in the exponent of Eq. (6)?
5.3 Would not $\sigma_{f} \sim N_{DoF}^{-1/2}$?
5.4 How does the Vlasov term reduces to a derivative in momentum space as required for the limit to the 'usual' Boltzmann-Vlasov equation is recovered?
5.5 Doesn't the collision term assume some BBGKY-like truncation ? Wouldn't the smallness of the system break even this "functional molecular chaos"?
5.6 $\vert M \vert^{2}$ is defined before it is even mentioned, in eq. 10.
5.7 The integration measures $dx_{1,2}$ and $d^{3}[k_{1,2,3}]$ are not defined

6.1 In Eq. (11), shoudn't $\hat{C}$ be $\mathcal{C}$? Why are there hats in $\mathcal{C}$ and $\mathcal{V}$?
6.2 The text below Eq. (11) has an incomplete sentence that impairs the understanding. "Away from a full ensemble average,"

7.1 In the last paragraph of p.6 in the phrase "(...) we do not know if the volume cell is being moved by microscopic pressure (...) $\bf{and}$ a macroscopic force ..." should not the highlighted 'and' be an 'or' in a 'either' ... 'or' sentence?

7.2 In the last phrase of p. 6, does the author mean $\mathcal{V}[f]$ ($\mathcal{V}^{\mu}[f_1,f_2]$ ?) or $\langle \mathcal{V}^{\mu}[f_1,f_2] \rangle$ ? I would expect that the average should have some redundancy, not the operator itself.

7.3 In Fig. 1, are $\delta f(x)$ and $\delta f(p)$ marginal distributions of deviations from local equilibrium (e.g. $\delta f(p)$ is $\delta f(x,p)$ integrated over x)? Please, emphasize.

8.1 What is the definition of $\Delta_{i}^{\mu}$ and integration measure, $d[f'_{i_{1} j_{1}}]$, in Eq. 12 ?
8.2 Is not there a missing index in $\partial f/\partial p$ in Eq. 12? Wouldn't the specific index change the discussion that follows?
8.3 Usually in random matrix theory, it is assumed that the ensemble is invariant under similarity transformations $M \mapsto M' = U^{-1}M U$, where U is unitary, and Eq. 13 reminds me of that. How do we see that $\mathcal{C}$ and $\mathcal{V}$ have the correct properties so that transformation (13) is valid?
8.4 Since the phase space has been discretized, is not Lorentz invariance also discretized? If yes, it is worth emphasizing it.
8.5 Is $\{f_{i_1 j_{2}} f'_{i_2 j_{2}} \mathcal{V}_{i_1 i_{2}} \} \propto \langle x - x' \rangle^{-2}$ an assumption or a result from random matrix theory? Please, this should be made clear.
8.6 What is being extremized so that Lagrange multipliers are considered in Eq. (14)? Is it functional (6) or some entropy functional, that is not defined?
8.7 The non-linearity of Eq. 12 should lead to the rising of multiplicative noise, right? Is one of the assumptions that these effects are small, even classically?

9.1 In Ref. 33 $\rho(\lambda)$is the eigenvalue density function, what would be its relation with $f_{ij}$ in the present case, the probability density of eigenvalues of $f_{ij}$?
9.2 Since $J$ is related to the maximum (and the minimum) eigenvalues in the continuum part of the distribution, I would expect $J_p$ and $J_x$ in Eq. 16 be related to the cut-off $\Lambda$, why it is instead $\sim \langle x \rangle $ and $\sim \langle p \rangle $?

10. How can one see that the highly non-linear combination of $N_{x,p}$ and $J_{x,p}$ are small (since the author claims that the RHS of 16(?) is negligible) and lead to a 'relaxation time' much smaller than the relaxation time of the Boltzmann equation, which would lead collision term to grow? Is there a compensation between the collision and the Vlasov terms?

11. Should there not be indices $j$ in the momenta in Eq. 17?

12. Would the ensemble on step (i) in section III be created with a Metropolis-like algorithm?

13. Is the universality evoked in the second paragraph of the discussion section related to an attractor-like universality, in which the system 'forgets' the non-hydrodynamic modes and an free-streaming, 'asymptotically ideal' hydro emerges?

14. In sec. 2 of the appendix, is the BBGKY hierarchy referred the common BBGKY or the functional one? Could you elaborate on what is "non-perturbative" in this context?

15. The common semi-classical expansion of quantum kinetic theory is motivated by the smallness of the wave packet? What is the suggestion of the present functional semi-classical expansion? The locality of the wave-functional packet? How is that different?

Best regards,

Anonymous Referee

Dear editors and author,

Below I describe the changes I would request from the author:

A. Address item 1.1 in the report;

B. Address item 1.2 in the report;

B.1 I recommend that the author starts a new section after the paragraph
ending in "and placing it within the more conventional
transport theory have been left to the appendix", before equation (1).

B.2 The beginning of such new section the author should discuss, at least
briefly, in one paragraph, the regime of validity of the assumptions. I think
this cannot be fully relegated to the appendices. A summary of section 2 of
the appendix should be enough.

C. Address possibly incomplete/ambiguous phrases in items 2, 6.2, 7.1, 7.2, 7.3 of the Report;

D. Address possible typos in items 3,4, 5.1, 5.2, 5.3, 6.1, of the report;

E. Address the notation problems pointed in items 5.6, 5.7,8.1,8.2,11 of the report;

F. Address questions in items 5.0,5.5, 8.3, 8.4, 8.5, 8.6, 8.7, 9.1, 9.2, 9.3, 10, 11, 12, 13, 14, 15 of the report.

Best regards,

Anonymous Referee