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

I thank the referees for their careful exhaustive reading which pointed out a lot of typos and bad editing, for which I also apologize.

Regarding the second referee report, I modified the text and added references in a way that I hope addresses their concerns. One issue is that the concept of Wilsonian renormalization was discussed in a somewhat misleading way in the text, and this has now been reworded.

The first referee report is much longer,and some additional comments are necessary in the reply. As part of their objections, they asked for a reorganization of the paper in a way that moves some motivating text from the appendix to the introduction. As a result a lot of the equation numbering got significantly changed, and I apologize for the confusion caused by this.

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.

A lot of the referee's points are well taken and I thank them for the careful reading and the comments, which help us improve the manuscript.

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.

We have added some discussion.

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.

As a result of this report we changed the layout of the paper, moving and editing the appendices into the introduction

1. 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.

The main idea of this was discussed in the paper https://arxiv.org/abs/nucl-th/0703031 (Now cited) Basically if microscopic physics is random one expects extra dynamical fluctuations beyond initial condition fluctuations proportional to the scale separation. So far no sign of this was ever found

1. If I understand correctly, in Eq. (1) ΛΛ 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?

It is a cut-off common to both equations.
Usually in (1) it is assumed to be the mass. We now clarified this. Note that the LHS of eq (4) and (1) is indeed identical, since in both cases the equation is about the evolution of f

1. On the last phrase of the paragraph below eq. (3), was f(x1,x2,...,fn)f(x1,x2,...,fn) 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.

It is a typo, now fixed.

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?

This is exactly the point. Functionals are a way to take into account all correlations without truncating. This work is trying to construct the limit of Wigner functionals (previously discussed in Mrowczynski and Muller,referenced) with h--->0. On the other hand Wigner functions of 1,2,...n particles also truncate (but keep the h expansion). The figure in the appendix, now moved to the introduction, makes this point.

5.1 Subscript missing in Eq. (5)? ⟨O⟩f2⟨O⟩f2 instead of ⟨O⟩⟨O⟩ ? 5.2 Would not there be a minus sign in the exponent of Eq. (6)? 5.3 Would not σf∼N−1/2DoFσf∼NDoF−1/2?

All of these are correct and have been fixed.

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?

The derivative only acts on f_1, all that happens is that there is an "ensemble" of configurations and the force on f_1 is an ensemble average. I believe this to be correct in the limit we focus on f_1 "as an observable". This is now mentioned in the text

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"?

I do not believe it does. This question is equivalent to the justification of a Gaussian functional for f'(x,p) even if more than 2-particle interactions are relevant. Renormalizeable lagrangians contain a maximum of 2--->2 reducible fundamental processes. The limit of h--->0 means that quantum fluctuations are small w.r.t. energy contained in a characteristic length scale. Thus, the Vlasov term contains a maximum of 2<---->2 potentials. Since the collision term is the UV completion of the Vlasov term, higher order processes should also be ignored there. We have added a comment on this issue in the text.

The only truncation left is classicality, lack of interference between processes, which is part of the h--->0 limit

5.6 |M|2|M|2 is defined before it is even mentioned, in eq. 10. 5.7 The integration measures dx1,2dx1,2 and d3[k1,2,3]d3[k1,2,3] are not defined

These were now shuffled around and clarified. The measure should be the same as in the Boltzmann equation.

6.1 In Eq. (11), shoudn't ^CC^ be CC? Why are there hats in CC and VV?

We have clarified this part.

6.2 The text below Eq. (11) has an incomplete sentence that impairs the understanding. "Away from a full ensemble average,"

This was an old paragraph I forgot to comment out, fixed.

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 (...) andand a macroscopic force ..." should not the highlighted 'and' be an 'or' in a 'either' ... 'or' sentence?

Yes, this was a misprint, fixed.

7.2 In the last phrase of p. 6, does the author mean V[f]V[f] (Vμ[f1,f2]Vμ[f1,f2] ?) or ⟨Vμ[f1,f2]⟩⟨Vμ[f1,f2]⟩ ? I would expect that the average should have some redundancy, not the operator itself.

Yes I have modified this discussion for clarity

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

This was now clarified in the caption

8.1 What is the definition of ΔμiΔiμ and integration measure, d[f′i1j1]d[fi1j1′], in Eq. 12 ? 8.2 Is not there a missing index in ∂f/∂p∂f/∂p in Eq. 12? Wouldn't the specific index change the discussion that follows?

These points were now clarified.

8.3 Usually in random matrix theory, it is assumed that the ensemble is invariant under similarity transformations M↦M′=U−1MUM↦M′=U−1MU, where U is unitary, and Eq. 13 reminds me of that. How do we see that CC and VV have the correct properties so that transformation (13) is valid?

Very good question, I believe so and have added a discussion of this point. I believe these transformations are related to the "gauge" symmetry under question.

8.4 Since the phase space has been discretized, is not Lorentz invariance also discretized? If yes, it is worth emphasizing it.

Yes, this is a non-relativistic limit. In the text I emphasized that causality is violated by the randomness of the matrices. I will also add Lorentz invariance is violated.

8.5 Is {fi1j2f′i2j2Vi1i2}∝⟨x−x′⟩−2{fi1j2fi2j2′Vi1i2}∝⟨x−x′⟩−2 an assumption or a result from random matrix theory? Please, this should be made clear.

The discussion has been improved. THis is a consequence of the assumption of a central force between DoFs which of course is not always valid (e.g. magnetic fields).

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?

A good point which highlights the limitations of this work. There is no straight-forward equivalent of the H-theorem, bar the fact that the likely outcome corresponds to local hydrodynamic equilibrium. See also the new section IVA which provides a link between the Crooks fluctuation theorem and the approach discussed here.

9.2 Since JJ is related to the maximum (and the minimum) eigenvalues in the continuum part of the distribution, I would expect JpJp and JxJx in Eq. 16 be related to the cut-off ΛΛ, why it is instead ∼⟨x⟩∼⟨x⟩ and ∼⟨p⟩∼⟨p⟩?

The referee is correct here.

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 ρ(λ)ρ(λ)is the eigenvalue density function, what would be its relation with fijfij in the present case, the probability density of eigenvalues of fijfij?

1. How can one see that the highly non-linear combination of Nx,pNx,p and Jx,pJx,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?

All three questions are very good and at the moment I do not have an exhaustive answer. I think section IVA gives a way to interpret these random matrices in a way that is similar to usual thermodynamics (via the fact that Random matrices ar also systems of equations with random linear coefficients). However, as everything in this work, it is speculative and conjectural.

1. Should there not be indices jj in the momenta in Eq. 17?

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

Yes. This was now emphasized

1. 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?

I do not think so. It is more like a Bayesian "inverse attractor" discussed in reference [1] where every system "looks like" the universal system when only a few parameters are sampled. I clarified this issue in the discussion now.

1. 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?

In a perturbation expansion usually n-point functions correspond to n-4 expansions in the coupling constant. This is somewhat of a sloppy argument and it is broken in many ways; In particular close to equilibrium resummations and thermal divergences show that orders get mixed up. Nevertheless, within a many body approach out of equilibrium it makes sense to think that higher order terms in the BBGKY hyerarchy would also correpond to a higher order perturbative expansion. Functional methods tend to be outside the perturbative expansion (see the paper by Kogan and co-authors cited) so it makes sense that the same thing is true for the BBGKY hyerarchy. We expanded this point in the text.

1. 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?

The smallness of the wavepacket in semi-classical kinetic theory is needed to ensure sequential scatterings. Hence, one needs the wavepacket to be small w.r.t. the scattering length. In the approach taken here the wavepacket and the scattering length can be of comparable size, so there is no scattering length. However, probabilities are "classical" (information is carried by a phase space distribution, not a density matrix or a Wigner function).
I agree this is a bit of a heuristic assumption, in fact that such a regime exists is the most speculative assumption of the paper, motivated mostly by the fact that "it might work". But within quantum chaos/the Eigenstate thermalization hypothesis it could make sense.

Best regards,

Anonymous Referee

Requested changes

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

-Reorganization of the paper, elimination of the appendices and putting the material there in an introductory section.
- Rewriting and expansion of the discussion via points suggested by the referees.