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

by Giorgio Torrieri

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Submission summary

Authors (as registered SciPost users): Giorgio Torrieri
Submission information
Preprint Link: scipost_202401_00013v2  (pdf)
Date accepted: 2024-02-28
Date submitted: 2024-02-05 01:22
Submitted by: Torrieri, Giorgio
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Fluid Dynamics
  • High-Energy Physics - Theory
Approach: Theoretical


We argue that one can model deviations from the ensemble average in non-equilibrium statistical mechanics by promoting the Boltzmann equation to an equation in terms of {\em functionals} , representing possible candidates for phase space distributions inferred from a finite observed number of degrees of freedom. We find that, provided the collision term and the Vlasov drift term are both included, a gauge-like redundancy arises which does not go away even if the functional is narrow. We argue that this effect is linked to the gauge-like symmetry found in relativistic hydrodynamics \cite{bdnk} and that it could be part of the explanation for the apparent fluid-like behavior in small systems in hadronic collisions and other strongly-coupled small systems\cite{zajc}. When causality is omitted this problem can be look at via random matrix theory show, and we show that in such a case thermalization happens much more quickly than the Boltzmann equation would infer. We also sketch an algorithm to study this problem numerically

Author comments upon resubmission

Em sex., 2 de fev. de 2024 às 12:09, Giorgio Torrieri escreveu:

Dear Editorial Board and Author,

I thank the author for editing his material. Besides, I have some replies on my previous comments.

I thank the referee for their second report, which also helped greatly in improving the paper. It must be said that while the previous report made many very good points, it was not easy to separate the main scientific objections as opposed to stilystic concerns. So I must say that I did not properly appreciate the concern with the Gaussian ansatz evident in the following comment, which I understand is crucial for the referee's final assessment:

I think this decorrelation at the functional level is not trivial, given the system is admittedly strong coupled. This gaussian functional approximation is acceptable, but should be made explicit.

I have now expanded the text (see discussion around the new Eq. (13) highlighted in blue) and added relevant references (Jona-Lasinio's work), which I hope make it clear the Gaussian ansatz has nothing to do with the weak coupling limit.
Also, my answer was overly "negative" in describing how "heuristic" was the approach in this work and vague w.r.t. where exactly the "heuristics" arises. In this regard I would like to point out that the referee's previous comments:

  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?

Are very similar to the other referee's comments which prompted the additional final paragraph of section IIB. (Now highlighted in blue and extended).

I responded to the previous report that "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" But in fact the well-definedness of the Gaussian functional integral makes something similar to the ansatz described here unavoidable (this is extensively discussed in, for example, Kogan+Kovner's work), so, while this paper is speculative, it is less heuristic than the referee suggests:

The heuristic assumption is that a regime exists where the quasi-particle/"color glass" type assumptions fail, one has to consider functionals rather than correlation functions, but classical probability in lieu of quantum density matrices can still be used. Such a regime might not exist. But if it does, the Gaussian assumption is pretty much inevitable, as I now explain in detail.

The typos the referee points out are well-taken and have been corrected

However, Given the above discussion I would like to ask for a re-evaluation of the conclusion below

The author has surely improved the work and made interesting points in the reply to the comments. Besides, I have mentioned in the previous report that this paper could "provide a novel and synergetic link between different research areas". Nevertheless, the number of heuristic steps and conjectures (which are admited by the author) render me very doubtful to fit this work in the guidelines of Scipost Physics. Thus, I would recommend, also based in the corresponding guidelines, that it is published in SciPost Physics Community Reports after the above problems are addressed.

First of all I hope it is clear that something like the Gaussian approxmation is unavoidable on convergence grounds, and is unrelated to weak coupling. Secondly, in the revision of the work I point some non-trivial numerical tests which could give rise to much more quantitative follow-up work. So while this work is heuristic, it is in many ways an "educated guess" rather than a wild hypothesis, and this guess leads to falsifiable predictions. For this reason, I would like a reconsideration for the appropriateness of this work for scipost physics.

Still some typos remain, some of which impair the understanding:

1) In p. 3, 2nd paragraph : which phrase contains "the latter" and which contains "the former"? The difference is crucial.

A: "The ideal hydrodynamic limit is reached by imposing a further local equilibrium condition on the Boltzmann collision term".

This phrase might imply that the collision term is modified so that local equilibrium is enforced, but rather, the local equilibrium ~involves~ the collision term.

(p.6 1st paragraph) A:"close to it's neighborhood"?

Shouldn't it's be 'its'?

"killing vector" written multiple times without capitalization in the 'K'

All these were now corrected

List of changes

- New equation (13) and reference [36], with 1.5 pages of discussion
- Several points and discussions expanded and sharpened
All changes are highlighted in blue

Published as SciPost Phys. 16, 070 (2024)

Reports on this Submission

Anonymous Report 1 on 2024-2-21 (Invited Report)


Dear Author and Editors,

After re-reading the author's changes and the addition of discussions, I reconsider my points and I recommend publication.

Best regards,

Anonymous referee

  • validity: ok
  • significance: ok
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: good

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