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The functional generalization of the BoltzmannVlasov equation and its Gaugelike 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:  20240228 
Date submitted:  20240205 01:22 
Submitted by:  Torrieri, Giorgio 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We argue that one can model deviations from the ensemble average in nonequilibrium 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 gaugelike redundancy arises which does not go away even if the functional is narrow. We argue that this effect is linked to the gaugelike symmetry found in relativistic hydrodynamics \cite{bdnk} and that it could be part of the explanation for the apparent fluidlike behavior in small systems in hadronic collisions and other stronglycoupled 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 torrieri@unicamp.br 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 (JonaLasinio'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:
 The common semiclassical expansion of quantum kinetic theory is motivated by the smallness of the wave packet? What is the suggestion of the present functional semiclassical expansion? The locality of the wavefunctional 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 welldefinedness 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 quasiparticle/"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 welltaken and have been corrected
However, Given the above discussion I would like to ask for a reevaluation 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 nontrivial numerical tests which could give rise to much more quantitative followup 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)