arxiv: 2604.12691 · v1 · submitted 2026-04-14 · ✦ hep-ph · nucl-th

Recognition: unknown

Hydrodynamic Initial Conditions in Small Systems from Proton Phase-Space Entropy

Gabriel Rabelo-Soares , Gojko Vujanovic , Giorgio Torrieri

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:50 UTC · model grok-4.3

classification ✦ hep-ph nucl-th

keywords Wehrl entropyhydrodynamic initial conditionsproton phase spacesmall collision systemsrelativistic hydrodynamicsentropy depositionpp and pA collisionscollective flow

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The pith

The Wehrl entropy from coarse-graining the proton's phase-space distribution supplies the entropy current required for relativistic hydrodynamics in small collision systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the mismatch between quantum descriptions of the proton, which are pure states with zero entropy, and the entropy current demanded by relativistic hydrodynamics for modeling collective flow in proton-proton and proton-nucleus collisions. It shows that coarse-graining the proton phase-space wave function using a Wehrl-like entropy produces a positive semi-classical quantity that measures the density of accessible microstates at a given scale. This quantity serves directly as the initial entropy deposition. A reader would care because it supplies a first-principles link from proton structure to fluid initial conditions, reducing reliance on ad-hoc parametrizations when interpreting flow data from small systems.

Core claim

The appropriate matching between proton wave function and classical hydrodynamics emerges from the coarse-graining of its phase-space distribution quantified by the Wehrl-like entropy. This entropy provides a semi-classical, positive-definite measure of the density of accessible microstates at a given resolution scale, and therefore constitutes the appropriate quantity to characterize entropy deposition in small collision systems.

What carries the argument

The Wehrl-like entropy obtained by coarse-graining the proton phase-space distribution, serving as a positive semi-classical entropy measure that matches the requirements of an entropy current in relativistic hydrodynamics.

Load-bearing premise

Coarse-graining the proton phase-space distribution with Wehrl entropy produces an entropy current that satisfies relativistic hydrodynamics without additional assumptions about resolution scale or mixing.

What would settle it

A explicit calculation in which the Wehrl entropy current violates positivity or conservation laws of relativistic hydrodynamics when applied to a solvable proton wave packet, such as a Gaussian state in a controlled boost-invariant setup.

read the original abstract

The experimental observation of collective behaviour in proton-proton and proton-nucleus collisions poses a fundamental theoretical question regarding the proper characterization of the initial state underlying hydrodynamic evolution. While relativistic hydrodynamics requires an initial condition (IC) characterized by an entropy current, corresponding to a maximally mixed state, the microscopic description of the proton is based on inherently quantum objects, that are projections of pure states. We show that the appropriate matching between proton wave function and classical hydrodynamics emerges from the coarse-graining of its phase-space distribution quantified by the Wehrl-like entropy. This entropy provides a semi-classical, positive-definite measure of the density of accessible microstates at a given resolution scale, and therefore constitutes the appropriate quantity to characterize entropy deposition in small collision systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proposes Wehrl entropy from proton phase-space coarse-graining as the source of hydrodynamic initial entropy in small systems, but the abstract supplies no derivation or check on the entropy current.

read the letter

The core idea is to coarse-grain the proton wave function into a Husimi distribution and use the Wehrl entropy S_W = -∫ Q ln Q to generate the initial entropy density for hydrodynamics in pp and pA collisions. This is meant to turn a pure quantum state into something that looks like a mixed state suitable for hydro initial conditions. That is the main new angle, and it is a reasonable way to frame the problem that has bothered people working on small-system collectivity at the LHC. The paper does spell out why a positive semi-classical entropy measure is preferable to just using the quantum state directly, and that part is clear enough from the abstract alone. Credit for identifying the mismatch and pointing to a concrete tool from quantum optics and information theory. The soft spot is exactly the one in the stress-test note. Hydrodynamics needs a future-directed timelike entropy current s^μ with s^μ u_μ = s > 0, not just a scalar density. The Wehrl construction gives a scalar at a chosen smearing scale, so an extra rule is still required to pick the local rest frame and to fix the coherent-state width from the proton wave function. Nothing in the pure-state description selects that scale uniquely. The abstract does not show how they close this gap, and if the resolution parameter ends up being tuned, the claim that the entropy “emerges” loses force. The full text may address this, but on the available material the mapping remains under-determined. This is for people already working on initial-state modeling and entropy production in small systems. A reader who knows Wehrl entropy or phase-space methods in QCD will get the most out of it and can judge whether the current construction holds up. It is worth sending to a serious referee because the problem is central and the proposal is specific enough to test, even if the current version needs more on the four-current and the scale choice.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes that the initial entropy current for relativistic hydrodynamics in small systems (pp and pA collisions) can be obtained by coarse-graining the proton phase-space wave function via a Wehrl-like entropy constructed from the Husimi distribution Q. This entropy is presented as a semi-classical, positive-definite measure of accessible microstates at a chosen resolution scale, providing the appropriate matching between the pure quantum state of the proton and the mixed-state requirements of hydrodynamics, thereby characterizing entropy deposition without ad-hoc assumptions.

Significance. If the construction is shown to yield a valid hydrodynamic entropy current, the work would supply a principled, first-principles route to initial conditions in small-system hydrodynamics, addressing the theoretical tension between quantum proton descriptions and collective flow observations. It could reduce reliance on phenomenological parametrizations and offer a falsifiable link between proton wave-function models and hydrodynamic evolution.

major comments (2)

[Derivation of the entropy current (likely §3 or equivalent)] The central claim requires that the Wehrl entropy directly supplies a relativistic entropy current s^μ satisfying the hydrodynamic constraints (future-directed timelike vector with s^μ u_μ = s > 0 and ∇_μ s^μ = 0 on the initial hypersurface). The manuscript must explicitly derive how the scalar S_W = −∫ Q ln Q is promoted to this four-current, including the choice of local rest frame and the fixing of the coherent-state width (resolution scale) from the proton wave function alone; the pure-state description does not uniquely select this scale, leaving the mapping under-determined unless an additional rule is introduced.
[Results or validation section] Table or figure showing the resulting initial entropy density (if present) should be accompanied by a demonstration that the constructed s^μ remains positive-definite and satisfies the ideal-hydro continuity equation without extra tuning; otherwise the claim that this is 'the appropriate quantity' for small-system ICs rests on an incomplete mapping.

minor comments (2)

[Introduction or §2] The notation for the Husimi distribution Q and the precise definition of the Wehrl entropy should be given in an explicit equation early in the text, with a reference to the original Wehrl construction for clarity.
[Numerical results] Any numerical examples or plots of the coarse-grained entropy should specify the numerical value chosen for the resolution scale and discuss its sensitivity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address each major comment below and will revise the manuscript to improve clarity and completeness.

read point-by-point responses

Referee: [Derivation of the entropy current (likely §3 or equivalent)] The central claim requires that the Wehrl entropy directly supplies a relativistic entropy current s^μ satisfying the hydrodynamic constraints (future-directed timelike vector with s^μ u_μ = s > 0 and ∇_μ s^μ = 0 on the initial hypersurface). The manuscript must explicitly derive how the scalar S_W = −∫ Q ln Q is promoted to this four-current, including the choice of local rest frame and the fixing of the coherent-state width (resolution scale) from the proton wave function alone; the pure-state description does not uniquely select this scale, leaving the mapping under-determined unless an additional rule is introduced.

Authors: We agree that the promotion from the scalar Wehrl entropy to the four-current requires more explicit steps. In the revised manuscript we will expand the relevant section to derive s^μ = s u^μ, where the entropy density s is obtained by normalizing S_W over the transverse area of the collision region and u^μ is the local four-velocity extracted from the first moment of the Husimi distribution Q. This choice automatically ensures s^μ is future-directed and timelike with s^μ u_μ = s > 0. The coherent-state width is fixed by the intrinsic transverse momentum scale present in the proton wave-function model (the saturation momentum Q_s that sets the resolution at which the phase-space distribution is sampled). We will state this rule explicitly and show that it is determined solely by the input proton model, thereby removing the under-determination. revision: yes
Referee: [Results or validation section] Table or figure showing the resulting initial entropy density (if present) should be accompanied by a demonstration that the constructed s^μ remains positive-definite and satisfies the ideal-hydro continuity equation without extra tuning; otherwise the claim that this is 'the appropriate quantity' for small-system ICs rests on an incomplete mapping.

Authors: We will add a new panel (or short subsection) in the results that explicitly verifies the properties of s^μ. Because the Husimi distribution Q is positive semi-definite by construction, the Wehrl entropy S_W is non-negative and the resulting s is positive for any non-trivial proton configuration. We will also demonstrate numerically that ∇_μ s^μ vanishes on the initial hypersurface for the ideal-hydro case (no source term is present at t=0 by definition of the initial condition). These checks will be performed on representative proton wave functions without additional tuning parameters. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external Wehrl entropy definition and standard hydro matching

full rationale

The provided abstract and skeptic summary describe a mapping from proton wavefunction via Wehrl coarse-graining to an entropy scalar, then to a hydrodynamic initial entropy current. No equation in the visible text reduces the output entropy current to a fitted parameter or self-citation by construction; the resolution scale is acknowledged as an input choice rather than derived from the target hydrodynamics. The central step is an ansatz that the Wehrl scalar supplies the required s, which is an external assumption rather than a tautology internal to the paper's own equations. No self-citation chain or renaming of a known result is exhibited in the given material.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on two domain assumptions about hydrodynamics and quantum proton states; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

domain assumption Relativistic hydrodynamics requires an initial condition characterized by an entropy current corresponding to a maximally mixed state.
Explicitly stated in the abstract as the requirement for hydrodynamic evolution.
domain assumption The microscopic description of the proton is based on inherently quantum objects that are projections of pure states.
Stated in the abstract as the basis for the proton wave function.

pith-pipeline@v0.9.0 · 5430 in / 1281 out tokens · 60091 ms · 2026-05-10T14:50:18.846722+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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