⚛️ nuclear theory

Hydrodynamical Initial State in Small Systems From the Phase-Space Entropy

This paper proposes that the appropriate initial condition for hydrodynamic evolution in small collision systems is characterized by the Wehrl entropy derived from the coarse-grained Husimi distribution of the partonic Wigner function, thereby bridging the gap between the quantum nature of hadronic structure and the classical requirements of hydrodynamics.

Original authors: Gabriel Rabelo-Soares, Gojko Vujanovic, Giorgio Torrieri

Published 2026-09-02

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CC BY 4.0

Original authors: Gabriel Rabelo-Soares, Gojko Vujanovic, Giorgio Torrieri

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For decades, physicists have been smashing heavy atomic nuclei together at nearly the speed of light to recreate a state of matter that existed just moments after the Big Bang. This fiery soup, known as the quark-gluon plasma, behaves less like a chaotic gas and more like a perfect, frictionless fluid. To understand how this fluid forms, scientists use a powerful mathematical tool called hydrodynamics, which describes how fluids flow. However, this tool has a strict requirement: it needs a starting point defined by entropy, a measure of how many different ways the tiny particles inside can be arranged. This works beautifully for collisions between large nuclei, where there are enough particles to smooth out the details. But recently, experiments have shown that even when smashing together much smaller objects, like single protons, this fluid-like behavior still appears. This creates a puzzle: a single proton is a tiny, rigid quantum object, described by a pure, unblurred state of matter, while hydrodynamics requires a messy, mixed-up starting point. The question has been how to bridge the gap between the rigid quantum world of a single proton and the fluid world of hydrodynamics.

A team of researchers has proposed a solution to this puzzle by changing how we look at the proton's internal structure. Instead of trying to force the proton to fit into the fluid model directly, they suggest that the fluid behavior emerges when we deliberately blur our view of the proton. In the quantum world, particles are described by wave functions that contain perfect information about position and momentum, but this information is so precise that it cannot be used to calculate the "messiness" or entropy needed for fluid dynamics. The researchers argue that to get the right starting conditions for the fluid, we must apply a specific kind of blur, or coarse-graining, to the proton's internal map. This process is similar to looking at a high-resolution photograph through a slightly frosted glass; the sharp, individual details of the quantum waves disappear, replaced by a smooth, positive distribution that looks like a statistical mixture.

By applying this blurring technique, the team found that the proton's internal structure transforms into a new type of distribution that naturally possesses the entropy required for hydrodynamics. They calculated this new entropy, which they call Wehrl entropy, and showed that it arises not from the fundamental quantum entanglement of the particles, but from the loss of information caused by the blurring process itself. This is a crucial distinction because it means the fluid behavior in these tiny collisions is a result of how the system is observed and resolved, rather than a hidden property of the proton's ground state. The researchers demonstrated that this entropy density can be directly linked to the number of particles produced in the collision, providing a clear, calculable starting point for the fluid equations.

The paper explicitly argues against the idea that standard maps of the proton's internal parts, which are often used to describe collisions, are sufficient to explain this phenomenon. Those standard maps describe the proton as a pure quantum state, which mathematically has zero entropy and cannot serve as a starting point for a fluid. The authors show that relying on these unblurred maps leads to contradictions and fails to capture the collective behavior seen in experiments. Instead, they prove that only by acknowledging the finite resolution of our measurements and the resulting loss of quantum detail can we derive a valid entropy that drives the fluid evolution. This approach suggests that the transition from a quantum particle to a classical fluid is a process of decoherence, where the sharp quantum features are washed out by the scale of the observation.

The findings offer a concrete way to test these ideas in future experiments. The researchers predict that if their model is correct, the flow patterns observed in collisions involving polarized protons—where the spin of the proton is aligned in a specific direction—should differ from those in unpolarized collisions, even if the total number of particles produced is the same. This is because the blurring process preserves specific spatial details of the proton's internal structure that are lost in other methods. By comparing these subtle differences in how the fluid swirls, scientists can verify whether the initial state is indeed governed by this coarse-grained entropy. The work provides a self-consistent bridge between the microscopic quantum description of a proton and the macroscopic laws of fluid dynamics, suggesting that the fluid behavior in small systems is a natural consequence of how quantum information is processed at the scale of the collision.

Technical Summary: Hydrodynamical Initial State in Small Systems From the Phase–Space Entropy

Problem Statement
The experimental observation of collective behavior in small collision systems (proton–proton and proton–nucleus) challenges the theoretical characterization of the initial state required for relativistic hydrodynamic evolution. While relativistic hydrodynamics necessitates an initial condition defined by an entropy current corresponding to a maximally mixed state, the microscopic description of the proton is based on pure quantum states (wave functions). This creates a fundamental mismatch: hydrodynamics requires a statistical ensemble, whereas the proton is described by a pure state with zero von Neumann entropy. The central problem addressed is how to properly characterize the initial entropy deposition in small systems where quantum fluctuations are significant and the system size is comparable to the nucleon radius, thereby bridging the gap between the quantum partonic structure and the classical hydrodynamic framework.

Methodology
The authors propose a formalism based on the coarse-graining of the partonic Wigner distribution to resolve the pure-state vs. mixed-state contradiction. The methodology proceeds through the following steps:

Phase-Space Formalism: The paper utilizes the light-cone wave function (LCWF) formalism to describe the hadron. It introduces the Wigner distribution, which provides a complete phase-space description but is not positive-definite, preventing a direct probabilistic interpretation.
Coarse-Graining to Husimi Distribution: To obtain a positive-definite phase-space distribution suitable for entropy calculations, the authors apply a Gaussian smearing (Weierstrass transform) to the Wigner function, resulting in the Husimi distribution ( $H_\ell$ ). This introduces a resolution scale $\ell$ , associated with a thermalization scale $\Lambda \sim \ell^{-1}$ .
Wehrl Entropy Definition: Instead of the von Neumann entropy (which vanishes for pure states), the authors define the initial entropy using the Wehrl entropy ( $S_W$ ). This is a semiclassical measure calculated from the Husimi distribution, representing the density of accessible microstates at a given resolution scale.
Connection to Hydrodynamics: The paper links this entropy to the initial conditions of relativistic hydrodynamics. It argues that for small systems, the initial state should be treated as a "maximally decohered" state in a coarse-grained sense, rather than a fully thermalized state. The entropy density is constructed by integrating the Husimi-weighted parton distributions over the transverse plane and longitudinal momentum fraction.
Fluctuating Hydrodynamics: The framework connects the calculated entropy to a Gaussian partition function (based on the fluctuating hydrodynamics approach of Ref. [27]), where the entropy content determines the initial fluctuations and the scale $\Lambda$ is fitted to the global multiplicity of the collision.

Key Contributions and Results

Resolution of the Pure/Mixed State Paradox: The paper shows that the appropriate matching between hadronic structure and classical hydrodynamics emerges naturally from the coarse-graining of the partonic Wigner distribution. This process transforms the pure quantum state into a semiclassical, positive-definite Husimi distribution.
Wehrl Entropy as Initial Condition: The authors establish that the Wehrl entropy, derived from the Husimi distribution, provides the correct quantity to characterize entropy deposition in small systems. Unlike the von Neumann entropy, the Wehrl entropy is non-zero for pure states due to the loss of phase information inherent in the coarse-graining process.
Limitations of GPDs and TMDs: The work clarifies that Generalized Parton Distributions (GPDs) and Transverse Momentum Distributions (TMDs) cannot be directly interpreted as entropy density. GPDs and TMDs represent projections of the pure state and lack the necessary coarse-grained structure to define an entropy density. The proposed model requires the full phase-space information encoded in the Husimi distribution.
Scaling and Multiplicity: The model relates the initial entropy deposition to the parton multiplicity. The localization scale $\Lambda$ is determined by the energy budget and the final multiplicity ($dN/dy$). The paper provides an estimate for the initial entropy deposition scale based on the relation between entropy and final multiplicity, suggesting $H^{IC}_\Lambda \sim 4\Lambda^3 \Delta y (dN/dy)$ , rather than deriving a strict functional relationship between the two.
Predictions for Anisotropic Flow: The framework suggests that the initial eccentricity ( $\epsilon_n$ ) and subsequent anisotropic flow ( $v_n$ ) in small systems are determined by the phase-space distribution rather than its projections. The paper posits that for the hydrodynamic description to hold, the scale $\Lambda$ must be parametrically larger than the inverse nucleon radius ( $1/R_h$ ); otherwise, fluctuations will wash out flow observables. It also predicts differences in $v_n$ between polarized ( $p^\uparrow A$ ) and unpolarized collisions in bins of equal multiplicity.

Significance and Claims
The paper claims to provide a self-consistent bridge between the microscopic quantum field theory description of the proton and the macroscopic framework of relativistic fluid dynamics in small collision systems. Its primary significance lies in redefining the initial condition for hydrodynamics not as a thermalized state, but as a state of "maximal decoherence" achieved through phase-space coarse-graining.

The authors assert that this approach resolves the theoretical inconsistency of applying hydrodynamics to systems described by pure quantum states. By identifying the Wehrl entropy as the relevant thermodynamic quantity, the framework allows for the construction of initial conditions that naturally incorporate quantum fluctuations and spatial resolution effects. The paper concludes that this formalism is testable, particularly through the analysis of anisotropic flow coefficients in low-multiplicity events and polarized collisions, offering a pathway to probe the transition from quantum coherence to hydrodynamic collectivity. The work emphasizes that hydrodynamization in this context is understood as a process of phase-space decoherence governed by the experimental and dynamical resolution scales.

Technical Summary: Hydrodynamical Initial State in Small Systems From the Phase–Space Entropy

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