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

This paper proposes that the appropriate initial conditions for hydrodynamic evolution in small collision systems can be derived by coarse-graining the proton's quantum wave function into a semi-classical, positive-definite Wehrl-like entropy, thereby bridging the gap between pure quantum states and the maximally mixed states required by relativistic hydrodynamics.

The Gist

Imagine you are trying to predict how a crowd of people will move after a sudden event, like a fire alarm going off in a stadium. To do this, you need to know exactly where everyone is standing and how they are connected at the very moment the alarm rings. This is the "initial condition" for your prediction.

For a long time, physicists have used this "crowd movement" logic (called hydrodynamics) to understand what happens when giant atomic nuclei smash into each other at nearly the speed of light. They create a super-hot, super-dense soup of particles called the Quark-Gluon Plasma (QGP), which flows like a perfect liquid.

But recently, scientists noticed something weird: even when they smash tiny protons (which are much smaller than nuclei) into other protons or nuclei, this "perfect fluid" behavior still happens. This is a puzzle.

The Big Problem: The Quantum vs. The Classical

Here is the conflict:

• The Proton (Quantum World): A proton is a tiny, fuzzy quantum object. It's not a solid ball; it's a cloud of possibilities. In quantum mechanics, if you know everything about a proton, it's in a "pure state"—meaning it has zero entropy (zero disorder). It's like a perfectly organized library where every book is in its exact spot.
• The Fluid (Classical World): To use hydrodynamics (the math of flowing liquids), you need a messy, disordered starting point. You need "entropy." You need a library where books are scattered, and you don't know exactly where they are.

The Question: How do you turn a perfectly ordered, fuzzy quantum proton into a messy, classical starting point for a fluid flow?

The Solution: The "Blurry Camera" Analogy

The authors of this paper propose a clever solution using a concept called Phase-Space Entropy.

Imagine taking a high-resolution photo of a proton. Because of the rules of quantum mechanics (the Uncertainty Principle), you can't see the proton's exact position and speed at the same time. It's like trying to take a photo of a hummingbird's wings; if you focus on the wings, the bird blurs, and if you focus on the bird, the wings blur.

The authors suggest we use a "blurry camera" (mathematically called a Husimi distribution or Wehrl entropy).

1. The Resolution Scale ($\ell$ or $\Lambda$): Think of this as the pixel size of your camera. If your pixels are huge, you can't see the tiny details of the proton. You just see a fuzzy blob.
2. Coarse-Graining: When you look at the proton through this "blurry lens," the perfect quantum order disappears. The "pure state" turns into a "mixed state." Suddenly, you don't know exactly where the parts of the proton are. You have lost some information.
3. The Resulting Entropy: This "loss of information" or "fuzziness" is what we call entropy. It's not that the proton is actually messy; it's that our view of it (at the scale of the collision) is messy.

The Analogy: The Orchestra vs. The Noise

• The Quantum Proton: Imagine a perfect orchestra playing a silent, coordinated piece of music. Every musician is in perfect sync. If you listen closely, it's a single, pure note. (Zero entropy).
• The Hydrodynamic Start: Now, imagine you are standing far away, or you are wearing noise-canceling headphones that only let in a muffled sound. You can't hear the individual musicians. You just hear a general "rumble" or "noise."
• The Paper's Insight: The authors say that the "rumble" you hear (the entropy) is the perfect starting point for the fluid dynamics. The more you "blur" your view (by changing the resolution scale of the collision), the more "rumble" (entropy) you get.

How It Works in the Experiment

When two protons smash together:

1. The collision happens at a specific energy scale (a specific "resolution").
2. This scale acts as the "blurry lens."
3. The authors calculate how much "fuzziness" (entropy) exists at that specific scale using a formula based on the proton's internal structure (its partons, or tiny building blocks).
4. This calculated "fuzziness" becomes the initial map for the fluid. It tells the computer simulation: "Start the fluid flow here, with this much disorder."

Why This Matters

This paper bridges a huge gap. It explains how a tiny, quantum object (a proton) can suddenly behave like a giant, classical fluid. It suggests that the "fluid" doesn't appear out of nowhere; it emerges naturally when we stop looking at the proton with a microscope and start looking at it with the "eye" of the collision itself.

In short: The paper says, "Don't try to force the quantum proton to be a classical fluid. Instead, realize that the act of colliding blurs the quantum details, creating just enough 'messiness' (entropy) to kickstart the fluid flow we see in experiments."

This helps scientists predict how these tiny collisions will behave, potentially unlocking new ways to understand the fundamental building blocks of our universe.

Technical Summary

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

1. Problem Statement
The paper addresses a fundamental theoretical challenge in high-energy nuclear physics: characterizing the initial conditions (IC) for relativistic hydrodynamic evolution in small collision systems (proton-proton and proton-nucleus collisions).

• The Conflict: Relativistic hydrodynamics requires an initial state defined by an entropy current, which corresponds to a maximally mixed (decohered) state. However, the microscopic description of the proton is based on quantum mechanics, where the proton is a pure state (a superposition of partonic Fock states) with vanishing Shannon entropy.
• The Gap: There is no direct mapping between the pure quantum wave function of a proton and the classical, statistical entropy required to initialize hydrodynamic simulations. The authors seek a method to bridge this gap by defining a semi-classical entropy measure derived from the proton's phase-space distribution.

2. Methodology
The authors propose a framework based on coarse-graining the proton's quantum phase-space distribution to generate a physical entropy density. The methodology proceeds through the following steps:

• Phase-Space Representation: The proton is described using the Wigner distribution in light-front quantization, which depends on longitudinal momentum fraction ($x$), intrinsic transverse momentum ($k_\perp$), and transverse position ($b_\perp$).
• Coarse-Graining (Husimi Distribution): Recognizing that the Wigner function cannot be interpreted as a probability density due to the uncertainty principle, the authors introduce a Gaussian smearing to obtain the Husimi distribution ($H_\ell$). This is defined by a resolution scale parameter $\ell$ (related to a thermalization scale $\Lambda \sim \ell^{-1}$).
  • Equation (1) defines the Husimi distribution as a convolution of the Wigner function with Gaussian kernels in position and momentum space.
• Entropy Definition (Wehrl Entropy): The authors utilize the Wehrl entropy ($\langle \ln H_\ell \rangle$) as a probe. Unlike the von Neumann entropy (which is zero for pure states), the Wehrl entropy is non-zero for pure states once coarse-grained, providing a semi-classical, positive-definite measure of the density of accessible microstates.
• Connecting to Partonic Structure: The authors link the Wehrl entropy to the Light-Cone Wave Function (LCWF) formalism.
  • They express the gluon structure function $G(x, Q^2)$ in terms of transverse coordinates using Parseval's theorem.
  • They define a transverse gluon distribution $\bar{G}(x, b_\perp, Q^2)$ (Eq. 3), which is independent of the phase of gluon fields, representing a "maximally decohered" quantity.
• Deriving the Entropy Density: By relating the resolution scale $Q^2$ to the breaking of quantum coherence, the authors propose that the Wehrl entropy density is proportional to the integral of the transverse gluon distribution up to the resolution scale $\Lambda$:
  $$\langle \ln W_\Lambda \rangle (x, b_\perp) = N \int_0^\Lambda d^2Q \, \bar{G}(x, b_\perp, Q^2)$$
  (Eq. 5). This equation posits that probing the proton at scale $Q^2$ breaks coherence, increasing accessible microstates and generating entropy approximated by the Husimi distribution.

3. Key Contributions

• Theoretical Bridge: The paper provides a rigorous theoretical mechanism to map the pure quantum state of a proton to the mixed state required for hydrodynamics via phase-space coarse-graining.
• Wehrl Entropy as IC: It identifies the Wehrl-like entropy derived from the Husimi distribution as the appropriate quantity to characterize entropy deposition in small systems, replacing ad-hoc geometric assumptions.
• Resolution Scale Dependence: It explicitly connects the entropy scale $\Lambda$ to the energy budget of the local thermalized cell and the uncertainty principle ($\Lambda \sim L^{-1}$), ensuring the entropy density scales correctly ($\sim \Lambda^3$).
• Extensivity in Collisions: The framework suggests that for $pp$ and $pA$ collisions, the total entropy deposition is the sum of the target and projectile entropies, allowing for the calculation of initial eccentricities.

4. Results and Predictions

• Qualitative Framework: The paper establishes that the localization of degrees of freedom (described by probability distributions in Eq. 3) serves as a proxy for the Husimi smearing of the Wigner functional.
• Predictive Capability: The authors argue that once the entropy density profile is defined, it can serve as the initial condition for hydrodynamic evolution.
  • This allows for the calculation of flow harmonics ($v_n$) in $pp$ and $pA$ collisions.
  • The total multiplicity of a collision can set the scale $\Lambda$, making the model potentially predictive without arbitrary fitting parameters.
• Future Work: The paper notes that a full quantitative analysis comparing calculated $v_n$ with experimental data is reserved for future work. However, it suggests that comparing flow harmonics in collisions with polarized protons could serve as a quantitative probe of this 3D structure.

5. Significance
This work is significant for the field of heavy-ion physics for several reasons:

• Resolving the "Small System" Paradox: It offers a solution to the long-standing question of how hydrodynamics, a classical theory, can emerge from the quantum pure states of small systems like protons.
• Microscopic Foundation: It moves beyond phenomenological models (like Glauber or CGC-based geometric models) by grounding the initial entropy in the fundamental quantum phase-space structure of the nucleon.
• New Observables: It proposes that the Wehrl entropy is a physically meaningful observable that links the microscopic partonic degrees of freedom to macroscopic collective flow, potentially explaining the "perfect fluid" behavior observed in small collision systems.
• Experimental Guidance: By suggesting that polarized proton collisions could test this 3D structure, the paper provides a concrete direction for future experimental verification at facilities like the LHC and RHIC.