At the Large Hadron Collider, not only nuclei but also individual protons collide. Even in such small systems, collective behavior resembling the flow of a liquid is observed. But the proton is a quantum object; its state is described by a wave function, not by temperature and pressure. How does the classical picture emerge from the quantum 'mosaic'?
Hydrodynamics requires initial conditions set by the entropy flux — a measure of disorder. The quantum state of a proton is a projection of a pure state, whose entropy is zero.
The authors propose using the Weyl entropy, which arises from coarse-graining phase space — averaging over small cells. Such entropy is always positive and provides a semiclassical measure of the number of accessible microstates at a given resolution. It is precisely this entropy that allows linking the quantum wave function of the proton to the classical initial conditions for hydrodynamics.
This is akin to translating a mosaic into a painting: if you look at the mosaic too closely, you see individual pieces (quantum states), but step back, and they merge into a unified canvas (hydrodynamic description). The Weyl entropy is the tool that measures the degree of this merging.
as written by the authors arXiv:2604.12691
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.
The Weyl entropy is named after the physicist who introduced the concept of coherent states — peculiar 'bridges' between the quantum and classical worlds.