---
title: "Heavy-ion matter and hydrodynamics"
description: "Explore expanding matter, Bjorken flow, hydrodynamic attractors and particle correlations in nine chapters with four experiments."
canonical_url: "https://sandboxphysics.com/stories/qgp-hydrodynamics/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/qgp-hydrodynamics/"
---

FR06 · A course with experiments

# Heavy-ion matter and hydrodynamics

Two nuclei collide, and the matter between them expands. Follow one small piece: find its rest frame, calculate its directional pressures, then ask when its evolution becomes predictable. Finally, use particle correlations to see how a transport inference depends on what we assume about the beginning.

[Begin with one moving patch →](https://sandboxphysics.com/stories/qgp-hydrodynamics/local-rest-frame/)

L1 → L2

## Describe the expanding patch

Define a clock and a stress tensor. Derive dilution, compare collisionless and ideal limits, and add finite shear relaxation.

L3

## Test prediction and shared motion

Separate hydrodynamic agreement from isotropy. Recalculate a limited published attractor equation, then connect shape with angular correlations.

L4

## Read a posterior with its assumptions

Fit a fixed synthetic event sample. Change geometry, nonflow and model error while preserving the observations.

## Read in order, pause for the experiments

1. 01

   L1

   ### [Find the rest frame of expanding matter](https://sandboxphysics.com/stories/qgp-hydrodynamics/local-rest-frame/)

   Define collision geometry, energy flow and directional pressure before describing a heavy-ion collision as a fluid.

2. 02

   L1

   ### [Follow the clock carried by the expanding matter](https://sandboxphysics.com/stories/qgp-hydrodynamics/bjorken-conservation/)

   Derive Bjorken energy conservation from proper time, comoving volume and longitudinal pressure work.

3. 03

   L2

   ### [Compare ideal flow with particles that never collide](https://sandboxphysics.com/stories/qgp-hydrodynamics/expansion-limits/)

   Derive conformal cooling and the finite-time free-streaming limit from one initial energy and pressure state.

4. 04

   L2

   ### [Let shear stress relax over a finite time](https://sandboxphysics.com/stories/qgp-hydrodynamics/causal-shear-relaxation/)

   Build the conformal MIS equations, define their transport coefficients and identify when the closure leaves its useful domain.

5. 05

   L3

   ### [Ask when fluid evolution works before pressures agree](https://sandboxphysics.com/stories/qgp-hydrodynamics/hydrodynamization-isotropy/)

   Separate a hydrodynamic residual, pressure isotropy and microscopic thermal equilibrium in an expanding system.

6. 06

   L3

   ### [Recalculate a limited MIS attractor benchmark](https://sandboxphysics.com/stories/qgp-hydrodynamics/attractor-benchmark/)

   Transform the published MIS equation, compare independent numerical variables and identify the limits of an attractor reproduction.

7. 07

   L3

   ### [Connect an initial shape to a final angular pattern](https://sandboxphysics.com/stories/qgp-hydrodynamics/eccentricity-flow-response/)

   Define eccentricity and flow harmonics, then study a declared linear-response model with viscous damping.

8. 08

   L3

   ### [Measure common flow through particle correlations](https://sandboxphysics.com/stories/qgp-hydrodynamics/multiparticle-correlations/)

   Derive distinct-particle pair correlations and four-particle cumulants, with explicit finite-sample and nonflow limits.

9. 09

   L4

   ### [Infer transport while changing the assumed initial state](https://sandboxphysics.com/stories/qgp-hydrodynamics/conditional-transport-inference/)

   Recalculate a frozen synthetic posterior and inspect how geometry, nonflow and model error move the viscosity constraint.

## Four experiments along the way

[Bjorken expansion ↗](https://sandboxphysics.com/labs/bjorken-expansion/) [Hydrodynamic attractor ↗](https://sandboxphysics.com/labs/hydrodynamic-attractor/) [Shape to collective flow ↗](https://sandboxphysics.com/labs/eccentricity-flow/) [Transport inference ↗](https://sandboxphysics.com/labs/qgp-transport-inference/)

The first two experiments solve a declared longitudinal model. The flow experiments use a separate limited response law and saved synthetic events. The chapters identify the published equations, numerical checks and source versions where each enters.

## Prepare as the next question requires

Begin with energy, momentum and basic algebra. The course introduces the matrix, derivative, angular moment and probability tools before using them. Every chapter includes a worked example, a prediction task and explained exercises. More advanced field theory and statistics can be studied when the later methods need them.

For collision-event context, visit the particle collider. For familiar viscous-flow intuition, compare Couette and Poiseuille flow. These are supporting examples, with different models and physical scales.

[Particle collider](https://sandboxphysics.com/labs/particle-collider/) [Couette–Poiseuille flow](https://sandboxphysics.com/labs/couette-poiseuille/)