---
title: "From Eccentricity to Collective Flow Interactive Lab | Sandbox Physics"
description: "Why do more particles leave along the short axis? Pressure gradients are steeper across the short direction of an elongated cloud."
canonical_url: "https://sandboxphysics.com/labs/eccentricity-flow/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/eccentricity-flow/"
---

FR06 · Geometry and emission

# From Eccentricity to Collective Flow

**Why do more particles leave along the short axis?** Pressure gradients are steeper across the short direction of an elongated cloud. A limited linear response converts this initial shape into an angular emission pattern; stronger viscous damping weakens the pattern, especially its higher harmonics.

Stretch and turn an initial density cloud, then watch its angular emission change. Compare the same shape under weak and strong viscous damping.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A Gaussian initial density feeds an assigned linear response for elliptic flow. Particle directions sample that angular distribution; their animated speed and travel distance are schematic. The model has no transverse hydrodynamic evolution, freeze-out surface or hadronic rescattering. Response and damping constants are illustrative.

Initial ellipticity $0.6$

Elliptic flow amplitude $0.232$

Response after damping $0.387$

Long to short width $2$

### Angular emission density

> Interactive response or posterior plot.

Teal shows the current model; the dashed line shows isotropic emission, with angles measured from the horizontal axis.

### The same cloud under different viscosities

> Interactive response or correlation comparison.

The marker shows the current viscosity; geometry and response amplitude stay fixed along the curve.

Model, units and sources

Natural units are used. The Gaussian density has unit mean squared radius per axis before deformation. Its short-axis direction defines the elliptic emission plane. This is an assigned response calculation, with no transverse fluid solver.

$$
v_n=\kappa_0\varepsilon_n e^{-1.8n^2(\eta/s)/(RT)},\qquad n=2,3
$$

The response coefficient and damping scale are illustrative. Higher harmonics and nonlinear mode coupling are omitted.

[Gardim et al. (2012)](https://arxiv.org/abs/1111.6538v2) · [Bilandzic et al. (2011)](https://arxiv.org/abs/1010.0233v2) · [Bernhard et al. (2019)](https://doi.org/10.1038/s41567-019-0611-8)

Physics tutorial

## From Eccentricity to Collective Flow

**Background** Stretch and turn an initial density cloud, then watch its angular emission change. Compare the same shape under weak and strong viscous damping.

**Why it matters** Pressure gradients are steeper across the short dimension of an elliptic density cloud. Collective expansion can therefore leave a momentum anisotropy opposite to the long spatial axis.

Start with the essentials

- **Focus question**: Why do more particles leave along the short axis?
- **One-sentence intuition**: An initial spatial ellipticity becomes an angular emission anisotropy through an assumed response; viscous damping reduces that response.

### Core mathematical model

#### Gaussian shape

$$
\varepsilon_2=\frac{\langle y^2\rangle-\langle x^2\rangle}{\langle y^2\rangle+\langle x^2\rangle}
$$

The coordinates follow the short and long principal axes. A circular cloud has zero ellipticity.

#### Assigned linear response

$$
v_n=\kappa_0\varepsilon_n\exp\!\left[-\frac{1.8n^2(\eta/s)}{RT}\right]
$$

The dimensionless damping coefficient is illustrative. The formula is used only for the second and third harmonics, with no nonlinear coupling.

#### Angular emission

$$
p(\phi)=\frac{1+2v_2\cos[2(\phi-\Psi_2)]}{2\pi}
$$

The second harmonic measures how strongly emission prefers the short-axis direction. The physical speed is not specified by this angular model.

### Common difficulties

#### What the estimate means

**Typical misconception** An elliptical picture already solves the collision dynamics.

**Better mental model** This experiment specifies a response law and samples its angular distribution. A transverse fluid calculation must evolve the energy-momentum tensor before predicting the emitted particles.

### Run the experiment

1. 01

   #### Drag the cloud

   Drag the bright handle outward, or use Initial ellipticity.

   **What to observe:** The density elongates and more model particles leave across its short dimension.

2. 02

   #### Rotate the preparation

   Choose Turn the cloud, then rotate the short axis.

   **What to observe:** The preferred emission direction rotates with the principal axes.

3. 03

   #### Hold the shape fixed

   Compare An elliptic cloud with Stronger damping.

   **What to observe:** The initial shape stays the same as the outgoing angular contrast weakens.

Physical conclusion

Spatial shape and emission direction are different observables connected by dynamics.

## Related experiments and articles

Heavy-ion matter and hydrodynamics · Reading path

### From directional pressure to transport inference

Articles on this path that use this experiment, followed by the other experiments they suggest.

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

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)