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.

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

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

The same cloud under different viscosities

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) · Bilandzic et al. (2011) · Bernhard et al. (2019)

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 misconceptionAn elliptical picture already solves the collision dynamics.

Better mental modelThis 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

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.
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.
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.

From Eccentricity to Collective Flow

Angular emission density

The same cloud under different viscosities