---
title: "Hydrodynamic Attractor Interactive Lab | Sandbox Physics"
description: "Can expanding matter forget its starting pressure before the pressures become equal? Yes."
canonical_url: "https://sandboxphysics.com/labs/hydrodynamic-attractor/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/hydrodynamic-attractor/"
---

FR06 · Forgetting the initial pressure

# Hydrodynamic Attractor

**Can expanding matter forget its starting pressure before the pressures become equal?** Yes. In a specified relaxation model, different starting pressures can approach the same scaled trajectory while longitudinal and transverse pressures still differ. The rate of expansion keeps a pressure anisotropy even after much of the initial-state memory has decayed.

Release a family of different initial shear stresses. Watch their pressure shapes approach each other, then compare the same trajectories in scaled time with the regular MIS attractor and gradient approximations.

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

Model assumptions

The ensemble uses a conformal, boost-invariant MIS closure with zero quadratic BRSSS coefficient. It illustrates model-specific loss of initial shear memory, not a universal QGP thermalization time. The attractor is the regular solution extrapolated from small scaled time; its early negative-pressure segment lies outside the positive-pressure domain. The residual to the second-order gradient approximation and pressure isotropy are separate diagnostics. The publication preset adopts the coefficients of arXiv:1503.07514v2 with its zero-quadratic simplification. No experimental data are displayed.

## A family of histories, one late-time path

Colored lines: MIS initial-state family. White: regular attractor. Amber: first-order gradient; dashed blue: second-order gradient. These are model solutions.

Pressure anisotropy in scaled time $w=\tau T,\quad\mathcal A=\frac{P_T-P_L}{e/3}$

> Computed model trajectories.

Hydrodynamic residual and pressure isotropy $\lvert\phi/e-r_{\mathrm{grad},2}\rvert,\quad\lvert P_L/P_T-1\rvert$

> Pressure and residual diagnostic trajectories.

Interpretation and units

Time is in units of the initial proper time; energy density is in units of its initial value. Natural units are used. The pressure shape above encodes normalized tensor components and carries no physical length scale.

Physics tutorial

## How initial pressure memory fades

**Background** After a heavy-ion collision, neighbouring slices of matter rush apart along the beam. In a boost-invariant description each slice has the same history when followed with its own clock.

**Why it matters** Different starting pressures can produce nearly the same later shear stress. Comparing those histories tells us which information the expansion retains.

Start with the essentials

- **Focus question**: Can expanding matter forget its starting pressure before the pressures become equal?
- **One-sentence intuition**: Loss of initial shear memory and equality of the pressures occur on different criteria.

### Core mathematical model

#### Energy conservation in a comoving slice

$$
\frac{de}{d\tau}=-\frac{e+P_L}{\tau},\qquad P_L=\frac e3-\phi,\quad P_T=\frac e3+\frac\phi2
$$

Proper time is the time measured along a comoving trajectory. Positive longitudinal pressure adds pressure work to dilution. The shear convention lowers longitudinal pressure when the shear variable is positive.

#### A shear stress with a finite response time

$$
\tau_\pi\frac{d\phi}{d\tau}=\frac{4\eta}{3\tau}-\phi-\frac{4\tau_\pi\phi}{3\tau},\quad\tau_\pi=\frac{C_{\tau\pi}}T,\quad\eta=C_\eta s_{\mathrm{eq}}
$$

The expansion drives shear stress and a finite relaxation time delays its response. The last term follows the conformal MIS closure. The quadratic BRSSS term is set to zero.

#### Scaled pressure anisotropy

$$
w=\tau T,\qquad\mathcal A=\frac{P_T-P_L}{e/3}=\frac{9\phi}{2e}
$$

Scaled time compares the expansion time with a microscopic thermal scale. Plotting against it reveals the regular MIS branch approached by many starting shear stresses.

### Common difficulties

#### Flow and equilibrium

**Typical misconception** A hydrodynamic description requires equal pressures at every instant.

**Better mental model** Viscous hydrodynamics predicts unequal pressures. The residual measures agreement with a chosen constitutive approximation; the pressure ratio measures isotropy.

### Run the experiment

1. 01

   #### Follow the family

   Choose Lose the initial memory and watch the pressure shapes.

   **What to observe:** Different initial stresses approach similar shapes as the transient shear mode decays.

2. 02

   #### Delay the approach

   Choose Long-lived memory, then compare the family spread at the same proper time.

   **What to observe:** A larger relaxation coefficient keeps more initial shear information over the finite observation window.

3. 03

   #### Compare two criteria

   Choose Early gradient failure and inspect the two diagnostics below the stage.

   **What to observe:** A large early gradient correction can fail even when the MIS trajectory is smooth. Pressure isotropy and agreement with a truncated gradient relation need not arrive together.

Physical conclusion

An attractor describes a family of histories in a specified model; its scaled trajectory can remain strongly anisotropic.

## 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 **Ask when fluid evolution works before pressures agree**](https://sandboxphysics.com/stories/qgp-hydrodynamics/hydrodynamization-isotropy/)
2. [Article **Recalculate a limited MIS attractor benchmark**](https://sandboxphysics.com/stories/qgp-hydrodynamics/attractor-benchmark/)

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)