---
title: "Bjorken Expansion Interactive Lab | Sandbox Physics"
description: "Why does a plasma cool differently when it can push along its expansion?"
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FR06 · Longitudinal expansion

# Bjorken Expansion

**Why does a plasma cool differently when it can push along its expansion?** Longitudinal pressure does work as the matter expands, so its energy density falls faster. Collisions replenish that pressure; freely streaming particles instead lose longitudinal momentum and exert progressively less pressure along the expansion.

Compare an initially isotropic free-streaming gas, an ideal fluid and a relaxing shear stress at equal initial energy. Follow their cooling and their unequal longitudinal and transverse pressures.

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

Model assumptions

This conformal, boost-invariant model has no transverse expansion, conserved charges or bulk viscosity. Pressure is one third of energy density in equilibrium. The MIS closure retains one shear stress with zero quadratic BRSSS coefficient; it is not a solution of a kinetic relaxation-time equation. Only the MIS branch accepts an adjustable initial shear. Temperature and entropy are energy-matched equilibrium proxies away from equilibrium. The pressure glyph is a tensor diagram, not the shape of a collision or a momentum distribution. Negative pressure flags a closure extrapolation.

## Pressure changes how the energy falls

Cyan: MIS relaxation. Amber: ideal fluid. Violet: initially isotropic free streaming. All begin at the same energy density; the initial shear control changes only MIS.

Energy density during expansion $e/e_0$

> Computed model trajectories.

Longitudinal pressure relative to transverse pressure $P_L/P_T$

> 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

## The work done by expanding matter

**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** Dilution removes energy from a unit volume. Longitudinal pressure also does work, so the cooling rate tells us about the stress along the beam.

Start with the essentials

- **Focus question**: Why does a plasma cool differently when it can push along its expansion?
- **One-sentence intuition**: Larger longitudinal pressure makes the energy density fall faster at the same instantaneous energy.

### 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.

#### The ideal conformal limit

$$
e(\tau)=e_0\left(\frac{\tau_0}{\tau}\right)^{4/3},\qquad T(\tau)=T_0\left(\frac{\tau_0}{\tau}\right)^{1/3}
$$

Set shear stress to zero and integrate energy conservation. The ideal branch has equal pressures at every time. The temperature is defined by energy matching in every branch.

### 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

   #### Match the starting energy

   Choose Three cooling histories and leave the initial MIS shear at zero.

   **What to observe:** All three start isotropic. Their subsequent longitudinal pressures differ, so their cooling histories separate.

2. 02

   #### Change the response time

   Compare Quickly relaxing shear and Long-lived anisotropy.

   **What to observe:** Relaxation and expansion compete. A slow stress response preserves anisotropy for longer.

3. 03

   #### Inspect one time

   Pause and drag the glowing time handle. Read the energy and pressure plots at the moving marker.

   **What to observe:** The ideal branch follows the analytic power law. Free streaming starts with finite longitudinal pressure and approaches a slower late-time dilution law.

Physical conclusion

An ideal fluid cools through dilution and pressure work. A collisionless gas gradually loses longitudinal pressure as its longitudinal momenta redshift.

## 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 **Find the rest frame of expanding matter**](https://sandboxphysics.com/stories/qgp-hydrodynamics/local-rest-frame/)
2. [Article **Follow the clock carried by the expanding matter**](https://sandboxphysics.com/stories/qgp-hydrodynamics/bjorken-conservation/)
3. [Article **Compare ideal flow with particles that never collide**](https://sandboxphysics.com/stories/qgp-hydrodynamics/expansion-limits/)
4. [Article **Let shear stress relax over a finite time**](https://sandboxphysics.com/stories/qgp-hydrodynamics/causal-shear-relaxation/)

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)