FR06 · L4 · Chapter 9 of 9

Infer transport while changing the assumed initial state

A posterior is a probability distribution conditional on observations, a response model and prior choices. The same observations can favour a different viscosity when the assumed initial geometry or nonflow changes. A frozen synthetic benchmark lets us calculate those shifts and compare them with the known generating parameters.

Before you beginChapters 7–8 define the response and distinct-particle pair signal. A likelihood scores how well a model predicts observations; a prior assigns parameter weights before those observations; Bayes’s rule combines them.

Sandbox Physics · 2026-10-27

Freeze observations before fitting anything

Benchmark version one saves 4,500 generated events, 1,500 in each of three classes. Every event has $M=128$ tracks. The random seed is 602710; the planted parameters are $C_\eta=0.16$ and $\kappa_0=0.34$ . Four identical-angle doublets per event add the specified nonflow. The file saves harmonic Q vectors sufficient for the implemented pair and four-particle diagnostics.

The three dimensionless size–temperature products are $RT=(3.5,2.8,2.2)$ . The reference second-harmonic eccentricities are $(0.22,0.31,0.41)$ ; the third-harmonic values are $(0.12,0.15,0.18)$ . Together with chapter 7’s damping law, they fix the generating one-particle distributions. The two harmonic planes share one randomly rotated orientation in this benchmark.

Changing an inference setting reads these same saved Q vectors. It does not regenerate the observations. The benchmark is synthetic teaching data; no ATLAS, ALICE or RHIC measurement is relabelled as its input.

Write the likelihood for the quantity that was saved

For each class and harmonic, average the exact distinct-particle pair estimator. These give a six-component observation vector $\mathbf y$ . With the declared doublet fraction $d$ , the predicted entry is $\mu_n=v_n^2+d(1-v_n^2)$ . The parameters are $\boldsymbol\theta=(C_\eta,\kappa_0)$ . In a covariance matrix, diagonal entries are variances of sample means; off-diagonal entries record how the two harmonics fluctuate together. Its inverse measures prediction errors in uncertainty units.

\log L(\boldsymbol\theta)=-\frac12(\mathbf y-\boldsymbol\mu(\boldsymbol\theta))^\mathsf T\Sigma^{-1}(\mathbf y-\boldsymbol\mu(\boldsymbol\theta))+\mathrm{constant}

The covariance retains the empirical two-harmonic covariance within each class, divided by the number of events. Independent classes form separate covariance blocks.

Add a diagonal model-discrepancy variance $\sigma_{\rm model}^2$ to those blocks; the default standard deviation is $0.00012$ in pair-correlation units. This is a declared allowance for prediction error. The demonstration uses a fixed covariance, so the omitted determinant is constant across the parameter grid. A parameter-dependent covariance would require retaining that determinant.

Normalise, marginalise and check the grid

Choose uniform priors $0.03\le C_\eta\le0.35$ and $0.25\le\kappa_0\le0.44$ . Bayes’s rule says $p(\boldsymbol\theta\mid\mathbf y)\propto L(\boldsymbol\theta)p(\boldsymbol\theta)$ . The Lab evaluates this on a 97-by-65 grid and uses trapezoidal edge weights when normalising and marginalising.

p(C_\eta\mid\mathbf y)=\int p(C_\eta,\kappa_0\mid\mathbf y)\,d\kappa_0

Integrating over the response scale carries its uncertainty into the viscosity marginal. Fixing that nuisance parameter at its best value would answer a different, more conditional question.

Grid refinement checks whether a numerical discretisation controls the interval. A closure check asks whether the known generating parameters are compatible with inference under the matching model. Passing one closure test verifies this pipeline for this benchmark; repeated synthetic datasets would be needed to study interval coverage.

Read the sensitivity before quoting an interval

Keep the observations fixed and change the geometry. A larger assumed elliptic deformation can be compensated by stronger viscous damping. Changing the pattern of eccentricities across classes and harmonics also changes the shape of the response degeneracy. This is why several harmonics and sizes are more informative than one amplitude.

Now ignore the doublets. The model tries to explain their excess pair correlation through a larger collective signal, and the inferred viscosity moves downward in this frozen dataset. Increasing the model-discrepancy allowance broadens the range of predictions treated as plausible. These are conditional calculations whose assumptions should accompany every reported interval.

Published QGP extractions use multi-stage event simulations, many experimental observables, transport through the hadronic stage and substantial uncertainty analysis. Bernhard and collaborators’ 2019 result and the frozen JETSCAPE study provide examples. This exercise reproduces a synthetic response inference, not their experimental posterior or a temperature-dependent QCD viscosity.

Worked example

Compare three fits of the same saved sample

With reference geometry, known doublets and the default discrepancy, the benchmark viscosity mean is about $0.1823$ ; its central 90-percent interval is approximately $[0.1267,0.2367]$ . The planted value $0.16$ lies inside that interval.

Keeping the sample fixed, the larger-ellipticity assumption moves the mean to about $0.2118$ . Ignoring nonflow instead moves it to about $0.1406$ . These numbers describe the version-one file and declared grid. They are not experimental estimates of QGP viscosity.

The posterior need not peak at the generating truth in a finite random sample. The useful comparison is between sampling uncertainty, model changes and the known truth, with every setting recorded.

Predict, then try the model

Predict which way viscosity should move if the assumed initial ellipticity grows but the observed correlations stay fixed.

Open transport inference. Switch between the reference and alternative geometry, then disable the nonflow correction. Export the observations and posterior settings for each comparison.

Transport parameter inference ↗

Compare your explanation

The file identity remains the same. Compare shifts of the marginal with its width and inspect goodness of fit; a narrow conditional interval can still move substantially under another model assumption.

Exercises with explanations

1. Why does a single measured harmonic amplitude leave a response–viscosity degeneracy?

Read the reasoning

Its logarithm obeys $\log(v_n/\varepsilon_n)=\log\kappa_0-n^2a_d C_\eta/(RT)$ . One equation fixes a line in two unknown parameters. Additional harmonics and sizes change the coefficients and can reduce that degeneracy.

2. Which research step would turn this into a real-data viscosity analysis?

Read the reasoning

A calibrated initial-state and event-evolution model, detector-compatible measured observables and covariances, treatment of model discrepancies, and external validation would all be needed. Choosing the real dataset, version and analysis cuts comes before fitting. The open-data landing page alone does not specify those inputs.

Repeat the calculation

Download the immutable version-one event statistics. The generator fixes the seed and single-particle distribution; the model notes define the inference grid, covariance and sensitivity checks.

Frozen synthetic event statistics Reproduce the synthetic sample Posterior integration source Model, covariance and benchmark checks

Sources and reading scope

Gardim et al., Mapping the hydrodynamic response to the initial geometry in heavy-ion collisions, arXiv:1111.6538v2
Version 2 (5 March 2012), §§II–IV: eccentricity estimators, linear response and nonlinear higher harmonics. The teaching exponential damping law is separately defined, not fitted from its event simulations.
Bilandzic, Snellings & Voloshin, Flow analysis with cumulants: direct calculations, arXiv:1010.0233v2
Version 2 (17 October 2011), Eqs. (3)–(4), (11)–(18): nonflow, Q vectors and exact distinct-particle correlators for uniform acceptance.
Bernhard, Moreland & Bass, Bayesian estimation of the specific shear and bulk viscosity of quark–gluon plasma (2019)
Published Nature Physics article, Figs. 2–4: a real-data multi-stage calibration, used as a methodological comparison. No experimental posterior or viscosity interval is imported into this course.
JETSCAPE Collaboration, Multi-system Bayesian constraints on the transport coefficients of QCD matter, arXiv:2011.01430v2
Version 2 (6 November 2020), §§II, V.3, VI and VIII: likelihood, model discrepancy, closure tests and model dependence. This version is frozen for reading, not represented as the latest analysis.