FR06 · L3 · Chapter 6 of 9

Recalculate a limited MIS attractor benchmark

Many initial stresses can evolve toward a narrow family of curves after a change of scale. We can test that statement in a specified MIS model by solving the same published dynamics in two different variables. The useful result is a reproducible comparison with known equations, coefficients and errors.

Before you beginChapters 4–5 introduce the relaxation equations and gradient comparison. You need the chain rule and a numerical method for a first-order differential equation. The source-reading target is Heller–Spaliński 2015, version 2, Eqs. (4), (5), (8)–(10) and Fig. 1.

Sandbox Physics · 2026-10-27

Freeze the paper version and the model

Use the simplified conformal MIS system with the quadratic coefficient set to zero. The publication preset uses the paper’s viscosity and relaxation constants below. These constants are inspired by a strongly coupled supersymmetric theory; they are not a measurement of QCD transport.

C_\eta=\frac1{4\pi},\qquad C_\tau=\frac{2-\log2}{2\pi},\qquad C_{\lambda_1}=0

The zero quadratic coefficient is the explicit simplifying choice made before the published equation (9). The default teaching preset uses different declared coefficients.

The reproduction target is the approach of finite-initial-time MIS solutions to a distinguished scaled curve, as in the 2015 paper’s first figure. We do not digitise its image, perform Borel resummation, or reproduce the 2020 paper’s large phase-space cloud and principal-component analysis.

Change variables before comparing solvers

Set $w=\tau T$ and $f=\tau\dot w/w$ . Since conformality gives $\dot T/T=\dot e/(4e)$ , energy conservation yields $f=2/3+r/4$ . The pressure-anisotropy variable is $\mathcal A=(P_T-P_L)/(e/3)=9r/2=18f-12$ .

C_\tau w f\frac{df}{dw}+4C_\tau f^2+\left(w-\frac{16C_\tau}{3}\right)f-\frac{4C_\eta}{9}+\frac{16C_\tau}{9}-\frac{2w}{3}=0

This is the frozen source’s equation (9). Integrating it directly in scaled time gives a check independent of the implementation that advances logarithmic energy and shear in proper time.

To compare, use the same initial point, evolve both solvers and interpolate onto shared scaled-time samples. Refine each step independently. A small difference after refinement supports the variable conversion and numerical evolution; matching two coarse steps can hide the same interpolation error.

An early-time mathematical branch needs a physical domain

At small positive $w$ , regularity leaves two roots. The branch used for the reference starts from the positive root below, then is integrated forward. Starting at a small nonzero value introduces an error that must also be checked by reducing that starting value.

f_\star(0)=\frac23+\frac13\sqrt{\frac{C_\eta}{C_\tau}},\qquad r_\star(0)=\frac43\sqrt{\frac{C_\eta}{C_\tau}}

For the chosen coefficients this early branch has negative longitudinal pressure. It is a mathematical extension of the closure, visibly marked outside the positive-pressure domain.

An attractor is also about a family, not just one attractive-looking line. Compare different initial stresses at a common scaled time and report how their spread changes. Before they converge, there is no reason to replace them by one universal prediction. Changing the closure or the state-space coordinates is a further scientific choice.

The 2020 phase-space paper broadens the question to how a cloud of states loses effective dimensions. Its first figure motivates looking at families of initial states. A few lines in this Lab demonstrate the reduced MIS calculation; they do not replace that paper’s full-dimensional kinetic analysis.

Worked example

Convert a plotted ordinate back to pressure

Suppose a trajectory at some scaled time has $r=0.1$ . Its transformed value is $f=2/3+0.1/4\simeq0.69167$ , and its anisotropy is $\mathcal A=0.45$ . Recovering the directional pressures gives:

\frac{P_L}{P_T}=\frac{1/3-0.1}{1/3+0.1/2}=\frac{14}{23}\simeq0.60870

Reading a curve near its asymptotic f value can conceal an appreciable pressure difference. The affine conversion to anisotropy makes that difference easier to see.

Predict, then try the model

Do trajectories that begin with very different shear already lie close together at the earliest scaled time?

Choose the publication-coefficient preset. Compare the family before and after its spread contracts. Inspect the marked early reference branch and export the trajectory samples.

Hydrodynamic attractor ↗

Compare your explanation

The numerical benchmark checks a limited published equation and a family’s convergence. Its extrapolated negative-pressure segment remains labelled even when it helps locate the mathematical reference.

Exercises with explanations

1. Why does direct integration in scaled time require a monotonic scaled clock?

Read the reasoning

The change of variable divides by $\dot w=fw/\tau$ . If this vanishes or the clock reverses, one scaled-time branch no longer uniquely identifies the proper-time trajectory. The positive-pressure states here have positive $f$ .

2. What additional evidence would be needed to call this a reproduction of the full 2020 phase-space paper?

Read the reasoning

Its specified ensemble of initial states, phase-space coordinates, sampling times, dimensionality-reduction procedure and numerical dataset would have to be frozen and reproduced. Agreement of a reduced MIS curve does not check those separate calculations.

Repeat the calculation

The frozen reference contains eight scaled-time samples from an independent SciPy DOP853 integration of the published equation. The script records its tolerances and initial condition. Compare these values with the trajectory export and refine both calculations.

Reference values and provenance Python reproduction script Model and error checks

Sources and reading scope

Heller & Spaliński, Hydrodynamics beyond the Gradient Expansion: Resurgence and Resummation, arXiv:1503.07514v2
Version 2 (28 August 2015), Eqs. (4)–(10), Fig. 1. Equation (9) and the zero-lambda-one choice define the limited numerical benchmark; no Borel resummation is reproduced.
Heller, Jefferson, Spaliński & Svensson, Hydrodynamic attractors in phase space, arXiv:2003.07368v2
Version 2 (23 September 2020), Fig. 1 and phase-space discussion. Reading reference for dimensional reduction; its 10,000-state cloud, PCA and kinetic-theory dataset are not reproduced.