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.

Proper time $2.46\,\tau_0$

Energy density $0.354\,e_0$

Longitudinal / transverse pressure $0.266$

Family spread in shear $0.122$

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.

Pressure anisotropy in scaled time

w=\tau T,\quad\mathcal A=\frac{P_T-P_L}{e/3}

Hydrodynamic residual and pressure isotropy

\lvert\phi/e-r_{\mathrm{grad},2}\rvert,\quad\lvert P_L/P_T-1\rvert

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.

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 misconceptionA hydrodynamic description requires equal pressures at every instant.

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

Run the experiment

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.
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.
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.

Hydrodynamic Attractor

A family of histories, one late-time path

How initial pressure memory fades

Core mathematical model

Energy conservation in a comoving slice

A shear stress with a finite response time

Scaled pressure anisotropy

Common difficulties

Flow and equilibrium

Run the experiment

Follow the family

Delay the approach

Compare two criteria

Related experiments and articles

From directional pressure to transport inference