Norway

3-dim. QGP Fluid Dynamics and Flow Observables

László Csernai (Bergen Computational Physics Lab., Univ. of Bergen)

Introduction

Strong flow is observed => - Early, local eq., - EoS - n q scaling – QGP flows - no flow in hadronic matter > simultaneous hadronization and FO (HBT, high strangeness abundance)

Relativistic Fluid Dynamics

Eg.: from kinetic theory. BTE for the evolution of phase-space distribution: Then using microscopic conservation laws in the collision integral C: These conservation laws are valid for any, eq. or non-eq. distribution,

f(x,p).

These

cannot be solved

, more info is needed!

Boltzmann H-theorem: (i) for arbitrary f, the entropy increases, (ii) for stationary, eq. solution the entropy is maximal,  

EoS

P = P (e,n)

Solvable

for local equilibrium!

Relativistic Fluid Dynamics

For any EoS,

P=P(e,n),

and any energy-momentum tensor in LE(!): Not only for high v!

Two theoretical problems

• Initial state – Fitted initial states > moderate insight • Final Freeze Out Realistic Model, Continuos FO, ST layer, Non-eq. distribution

Stages of a Collision

Freeze Out >>> Detectors Hadronization, chemical FO, kinetic FO ------------- One fluid >>>

E O S

Fluid components, Friction Local Equilibration, Fluids Collective flow reveals the EoS if we have dominantly one fluid with local equilibrium in a substantial part of the space-time domain of the collision !!!

Heavy Colliding System

Idealizations FO Layer FO HS

QGP EoS One fluid

Hadronization Chemical Freeze Out Kinetic Freeze Out

time

Fire streak picture -

Only in 3 dimensions!

Myers, Gosset, Kapusta, Westfall

String rope --- Flux tube --- Coherent YM field

3 rd flow component

Initial state

3-Dim Hydro for RHIC (PIC)

.

3-dim Hydro for RHIC Energies

Au+Au E CM =65 GeV/nucl. b=0.5 b max A σ =0.08 => σ~10 GeV/fm e [ GeV / fm 3 ] T [ MeV] .

t=0.0 fm/c, T max = 420 MeV, e max = 20.0 GeV/fm 3,

8.7 x 4.4 fm

EoS: p= e/3 - B/3, L x,y = 1.45 fm, L z =0.145 fm B = 397 MeV/fm 3

.

Au+Au E CM =65 GeV/nucl. b=0.5 b max A σ =0.08 => σ~10 GeV/fm e [ GeV / fm 3 ] T [ MeV] .

t=9.1 fm/c, T max = 417 MeV, e max = 19.6 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm

20.3 x 5.8 fm

.

Au+Au E CM =65 GeV/nucl. b=0.5 b max A σ =0.08 => σ~10 GeV/fm e [ GeV / fm 3 ] T [ MeV] .

t=18.2 fm/c, T max = 417 MeV, e max = 19.4 GeV/fm 3, L x,y = 1.45 fm, L z =0.145 fm 34.8 x 8.7 fm

Global Flow patterns:

3 rd flow component (anti - flow) Directed Transverse flow Elliptic flow Squeeze out

3 rd flow component Csernai & Röhrich [Phys.Lett.B458(99)454]

Hydro [Csernai, HIPAGS’93]

Preliminary

“Wiggle”, Pb+Pb, E lab =40 and 158GeV [NA49] A. Wetzler 158 GeV/A v

1

< 0 The “wiggle” is there!

Flow is a diagnostic tool

Impact par.

Transparency – string tension Equilibration time Consequence: v 1 (y), v 2 (y), …

Freeze Out FOHS - Movies:

B=0, T-fo = 139MeV B=0.4, T-fo = 139MeV B=0, T-fo = 180MeV B=0.4, T-fo = 180MeV [Bernd Schlei, Los Alamos, LA-UR-03-3410]

• (B) - Freeze out over FOHS - post FO distribution? = 1 st .: n, T, u, cons. Laws !

= 2 nd .: non eq. f(x,p) !!! -> (C) • (Ci) Simple kinetic model • (Cii) Covariant, kinetic F.O. description • (Ciii) Freeze out form transport equation • Note: ABC together is too involved!

B & C can be done separately -> f(x,p)

The Boltzmann Transport Equation and Freeze Out

Freeze out is : • • • • Strongly

directed Delocalized:

process: The m.f.p. - reaches infinity Finite

characteristic length The change is not negligible in the FO direction Modified Boltzmann Transport Equation for Freeze Out description

Early models: 1 The invariant “ Escape” probability in finite layer The escape form the

int

to

free

component • Not to collide, depends on remaining distance • If the particle momentum is not normal to the surface, the spatial distance increases

The invariant “ Escape” probability A t’ x’ D [RFG] B E C F Escape probability factors for different points on FO hypersurface, in the RFG . Momentum values are in units of [mc]

Results – the cooling and retracting of the interacting matter

Space-Like FO Time-Like FO [RFF] [RFF]

cooling



Cut-off factor retracting flow velocity No Cut-off

Results – the contour lines of the FO distribution, f(p) jump in [RFF]

With different initial flow velocities [RFF]

Recent open, flow related issues

• Is QGP a “perfect fluid” ? – Small (?) viscosity, but strong interaction (?) Laminar flow, not turbulent -> large viscosity Cascades need high cross section to reproduce flow • Comprehensive flow assessment v1, v2, v3 … should be evaluated on equal footing There is one reaction plane,  , (not  1  2  3 … ) y,  , pT correlations are equally important (y ?) • Solution: Event by Event flow evaluation