norway relativistic hydrodynamics and freeze-out lászló csernai (bergen computational physics...

Norway

Relativistic Relativistic Hydrodynamics and Hydrodynamics and

Freeze-outFreeze-out

Relativistic Relativistic Hydrodynamics and Hydrodynamics and

Freeze-outFreeze-out

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

Relativistic Fluid DynamicsRelativistic 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,

P = P (e,n) Solvable for local equilibrium!

EoSEoS

Relativistic Fluid DynamicsRelativistic Fluid DynamicsFor any EoS, P=P(e,n), and any energy-momentum tensor in LE(!):

Not only for high v!

Local Equilibration, Fluids

Fluid components, Friction

-------------- One fluid >>> E O E O SS -------------- One fluid >>> E O E O SS

Hadronization, chemical FO, kinetic FO

Freeze Out >>> Detectors

Stages of a CollisionStages of a CollisionStages of a CollisionStages of a Collision

Collective flow reveals the EoS ifwe have dominantly one fluid with local equilibrium in a substantial part of the space-time domain of the collision !!!

QGP EoS QGP EoS One fluidOne fluid

HadronizatiHadronizationon Chemical Freeze Chemical Freeze

OutOut Kinetic Freeze OutKinetic Freeze Out

Initi

al st

ate

time

Heavy Colliding SystemHeavy Colliding SystemHeavy Colliding SystemHeavy Colliding System

IdealizatioIdealizationsnsFO LayerFO Layer

FO HSFO HS

Small SystemSmall SystemSmall SystemSmall System

timeInitial state, Pre-equilibrium, cascade, Multi

Component Fluidno unique EoS Hadronization & Freeze Out

(One-Fluid)

FDFD

Multi Module ModelingMulti Module ModelingMulti Module ModelingMulti Module Modeling

• A: Initial state - Fitted to measured data (?)• B: Initial state - Pre-equilibrium: Parton

Cascade; Coherent Yang-Mills [Magas]• Local Equilibrium Hydro, EoS• Final Freeze-out: Kinetic models, measurables.

- If QGP Sudden and simultaneous hadronization and freeze out (indicated by HBT, Strangeness, Entropy puzzle)

Landau (1953), Milekhin (1958), Cooper & Frye (1974)

Fire streak picture - Only in 3 dimensions!Fire streak picture - Only in 3 dimensions!

Myers, Gosset, Kapusta, Westfall

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

Initial state

3rd flow component

3-dim Hydro for RHIC Energies

Au+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10 GeV/fm

e [ GeV / fm3 ] T [ MeV]

t=0.0 fm/c, Tmax= 420 MeV, emax= 20.0 GeV/fm3, Lx,y= 1.45 fm, Lz=0.145 fm

. .

EoS: p= e/3 - 4B/3, B = 397 MeV/fm3 8.7 x 4.4 fm

3-dim Hydro for RHIC EnergiesAu+Au ECM=65 GeV/nucl. b=0.5 bmax Aσ=0.08 => σ~10 GeV/fm



. .

11.6 x 4.6 fm




. .

14.5 x 4.9 fm




. .

17.4 x 5.5 fm




. .

20.3 x 5.8 fm




. .

23.2 x 6.7 fm




. .

26.1 x 7.3 fm




. .

31.9 x 8.1 fm




. .

34.8 x 8.7 fm

Global Flow patterns:Directed

Transverse flow

Elliptic flow

3rd flow component(anti - flow)

Squeeze out

3rd flow component

Hydro

[Csernai, HIPAGS’93]

[Phys.Lett.B458(99)454]Csernai & Röhrich

“Wiggle”, Pb+Pb, Elab=40 and 158GeV [NA49]

A. Wetzler

Preliminary

158 GeV/A

The “wiggle” is there!

v1 < 0

Flow is a diagnostic toolFlow is a diagnostic toolFlow is a diagnostic toolFlow is a diagnostic tool

Impact Impact par.par.

Transparency – Transparency – string tensionstring tension

EquilibrationEquilibrationtimetime

Consequence:Consequence:vv11(y), v(y), v22(y), …(y), …

• Kinetic freeze out: Strongly interacting matter becomes dilute and cold, the momentum distr. of particles in the absence of collisions freezes out and the particles propagate toward the detectors.

• Sometimes sequential FO is assumed: Chemical + Kinetic FO

• Now: Rapid, simultaneous FO and hadronization from super-cooled QGP in a thin layer (2-3 fm). The process of gradual FO is followed by kinetic description.

What is Freeze Out (FO) ???What is Freeze Out (FO) ???What is Freeze Out (FO) ???What is Freeze Out (FO) ???

(A)- identification of the Freeze Out Hyper-Surface (FOHS) [B. Schlei]

(B)- idealized FO over the FOHS

(C)- FO over a finite LAYER & described by kinetic theory, by the MBTE.

(A) – Movies:

B=0, T-fo = 139MeV

B=0, T-fo = 180MeV

B=0.4, T-fo = 139MeV

B=0.4, T-fo = 180MeV

[Bernd Schlei, Los Alamos, [Bernd Schlei, Los Alamos, LA-UR-03-3410]

• (B) - Freeze out over FOHS- post FO distribution?= 1st.: n, T, u, cons. Laws != 2nd.: 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)

(A) Freeze out as a (A) Freeze out as a discontinuitydiscontinuity

• Theory of discontinuities in relativistic flow (only space-like), [Taub, 1948]

• Generalization for both time-like and space-like discontinuities, [1987]

• 1st problem: We must use the correct parameters of the matter in the Post FO distribution

Matching conditions:

Discontinuity = where the properties of matter change suddenly

normal vector

=> n, T, u=> n, T, u

(A) Modified Cooper-Frye (A) Modified Cooper-Frye FormulaFormula

• The number of particles crossing,

• The kinetic definition of the particle four-flow

• Cooper-Frye formula, Cooper and Frye, 1975

• 2nd : Problem of negative contributions, on the space-like part of the hyper-surface, Sinyukov, et al.

• Sharp cut-off proposed, Bugaev, 1996• Solved, kinetic model, Anderlik et al. 1999,

Magas et al. 1999

Not known form hydro !

Kinetic freeze out models,Kinetic freeze out models,(a) just FO(a) just FO

Assumptions:• only short range interactions• one dimensional geometry• assume stationary flow, no explicit time

dependence

• Simple kinetic model

• Kinetic model (b) with re-scattering, re-thermalization:

Reference FramesReference Frames

• Pre FO – Local Rest frame of the gas, where the matter is at rest = Rest Frame of the Gas (RFG), moving with the peak of the Pre-FO distribution

• Post FO – Rest Frame of the Front, (RFF) attached to the FO front

p’x

(a)(a) Reproduced the Reproduced the cut Juttnercut Juttner distribution as the post FO distribution as the post FO distribution, --- but: the other half of distribution, --- but: the other half of the distribution remained there, the distribution remained there, interacting for everinteracting for ever

(b)(b) Gradually all matter froze out, no Gradually all matter froze out, no negative contribution, result negative contribution, result approximated by CJ distribution --- approximated by CJ distribution --- but: it took infinite time – unrealistic. but: it took infinite time – unrealistic.

Bondorf et al. [NP A296 (‘78) 320.]: Sph. Expansion -> Bondorf et al. [NP A296 (‘78) 320.]: Sph. Expansion -> increasing divergence & adiabatic cooling -> descreasing increasing divergence & adiabatic cooling -> descreasing random thermal flux <vrandom thermal flux <vrel FO without any collision beyond some radius !!!

22ndnd problem is not finished yet !!! problem is not finished yet !!!22ndnd problem is not finished yet !!! problem is not finished yet !!!

Feasibility: a realistic hydro model + a realistic FOHS modelwith Cooper Fry type FO is theonly manageable model (B).

However, a realistic post FO distribution should be used (!), and this should be investigated (C).

The Boltzmann Transport Equation and Freeze OutThe Boltzmann Transport Equation and Freeze Out

Freeze out is :

• Strongly directed process: • Delocalized:• The m.f.p. - reaches infinity • Finite characteristic length

Modified Boltzmann Transport Equation for Freeze Out

description

The change is not negligible in the FO direction

The Boltzmann Transport EquationThe Boltzmann Transport Equation

A nonlinear equation for dilute gasses, with the following assumptions:

1. Only binary collisions2. The molecular chaos – no correlations: gives the

number of collisions at the respective point

3. A smoothly varying function compared to the m.f.p.

1

2

3

4

Gain term Loss term

The Modified Boltzmann Transport EquationThe Modified Boltzmann Transport Equation

• Introducing, and the FO probability –

which feeds the free component

FO probablity not included !!!

re-thermalization term

free component

interacting component

MBTE ->MBTE -> MBTE ->MBTE ->

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

Early models:

1

The invariant “ Escape” probabilityThe invariant “ Escape” probability

Escape probability factors for different points on FO hypersurface, in the RFG. Momentum values are in units of [mc]

A B C

D E F

t’

x’

[RFG][RFG]

Results – the cooling and retracting of the interacting matterResults – the cooling and retracting of the interacting matter

Space-Like FO Time-Like FO

cooling

retracting

Cut-off factor flow velocity No Cut-off

[RFF] [RFF]

Results – the momentum distributionResults – the momentum distribution


asymmetric elongated in FO direction

curved due to the FO process

[RFF] [RFF]

Results – the contour lines of the FO distribution, f(p)Results – the contour lines of the FO distribution, f(p)


jump in [RFF]

With different initial flow velocities

[RFF] [RFF]

Time-like FO

• Post FO distributions are non – thermal !• Conservation laws must be satisfied !• Post FO distributions must be calculated

from transport theory, or can be approximated with adequate ansatz (Cancelling Juttner distribution)

• Note(!) BTE is not applicable, molecular chaos, and smoothness of the phase space distribution are not applicable for the FO process. Adequate MD models or MBTE models should be applied.

Conclusions

Posters: 2/39 Bravina 2/56 Zschocke 4/125 Zabrodin 5/150 Manninen10/270 Magas 10/272 Molnar, E.

norway relativistic hydrodynamics and freeze-out lászló csernai (bergen computational physics...

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