2008-04-21M. Csanád, T. Csörg ő, M.I. Nagy New analytic results in hydrodynamics UTILIZING THE...

2008-04-21

M. Csanád, T. Csörgő, M.I. Nagy

New analytic results in hydrodynamics

UTILIZING THE FLUID NATURE OF QGP

M. Csanád, T. Csörgő, M. I. Nagy

ELTEMTA KFKI RMKI

Budapest, Hungary

Hydrodynamics at RHIC and QCD EOS Workshop, BNL, USA

April 21, 2008

2008-04-21


High temperature superfluidity at RHIC!

All “realistic” hydrodynamic calculations for RHIC fluids to date have assumed zero viscosity= 0 →perfect fluid

a conjectured quantum limit:P. Kovtun, D.T. Son, A.O. Starinets, hep-th/0405231 How “ordinary” fluids compare to this limit?(4 ) η/s > 10

RHIC’s perfect fluid (4 ) η/s ~1 !

T > 2 TerakelvinThe hottest

& most perfect fluid

ever made… (( 44

sπ

Density)(Entropyπ

η44

R. Lacey et al., Phys.Rev.Lett.98:092301,2007

2008-04-21


Relativistic hydrodynamics

Energy-momentum tensor:

Relativistic

Euler equation:

Energy conservation:

Charge conservation:

Consequence is entropy conservation:

2008-04-21


Context

Renowned exact solutions

Landau-Khalatnikov solution: dn/dy ~ Gaussian

Hwa solution (PRD 10, 2260 (1974)) - Bjorken 0 estimate (1983)

Chiu, Sudarshan and Wang: plateaux

Baym, Friman, Blaizot, Soyeur and Czyz: finite size parameter

Srivastava, Alam, Chakrabarty, Raha and Sinha: dn/dy ~ Gaussian

Revival of interest: Buda-Lund model + exact solutions,

Biró, Karpenko+Sinyukov, Pratt (2007),

Bialas+Janik+Peschanski, Borsch+Zhdanov (2007)

New simple solutions

Evaluation of measurables

Rapidity distribution Advanced initial energy density

HBT radii Advanced life-time estimation

2008-04-21


Goal

Need for solutions that are:explicit

simple

accelerating

relativistic

realistic / compatible with the data:

lattice QCD EoS

ellipsoidal symmetry (spectra, v2, v4, HBT)

finite dn/dy

Report on a new class that satisfies each of these criteria

but not simultaneously

M.I. Nagy, T. Cs., M. Csanád, arXiv:0709.3677v1 , PRC77:024908 (2008)

T. Cs, M. I. Nagy, M. Csanád, arXiv:nucl-th/0605070v4, PLB (2008)

M. Csanád, M. I. Nagy, T. Cs, arXiv:0710.0327v3 [nucl-th] EPJ A (2008)

2008-04-21


Self-similar, ellipsoidal solutions

Publication (for example):T. Csörgő, L.P.Csernai, Y. Hama, T. Kodama, Heavy Ion Phys. A 21 (2004) 73

3D spherically symmetric HUBBLE flow:No acceleration:

Define a scaling variable for self-similarly expanding ellipsoids:

EoS: (massive) ideal gas

Scaling function (s) can be chosen freely.

Shear and bulk viscous corrections in NR limit: known analytically.

2008-04-21


Some general remarks

Hydrodynamics=

Initial conditions dynamical equations freeze-out conditionsExact solution = formulas solve hydro without approximation

Parametric solution = shape parameters introduced,

time dependence given by ordinary coupled diff. eqs.

Hydro inspired parameterization

= shape parameters determined only at the freeze-out

their time dependence is not considered

Report on new class of exact, parametric solution of relativistic hydro

M.I. Nagy, T. Cs., M. Csanád, arXiv:0709.3677v1 , PRC77:024908 (2008)

T. Cs, M. I. Nagy, M. Csanád, arXiv:nucl-th/0605070v4, PLB (2008)

M. Csanád, M. I. Nagy, T. Cs, arXiv:0710.0327v3 [nucl-th] EPJ A (2008)

Initial conditions: pressure and velocity on = 0 = const

EoS: - B = (p+B) cs2 = 1/

Freeze-out condition: T= Tf (= 0), local simultaneity, n = u

2008-04-21


New, simple, exact solutions

If If = d = 1 , general solution is obtained, for = d = 1 , general solution is obtained, for ARBITRARY initial conditions. It is initial conditions. It is STABLESTABLE ! !

Possible cases (one row of the table is one Possible cases (one row of the table is one solution):solution):

Hwa-Bjorken, Buda-Lund Hwa-Bjorken, Buda-Lund typetype

New, accelerating, d New, accelerating, d dimensiondimensiond dimensional withd dimensional with p=p(p=p(,,) ) (thanks T. S. (thanks T. S. Biró)Biró)

Special EoS, but general Special EoS, but general velocityvelocity

Nagy,CsT, Csanád:Nagy,CsT, Csanád: arXiv:0709.3677v1

2008-04-21



Different final states from similar initial states are reached by varying

2008-04-21



Similar final states from different initial states are reached by varying

2008-04-21


Rapidity distribution

Rapidity distribution from the 1+1 dimensional solution, for > 1.

Tf: slope parameter.

2008-04-21


Pseudorapidity distributions

BRAHMS data fitted with the analytic formula ofAdditionally: yη transformation

2008-04-21


BRAHMS rapidity distribution

BRAHMS dn/dy data fitted with the analytic formula

2008-04-21


Advanced energy density estimate

Fit result: Fit result: > 1> 1

Flows accelerate: Flows accelerate: do workdo work

initial energy density is higher than initial energy density is higher than Bjorken’s Bjorken’s

Work and acceleration. Work and acceleration.

FYI:FYI:

For For > 1 (accelerating) flows, both factors > 1> 1 (accelerating) flows, both factors > 1

2008-04-21


Advanced energy density estimate

Correction depends on timescales, dependence is:Correction depends on timescales, dependence is:

With a typical With a typical ff//00 of ~8-10, one gets a correction of ~8-10, one gets a correction factor of 2!

2008-04-21


Conjecture: EoS dependence of 0

Four constraintsFour constraints

1) 1) BjBj is independent of EoS ( is independent of EoS (= 1 case)= 1 case)

2) c2) css22= 1 case is solved for any = 1 case is solved for any > 0.5> 0.5

Corrections due to respect these limits.Corrections due to respect these limits.

3) c3) css22 dependence of dependence of is known in NR limit is known in NR limit

arXiv:hep-ph/0111139v2

4) Numerical hydro results, 4) Numerical hydro results, e.g. K. Morita, arXiv:nucl-th/0611093v2 Conjectured formula – given by the principle of Occam’s Conjectured formula – given by the principle of Occam’s razor:razor:

Using Using = 1.18, c = 1.18, css = 0.35, = 0.35, ff//00 = 10, we get = 10, we get ccss//BjBj = 2.9 = 2.9

0 = 14.5 GeV/fm3 in 200 GeV, 0-5 % Au+Au at RHIC in 200 GeV, 0-5 % Au+Au at RHIC

2008-04-21


Conjectured EoS dependence of 0

Using Using = 1.18, and = 1.18, and ff//00 = 10 as before = 10 as before

and cand css = 0.35, = 0.35, [PHENIX,[PHENIX, arXiv:nucl-ex/0608033v1 ] we get we get ccss//BjBj = 2.9 = 2.9

0 = 14.5 GeV/fm3 in 200 GeV, 0-5 % Au+Au at RHIC in 200 GeV, 0-5 % Au+Au at RHIC

2008-04-21


Advanced life-time estimate

Life-time estimation: for Hwa-Bjorken type of flows

Makhlin & Sinyukov, Z. Phys. C 39, 69 (1988)

Underestimates lifetime (Renk, CsT, Wiedemann, Pratt, …)

Advanced life-time estimate:

width of dn/dy related to acceleration and work

At RHIC energies: correction is about +20%

2008-04-21


Conjectured EoS dependence of c

Using Using = 1.18, and c = 1.18, and css = 0.35, = 0.35, we get we get cs/Bj = 1.36

in 200 GeV, 0-5 % Au+Au at RHIC in 200 GeV, 0-5 % Au+Au at RHIC

2008-04-21


Conclusions

Explicit simple accelerating relativistic hydrodynamicsAnalytic (approximate) calculation of observables Realistic rapidity distributions; BRAHMS data well described

No go theorem: same final states, different initial states

New estimate of initial energy density: c/Bj at least 2 @ RHIC

dependence on cs estimated, c/Bj ~ 3 for cs = 0.35

Estimated work effects on lifetime: at least 20% increase @ RHIC

dependence on cs estimated, c/Bj ~ 1.4 for cs = 0.35

A lot to do …more general EoSless symmetry, ellipsoidal solutionsasymptotically Hubble-like flows

2008-04-21


New simple solutions in 1+D dim

Fluid trajectories of the 1+D dimenisonal new solution

THANK YOU!

2008-04-21


Some comments

2008-04-21


RHIC and the Phase “Transition”The lattice tells us that collisions at RHIC map out

the interesting region

from Bj ~ 5 GeV/fm3

for flat dn/dy

to

c ~ 15 GeV/fm3

for finite dn/dy

~ from RHIC

to LHC

What about SPS?

2008-04-21


Models that pass the HBT test

Models with acceptable results:

nucl-th/0204054 Multiphase Trasport model (AMPT)Z. Lin, C. M. Ko, S. Pal

nucl-th/0205053 Hadron cascade modelT. Humanic

hep-ph/9509213 Buda-Lund hydro modelnucl-th/0305059 T. Csörgő, B. Lörstad, A. Ster

hep-ph/0209054 Cracow (single freeze-out, thermal) W. Broniowski, W. Florkowski

nucl-ex/0307026 Blast wave modelF. Retiére for STAR

arXiv:0801.4361v1 2 + 1 boost invariant rel. hydrodynamical solution,

Gaussian IC, lattice QCD EoS, resonance decays

W. Broniowski, M. Chojnacki, W. Florkowski, A. Kisiel

2008-04-21


Comments on RHIC HBT puzzle

Spectra, v2 and HBT radii described by ideal hydro + resonance decays

using Gaussian initial pressure profile and directional Hubble flow

W. Broniowski et al, arXiv:0801.4361v1

2008-04-21


Back-up Slides

2008-04-21


How Perfect is Perfect? Measure η/s !Damping (flow, fluctuations, heavy quark motion)

~ η/sFLOW: Has the QCD Critical Point Been

Signaled by Observations at RHIC?, R. Lacey et al., Phys.Rev.Lett.98:092301,2007 (nucl-ex/0609025)

The Centrality dependence of Elliptic flow, the Hydrodynamic Limit, and the Viscosity of Hot QCD, H.-J. Drescher et al., (arXiv:0704.3553)

FLUCTUATIONS: Measuring Shear Viscosity Using Transverse Momentum Correlations in Relativistic Nuclear Collisions, S. Gavin and M. Abdel-Aziz, Phys.Rev.Lett.97:162302,2006 (nucl-th/0606061)

DRAG, FLOW: Energy Loss and Flow of Heavy Quarks in Au+Au Collisions at √sNN = 200 GeV (PHENIX Collaboration), A. Adare et al., Phys.Rev.Lett.98:172301,2007 (nucl-ex/0611018)

π)±±(=

s

η

4

1 .2 1 .2 0 .1 1

π)(=

s

η

4

1 .8 3 .0 1

π)(=

s

η

4

1 .0 2 .3 1

π)(=

s

η

4

1 .5 2 .9 1

CHARM!

2008-04-21


Landau-Khalatnikov solutionPublications:

L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51

I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529

L.D.Landau and S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) 309

Implicit 1D solution with approx. Gaussian rapidity distribution

Basic relations:

Unknown variables:

Auxiliary function:

Expression of is a true „tour de force”

1 1cosh sinh , sinh cosht x

T T T T

1 1

cosh sinh , sinh cosht xT T T T

0 1cosh ( , ), sinh ( , )

( , ), ( , )

( , )

u t x u t x

T t x t x

T

0 1cosh ( , ), sinh ( , )

( , ), ( , )

( , )

u t x u t x

T t x t x

T

( , )T ( , )T

2008-04-21


Landau-Khalatnikov solution

Temperature distribution (animation courtesy of T. Kodama)

„Tour de force” implicit solution: t=t(T,v), r=r(T,v)

2008-04-21


Hwa-Bjorken solution

The Hwa-Bjorken solution / Rindler coordinates

2008-04-21


Hwa-Bjorken solution

The Hwa-Bjorken solution / Temperature evolution

2008-04-21


Bialas-Janik-Peschanski solutionPublications:

A. Bialas, R. Janik, R. Peschanski, arXiv:0706.2108v1

Accelerating, expanding 1D solution

interpolates between Landau and Bjorken

Generalized Rindler coordinates:

2008-04-21


0

0

( ), ( ), ( ), , 0

( ) 0

( )

x z dp p s s u v

t dts

sus

T T

v

0

0

( ), ( ), ( ), , 0

( ) 0

( )

x z dp p s s u v

t dts

sus

T T

v

Hwa-Bjorken solutionPublications:

R.C. Hwa, Phys. Rev. D10, 2260 (1974)

J.D. Bjorken, Phys. Rev. D27, 40(1983)

Accelerationless, expanding 1D simple boost-invariant solution

Rindler coordinates:

Boost-invariance (valid for asymptotically high energies):

2 2

cosh , sinh

arctanh ,

t r

rt r x x

t

2 2

cosh , sinh

arctanh ,

t r

rt r x x

t

00, , 1 ( )p D D T T

00, , 1 ( )p D D T T

depends on EoS, e.g.depends on EoS, e.g.

2008-04-21


New simple solutions in 1+d dim

.A constThe fluid lines (red) and the pseudo-orthogonal freeze-out surface (black)

2008-04-21


1 1

Rapidity distribution

Rapidity distribution from the 1+1 dimensional solution, for .

2008-04-21


1st milestone: new phenomena

Suppression of high pt particle production in Au+Au collisions at RHIC

2008-04-21


2nd milestone: new form of matter

d+Au: no suppression

Its not the nuclear effect

on the structure functions

Au+Au:

new form of matter !

2008-04-21


3rd milestone: Top Physics Story 2005

http://arxiv.org/abs/nucl-ex/0410003

PHENIX White Paper: second most cited in nucl-ex during 2006

2008-04-21


Strange and even charm quarks participate in the flow Strange and even charm quarks participate in the flow

vv22 for the φ follows that for the φ follows that

of other mesonsof other mesons

vv22 for the D follows that for the D follows that

of other mesonsof other mesonsv2hadron KET

hadron nv2quark KET

quark

KEThadron nKE

Tquark

4th Milestone: A fluid of quarks

2008-04-21


Predictions of the Buda-Lund model

Hydro predicts scaling (even viscous)

What does a scaling mean? See Hubble’s law – or Newtonian gravity:

Cannot predict acceleration or height

Collective, thermal behavior →

Loss of information

Spectra slopes:

Elliptic flow:

HBT radii:

2t gh 2t gh

2eff 0 tT T mu 2eff 0 tT T mu

12

0

( ), ~

( ) T

I wv w KE

I w 1

20

( ), ~

( ) T

I wv w KE

I w

2 2 2side long out

1~

t

R R Rm

2 2 2side long out

1~

t

R R Rm

2008-04-21


data

Axial Buda-Lund

Ellipsoidal Buda-Lund

Perfectnon-

relativistic solutions

Relativistic solutions

w/o acceleration

Relativistic solutions

w/acceleration

Dissipativenon-

relativistic solutions

HwaBjorkenHubble

What does the data tell us

2008-04-21


BudaLund fits to 130 GeV RHIC data

M. Csanád, T. Csörgő, B. Lörstad, A. Ster, nucl-th/0311102, ISMD03

2008-04-21


BudaLund fits to 200 GeV RHIC data

M. Csanád, T. Csörgő, B. Lörstad, A. Ster, nucl-th/0403074, QM04

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2008-04-21M. Csanád, T. Csörg ő, M.I. Nagy New analytic results in hydrodynamics UTILIZING THE...

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