Exact results in analytic hydrodynamics

2008-02-08

M. Csanád, T. Csörgő, M.I. Nagy

Exact results in analytic hydrodynamics

UTILIZING THE FLUID NATURE OF QGP

M. Csanád, T. Csörgő, M. I. Nagy

ELTEMTA KFKI RMKI

Budapest, Hungary

Quark Matter 2008, Jaipur, Rajastan, India February 8, 2008

2008-02-08


High temperature superfluidity at RHIC!

All “realistic” hydrodynamic calculations for RHIC fluids to date have assumed zero viscosity= 0 →perfect fluidBut there is a conjectured quantum limit:

“A Viscosity Bound Conjecture”, P. Kovtun, D.T. Son, A.O. Starinets,

hep-th/0405231 Where do “ordinary” fluids sit wrt this limit?(4 ) η/s > 10 !

RHIC’s perfect fluid (4 ) η/s ~1

on this scale:The hottest (T > 2 Terakelvin)and the most perfect fluid ever made…

(( 44

sπ

Density)(Entropyπ

η44

2008-02-08


Equations of relativistic hydroFour-momentum tensor:

Relativistic

Euler equation:

Energy conservation:

Charge conservation:

Consequence is entropy conservation:

2008-02-08


ContextReknown 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

2008-02-08


Goal

Need for solutions that are:explicitsimpleacceleratingrelativisticrealistic / compatible with the data:

lattice QCD EoSellipsoidal symmetry (spectra, v2, v4,

HBT)finite dn/dy

Report on a new class that satisfies these criteriabut not all simultaneouslyarXiv:0709.3677v1 [nucl-th] PRC(2008) in press

2008-02-08


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 velocity profile:

No acceleration, i.e.

Define a scaling variable (compatible to flow):

Self-similarly expanding ellipsoids with principal axes of at, bt and ct

Use EoS of a (massive) ideal gas

Scaling function can be chosen freely

u x u x

2 2 2

2 2 2 2 2 2

ss 0

x y z d

a t b t c t dt

2 2 2

2 2 2 2 2 2

ss 0

x y z d

a t b t c t dt

0 ,nm p p nT 0 ,nm p p nT

0u u 0u u

3 3/ 3 3/

0 0 00 0 0 0 0 0

1, s , ,

sp p n n T T p n T

3 3/ 3 3/

0 0 00 0 0 0 0 0

1, s , ,

sp p n n T T p n T

s s

2008-02-08


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 STABLE !ARBITRARY initial conditions. It is STABLE !

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

2008-02-08



Different final states from similar initial states are reached by

varying

2008-02-08



Similar final states from different initial states are reached by

varying

2008-02-08


Rapidity distribution

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

2008-02-08


Pseudorapidity distribution

BRAHMS data fitted with the analytic formula ofAdditionally: yη transformation

dn

dy

dn

dy

2008-02-08



BRAHMS data fitted with the analytic formula ofdn

dy

dn

dy

2008-02-08


Advanced energy density estimate

Fit result: Fit result: > 1> 1

Flows accelerate: Flows accelerate: do workdo work

initial energy density initial energy density >> Bjorken’s Bjorken’s

Corrections due toCorrections due to work work && acceleration. Ref: acceleration. Ref:

At RHIC energies the correction can be as high as a factor of At RHIC energies the correction can be as high as a factor of ~2!~2!

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

2008-02-08


Advanced energy density estimate

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

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

2008-02-08


Advanced life-time estimateLife-time estimation: for Hwa-Bjorken type of flows

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

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

New correction:

dn/dy width related to acceleration and work

At RHIC energies: correction is about +20%

flong Bj

T

TR

m f

long BjT

TR

m

c Bj c Bj

2008-02-08


Conclusions

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

described

New estimate of initial energy density: c/Bj up by factor of 2 @ RHIC

dependence on cs estimated

Estimate of work effects on lifetime: increase by 20% @ RHIC

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

2008-02-08


New simple solutions in 1+D dim

Fluid trajectories of the 1+D dimenisonal new solution

2008-02-08


Back-up Slides

2008-02-08


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

2008-02-08


Landau-Khalatnikov solution

Temperature distribution (animation courtesy of T.

Kodama)

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

2008-02-08


Hwa-Bjorken solution

The Hwa-Bjorken solution / Rindler coordinates

2008-02-08


Hwa-Bjorken solution

The Hwa-Bjorken solution / Temperature evolution

2008-02-08


Bialas-Janik-Peschanski solution

Publications:

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

Accelerating, expanding 1D solution

interpolates between Landau and Bjorken

Generalized Rindler coordinates:

2008-02-08


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.

2008-02-08


New simple solutions in 1+d dim

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

(black)

2008-02-08


1 1


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

2008-02-08


1st milestone: new phenomena

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

2008-02-08


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 !

2008-02-08


3rd milestone: Top Physics Story 2005

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

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

2008-02-08


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 mesons

4th Milestone: A fluid of quarks

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Exact results in analytic hydrodynamics

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