t. csörgő, m. csanád and y. hama mta kfki rmki, budapest, hungary

2007-Aug-1 T. Csörgő @ WPCF

T. Csörgő, M. Csanád and Y. Hama MTA KFKI RMKI, Budapest, Hungary

ELTE University, Budapest, Hungary

USP, Sao Paulo, Brazil

from new solutions of Navier-Stokes equations

Introduction:

Observed scaling of spectra, elliptic flow and HBT radii

but what are the viscous corrections?

Hydrodynamical scaling observed in RHIC/SPS data

Appear in beautiful, exact family of solutions of fireball hydro

non-relativistic, perfect and viscous exact solutions

relativistic, perfect, accelerating solutions

Their application to data analysis at RHIC energies -> Buda-Lund

Exact results:

what can (or can not) be learned from data?

Scaling properties of HBT radii and v2


Inverse slopes T of pt distribution increase ~ linearly with mass:

T = T0 + m<ut>2

Increase is stronger in more head-on collisions. Suggests collective radial flow, local thermalization and hydrodynamics

Nu Xu, NA44 collaboration, Pb+Pb @ CERN SPST. Cs. and B. Lörstad, hep-ph/9509213

Successfully predicted by Buda-Lund hydro model (T. Cs et al, hep-ph/0108067)

An observation:

PHENIX, Phys. Rev. C69, 034909 (2004)


data

Buda-Lund & exact hydrodynamics

Ellipsoidal Buda-LundPerfect

non-relativistic solutions

Axial Buda-LundRelativistic

solutionsw/o

acceleration

Relativistic solutions

w/acceleration

Dissipativenon-

relativistic solutions

HwaBjorkenHubble


Old idea: Quark Gluon PlasmaParadigm shift: Liquid of quarks

Input from lattice: EoS of QCD Matter

Tc=176±3 MeV (~2 terakelvin)(hep-ph/0511166)

at = 0, a cross-over

Aoki, Endrődi, Fodor, Katz, Szabó

hep-lat/0611014

Lattice QCD EoS for hydro: p(,T)but in RHIC region: p~p(T)

cs2 = p/e = cs

2(T) = 1/(T)

This EoS is in the Buda-Lund family of

exact hydrodynamical solutions!

Tc


Notation for fluid dynamics

Non-relativistic dynamicst: time,

r: coordinate 3-vector, r = (rx, ry, rz),

m: mass,

(t,r) dependent variablesn: number density,

: entropy density,

p: pressure,

: energy density,

T: temperature,

v: velocity 3-vector, v = (vx, vy, vz)


Non-rel perfect fluid dynamicsEquations of nonrelativistic hydro:

local conservation of

charge: continuity

momentum: Euler

energy

EoS needed:

Perfect fluid: 2 equivalent definitions, term used by PDG # 1: no bulk and shear viscosities, and no heat conduction.

# 2: T = diag(e,-p,-p,-p) in the local rest frame.

Ideal fluid: ambiguously defined term, discouraged

#1: keeps its volume, but conforms to the outline of its container

#2: an inviscid fluid


Dissipative, Navier-Stokes fluids

Navier-Stokes equations: dissipative, nonrelativistic:

EoS needed:

Shear and bulk viscosity, heat conductivity:


Parametric perfect hydro solutions

Ansatz: the density n (and T and ) depend on coordinates only through a scale parameter „s”

● T. Cs. Acta Phys. Polonica B37 (2006), hep-ph/0111139

Principal axis of ellipsoid:(X,Y,Z) = (X(t), Y(t), Z(t))

Density=const on ellipsoids. Directional Hubble flow. g(s): arbitrary scaling function. Notation: n ~ (s), T ~ (s) etc.


Family of perfect hydro solutions

T. Cs. Acta Phys. Polonica B37 (2006) hep-ph/0111139 Volume is V = XYZ

= (T) exact solutions: T. Cs, S.V. Akkelin, Y. Hama, B. Lukács, Yu. Sinyukov,hep-ph/0108067,

Phys.Rev.C67:034904,2003or see the sols of Navier-Stokes

later.

The dynamics is reduced to ordinary differential equations for the scales X,Y,Z:

PARAMETRIC solutions.

Ti: constant of integration

Many hydro problems can be easily illustrated and understood on the equivalent problem: a classical potential motion of a mass-point in (a shot)!Note: temperature scaling function (s) arbitrary!

(s) depends on (s). -> FAMILY of solutions.


From the new family of exact solutions, the initial conditions:

Initial coordinates: (nuclear geometry +

time of thermalization)

Initial velocities: (pre-equilibrium+ time of thermalization)

Initial temperature:

Initial density:

Initial profile function: (energy deposition

and pre-equilibrium process)

Initial profile = const of motion = final, observable profile!

Initial boundary conditions


From the new exact hydro solutions,the quantities that determine soft hadronic observables:

Freeze-out temperature: (from small corrections to HBT radii)

Final coordinates: (from small corrections to HBT radii)

Final velocities: (from slopes of particle spectra)

Final density: (enters as normalization factor)

Final profile function: (= initial profile function! from solution)

Final (freeze-out) boundary conditions


The time evolution (trajectory) depends on a „potential term”

through p = 1/cs2, related to the speed of sound:

Role of the Equation of States:

g

Time evolution of the scales (X,Y,Z) follows a classic potential motion.Scales at freeze out determine the hadronic observables. Info on history LOST!No go theorem - constraints on initial conditions (information on spectra, elliptic flow of penetrating probels) indispensable.

The arrow hits the target, but can one determine gravity from this information??


Illustration, (in)dependence on EOS

The initial conditions and the EoScan covary so that

the freeze-out distributions are unchanged(T/m = 180/940)


Initial and Freeze-out conditions:

Differentinitial conditions

lead to

same freeze-outconditions.

Ambiguity!

Penetratingprobesradiatethroughthe time evolution!


Family of viscous hydro solutions

T. Cs.,Y. Hama in preparation Volume is V = XYZ

Similar to hep-ph/0108067

The dynamics is reduced to

non-conservative equations of motion

for the parameters X,Y,Z:

n <-> s, m cancels from new terms:

depends on /s and /s


Dissipative, heat conductive hydro solutions

T. Cs. and Y. Hama, in preparationIntroduction of ‘kinematic’ heat conductivity:

Navier-Stokes, for small heat conduction,solved by the directional Hubble ansatz!Only new eq. from the energy equation:

Asymptotic (large t) role of heat conduction- same order of magnitude (1/t2) as bulk viscosity (1/t2) - shear viscosity term is one order of magnitude smaller

(1/t3) - valid only for nearly constant densities, - destroys self-similarity of the solution (if hot spots)


Illustration, bulk and shear viscosity

The initial conditions and the EoScan covary even in viscous case so that

exactly the same freeze-out distributions(T/m = 180/940, /n = 0.1 and /n = 0.1)


The time evolution (trajectory) depends on the fricition of the air (velocity dependent terms)!

Role of shear and bulk viscosity:

S, B

Time evolution of scales (X,Y,Z) is modified in case of viscosity/friction.

Scales at freeze out determine the hadronic observables. Info on history is LOST in the viscous case too!No go theorem - constraints on initial conditions (information from penetrating probes) indispensable.

The arrow hits the target, but can one determine air friction from this information??


Scaling predictions for (viscous) fluid dynamics

- Slope parameters increase linearly with mass- Elliptic flow is a universal function its variable w is proportional to transverse kinetic energy and depends on slope differences.

Gaussian approx:Inverse of the HBT radii increase linearly with massanalysis shows that they are asymptotically the same

Relativistic correction: m -> mt

hep-ph/0108067,nucl-th/0206051


Buda-Lund hydro prediction: Exact non-rel. hydro

PHENIX data:

Hydro scaling of slope parameters


Universal hydro scaling of v2

Black line:Theoretically predicted, universalscaling functionfrom analytic workson perfect fluid hydrodynamics:

hep-ph/0108067, nucl-th/0310040nucl-th/0512078


Hydro scaling of Bose-Einstein/HBT radii

Rside/Rout ~ 1 Rside/Rlong ~ 1 Rout/Rlong ~ 1

1/R2side ~ mt 1/R2

out ~ mt 1/R2long ~ mt

same slopes ~ fully developed, 3d Hubble flow

1/R2eff=1/R2

geom+1/R2thrm

and 1/R2thrm ~mt

intercept is nearly 0, indicating 1/RG

2 ~0,

thus (x)/T(x) = const!

reason for success of thermal models @ RHIC!


Summary

● Buda-Lund model● Was a parameterization● Is an interpolator btwn analytic, exact hydro

solutions with● Lattice QCD EOS● Shear and Bulk viscosity (NR), heat conductivity (NR)● Relativistic acceleration

● Scaling predictions of viscous hydrodynamics● Scaling properties of slope parameters do not

change● Scaling properties of elliptic flow do not change● Scaling properties of HBT radii are the same

● If shear and bulk viscosities are present in Navier-Stokes eqs.

● Asymptotic analysis:● Initially: shear > bulk > perfect fluid effects● At late times: perfect fluid > bulk > shear effects


Back-up Slides


Hydro problem equivalent to potential motion (a shot)!

Hydro: Shot of an arrow:Desription of data Hitting the target

Initial conditions (IC) Initial position and velocity

Equations of state Strength of the potential

Freeze-out (FC) Position of the target

Data constrain EOS Hitting the target tells the potential (?)

Different IC yields same FC Different archers can hit the target

EoS and IC can co-vary IC and potential co-varied

Universal scaling of v2 F/ma = 1

Viscosity effects Drag force of air

numerical hydro fails (HBT) Arrow misses the target (!)

Understanding hydro results


Some analytic Buda-Lund results

HBT radii widths:

Slopes, effective temperatures:

Flow coefficients are universal:


Role of initial temperature profiles

Determines density profile!Examples of density profiles- Fireball- Ring of fire- Embedded shells of fireExact integrals of hydroScales expand in time

Time evolution of the scales (X,Y,Z)- potential motion.

Scales at freeze out -> observables.- info on history LOST!

No go theorem: - constraints on initial conditions

(penetrating probels) indispensable.


#2: Our last work: inflation at RHIC

nucl-th/0206051


1st milestone: new phenomena

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


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 !


3rd milestone: Top Physics Story 2005

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

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


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


● This one figure encodes rigorous control of systematics

● in four different measurements over many orders of magnitude

Precision Probes

centralNcoll

= 975 94

== ==


Motion Is Hydrodynamic

x

yz

● When does thermalization occur? ● Strong evidence that final state bulk

behavior reflects the initial state geometry

● Because the initial azimuthal asymmetry persists in the final state dn/d ~ 1 + 2 v2(pT) cos (2) + ...

2v2


The “Flow” Knows Quarks● The “fine structure” v2(pT) for different mass

particles shows good agreement with ideal (“perfect fluid”) hydrodynamics

● Scaling flow parameters by quark content nq resolves meson-baryon separation of final state hadrons

baryonsbaryons

mesonsmesons


Dynamics of pricipal axis:

Canonical coordinates, canonical momenta:

Hamiltonian of the motion (for EoS cs

2 = 1/ = 2/3):

The role of initial boundary conditions, EoS and freeze-out in hydro can be understood from potential motion!

From fluid expansion to potential motion

i


Role of initial temperature profile

● Initial temperature profile = arbitrary positive function

● Infinitly rich class of solutions● Matching initial conditions for the density profile

● T. Cs. Acta Phys. Polonica B37 (2006) 1001, hep-ph/0111139

● Homogeneous temperature Gaussian density

● Buda-Lund profile: Zimányi-Bondorf-Garpman profile:


Illustrations of exact hydro results

● Propagate the hydro solution in time numerically:


Principles for Buda-Lund hydro model

● Analytic expressions for all the observables

● 3d expansion, local thermal equilibrium, symmetry

● Goes back to known exact hydro solutions:

● nonrel, Bjorken, and Hubble limits, 1+3 d ellipsoids

● but phenomenology, extrapolation for unsolved cases

● Separation of the Core and the Halo

● Core: perfect fluid dynamical evolution

● Halo: decay products of long-lived resonances

● Missing links: phenomenology needed

● search for accelerating ellipsoidal rel. solutions

● first accelerating rel. solution: nucl-th/0605070


A useful analogy

● Core Sun● Halo Solar wind● T0,RHIC ~ 210 MeV T0,SUN ~

16 million K ● Tsurface,RHIC ~ 100 MeV Tsurface,SUN

~6000 K

Fireball at RHIC our Sun


Buda-Lund hydro model

The general form of the emission function:

Calculation of observables with core-halo correction:

Assuming profiles for

flux, temperature, chemical potential and flow


The generalized Buda-Lund model

The original model was for axial symmetry only, central coll.In its general hydrodynamical form:

Based on 3d relativistic and non-rel solutions of perfect fluid dynamics:

Have to assume special shapes:Generalized Cooper-Frye prefactor:

Four-velocity distribution:

Temperature:

Fugacity:


Buda-Lund model is based on fluid dynamics

First formulation: parameterization based on the flow profiles of

•Zimanyi-Bondorf-Garpman non-rel. exact sol.•Bjorken rel. exact sol.•Hubble rel. exact sol.

Remarkably successfull in describing

h+p and A+A collisions at CERN SPS and at RHIC

led to the discovery of an incredibly rich family ofparametric, exact solutions of•non-relativistic, perfect hydrodynamics•imperfect hydro with bulk + shear viscosity + heat conductivity•relativistic hydrodynamics, finite dn/d and initial acceleration•all cases: with temperature profile !

Further research: relativistic ellipsoidal exact solutionswith acceleration and dissipative terms


Scaling predictions: Buda-Lund hydro

- Slope parameters increase linearly with transverse mass- Elliptic flow is same universal function. - Scaling variable w is prop. to generalized transv. kinetic energy and depends on effective slope diffs.

Inverse of the HBT radii increase linearly with massanalysis shows that they are asymptotically the same


hep-ph/0108067,nucl-th/0206051


Scaling and scaling violations

Universal hydro scaling breakswhere scaling with number of

VALENCE QUARKSsets in, pt ~ 1-2 GeV

Fluid of QUARKS!!

R. Lacey and M. Oldenburg, proc. QM’05A. Taranenko et al, PHENIX: nucl-ex/0608033


Exact scaling laws of NR hydro

- Slope parameters increase linearly with mass- Elliptic flow is a universal function and variable w is proportional to transverse kinetic energy and depends on slope differences.

Inverse of the HBT radii increase linearly with massanalysis shows that they are asymptotically the same


hep-ph/0108067,nucl-th/0206051


Hydro scaling of elliptic flow

G. Veres, PHOBOS data, proc QM2005Nucl. Phys. A774 (2006) in press


Hydro scaling of v2 and dependence

PHOBOS, nucl-ex/0406021PHOBOS, nucl-ex/0406021

s


Universal scaling and v2(centrality,)

PHOBOS, nucl-ex/0407012PHOBOS, nucl-ex/0407012


Universal v2 scaling and PID dependence

PHENIX, PHENIX, nucl-ex/0305013nucl-ex/0305013


Universal scaling and fine structure of v2

STAR, nucl-ex/0409033STAR, nucl-ex/0409033


Solution of the “HBT puzzle”

HBT volumeHBT volumeFull volume

Geometrical sizes keep on increasing. Expansion velocities tend to constants. HBT radii Rx, Ry, Rz approach a direction independent constant.

Slope parameters tend to direction dependent constants.General property, independent of initial conditions - a beautiful exact result.

t. csörgő, m. csanád and y. hama mta kfki rmki, budapest, hungary

Documents

conductive hydro solutions

t exact solutions

family of solutions

parametric solutions

hydro phenix data

density n

energy density

small heat conduction