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from new solutions of Navier-Stokes eq uation s. Scaling properties of HBT radii and v 2. T. Csörgő, M. Csanád and Y. Hama MTA KFKI RMKI, Budapest, Hungary ELTE University, Budapest, Hungary USP, Sao Paulo, Brazil. Introduction: Observed scaling of spectra, elliptic flow and HBT radii - PowerPoint PPT Presentation
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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
2007-Aug-1 T. Csörgő @ WPCF
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)
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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)
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
Dissipative, Navier-Stokes fluids
Navier-Stokes equations: dissipative, nonrelativistic:
EoS needed:
Shear and bulk viscosity, heat conductivity:
2007-Aug-1 T. Csörgő @ WPCF
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.
2007-Aug-1 T. Csörgő @ WPCF
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.
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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??
2007-Aug-1 T. Csörgő @ WPCF
Illustration, (in)dependence on EOS
The initial conditions and the EoScan covary so that
the freeze-out distributions are unchanged(T/m = 180/940)
2007-Aug-1 T. Csörgő @ WPCF
Initial and Freeze-out conditions:
Differentinitial conditions
lead to
same freeze-outconditions.
Ambiguity!
Penetratingprobesradiatethroughthe time evolution!
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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)
2007-Aug-1 T. Csörgő @ WPCF
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)
2007-Aug-1 T. Csörgő @ WPCF
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??
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
Buda-Lund hydro prediction: Exact non-rel. hydro
PHENIX data:
Hydro scaling of slope parameters
2007-Aug-1 T. Csörgő @ WPCF
Universal hydro scaling of v2
Black line:Theoretically predicted, universalscaling functionfrom analytic workson perfect fluid hydrodynamics:
hep-ph/0108067, nucl-th/0310040nucl-th/0512078
2007-Aug-1 T. Csörgő @ WPCF
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!
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
Back-up Slides
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
Some analytic Buda-Lund results
HBT radii widths:
Slopes, effective temperatures:
Flow coefficients are universal:
2007-Aug-1 T. Csörgő @ WPCF
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.
2007-Aug-1 T. Csörgő @ WPCF
#2: Our last work: inflation at RHIC
nucl-th/0206051
2007-Aug-1 T. Csörgő @ WPCF
1st milestone: new phenomena
Suppression of high pt particle production in Au+Au collisions at RHIC
2007-Aug-1 T. Csörgő @ WPCF
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 !
2007-Aug-1 T. Csörgő @ WPCF
3rd milestone: Top Physics Story 2005
http://arxiv.org/abs/nucl-ex/0410003
PHENIX White Paper: second most cited in nucl-ex during 2006
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
● This one figure encodes rigorous control of systematics
● in four different measurements over many orders of magnitude
Precision Probes
centralNcoll
= 975 94
== ==
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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:
2007-Aug-1 T. Csörgő @ WPCF
Illustrations of exact hydro results
● Propagate the hydro solution in time numerically:
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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:
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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
Relativistic correction: m -> mt
hep-ph/0108067,nucl-th/0206051
2007-Aug-1 T. Csörgő @ WPCF
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
2007-Aug-1 T. Csörgő @ WPCF
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
Relativistic correction: m -> mt
hep-ph/0108067,nucl-th/0206051
2007-Aug-1 T. Csörgő @ WPCF
Hydro scaling of elliptic flow
G. Veres, PHOBOS data, proc QM2005Nucl. Phys. A774 (2006) in press
2007-Aug-1 T. Csörgő @ WPCF
Hydro scaling of v2 and dependence
PHOBOS, nucl-ex/0406021PHOBOS, nucl-ex/0406021
s
2007-Aug-1 T. Csörgő @ WPCF
Universal scaling and v2(centrality,)
PHOBOS, nucl-ex/0407012PHOBOS, nucl-ex/0407012
2007-Aug-1 T. Csörgő @ WPCF
Universal v2 scaling and PID dependence
PHENIX, PHENIX, nucl-ex/0305013nucl-ex/0305013
2007-Aug-1 T. Csörgő @ WPCF
Universal scaling and fine structure of v2
STAR, nucl-ex/0409033STAR, nucl-ex/0409033
2007-Aug-1 T. Csörgő @ WPCF
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.