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 v 2
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T. Csörgő, M. Csanád and Y. Hama MTA KFKI RMKI, Budapest, Hungary
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
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
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.