exact results in analytic hydrodynamics
DESCRIPTION
Exact results in analytic hydrodynamics. UTILIZING THE FLUID NATURE OF QGP M. Csanád, T. Csörgő , M. I. Nagy ELTE MTA KFKI RMKI Budapest, Hungary Quark Matter 2008, Jaipur, Rajastan, India February 8, 200 8. High temperature superfluidity at RHIC!. - PowerPoint PPT PresentationTRANSCRIPT
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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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
Equations of relativistic hydroFour-momentum tensor:
Relativistic
Euler equation:
Energy conservation:
Charge conservation:
Consequence is entropy conservation:
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
New simple solutions
Different final states from similar initial states are reached by
varying
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
New simple solutions
Similar final states from different initial states are reached by
varying
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
Rapidity distribution
Rapidity distribution from the 1+1 dimensional solution, for .1 1
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
Pseudorapidity distribution
BRAHMS data fitted with the analytic formula ofAdditionally: yη transformation
dn
dy
dn
dy
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
Rapidity distribution
BRAHMS data fitted with the analytic formula ofdn
dy
dn
dy
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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!
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
New simple solutions in 1+D dim
Fluid trajectories of the 1+D dimenisonal new solution
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
Back-up Slides
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
Landau-Khalatnikov solution
Temperature distribution (animation courtesy of T.
Kodama)
„Tour de force” implicit solution: t=t(T,v), r=r(T,v)
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
Hwa-Bjorken solution
The Hwa-Bjorken solution / Rindler coordinates
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
Hwa-Bjorken solution
The Hwa-Bjorken solution / Temperature evolution
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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:
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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.
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
New simple solutions in 1+d dim
.A constThe fluid lines (red) and the pseudo-orthogonal freeze-out surface
(black)
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
1 1
Rapidity distribution
Rapidity distribution from the 1+1 dimensional solution, for .
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
1st milestone: new phenomena
Suppression of high pt particle production in Au+Au collisions at RHIC
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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 !
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
3rd milestone: Top Physics Story 2005
http://arxiv.org/abs/nucl-ex/0410003
PHENIX White Paper: second most cited in nucl-ex during 2006
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2008-02-08
M. Csanád, T. Csörgő, M.I. Nagy
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