Hydrodynamics as tool for diagnosing Hydrodynamics as tool for diagnosing
heavy ion collisions
heavy ion collisions - - Ib Ib
Sudden and simultaneous Hadronization & Freeze
Sudden and simultaneous Hadronization & Freeze- -out out – –
CNQ - CNQ - Scaling Scaling
Multi Module Modeling
• Initial state - pre-equilibrium: Parton Cascade; Coherent Yang-Mills [Magas]
• Local Equilibrium Hydro, EoS
• Final Freeze-out: Kinetic models, measurables
• If QGP Sudden and simultaneous
hadronization and freeze out (indicated
by HBT, Strangeness, Entropy puzzle)
Freeze out
Freeze out
Hypersurface
“Cooper-Frye” formula
Conservation Laws across hypersurface
Matching Conditions
Conservation laws Conservation laws
Nondecreasing entropy Nondecreasing entropy
Consequences of conservation laws – Problem I
Aside: Taub-adiabat and Rayleigh line
Perfect fluid
on both sides
of the front!
Aside: Taub-adiabat and Rayleigh line
Aside: Taub-adiabat and Rayleigh line
Aside: Taub-adiabat and Rayleigh line
Goal: scalar equations
I: Parallel Projection
Aside: Taub-adiabat and Rayleigh line
Taub [‘48]
missed the sign was not applicable for freeze- out.
The Rayleigh line is a straight line in the [P,X] plane. It gives the locus of
final states “2” if the initial state “1” is known. The slope, j, is given
by the current across the front.
Aside: Taub-adiabat and Rayleigh line
II: Orthogonal Projection
To obtain scalar eq.
….
Then to obtain scalar: (one of the cross terms and the last term cancel)
Aside: Taub-adiabat and Rayleigh line
Comparing the two equations for the current, j , :
So, we obtain the Taub adiabat :
The locus of the possible final states, “2”, lies on the Taub adiabat.
If the initial state and the EoS of the final state is known the Taub
adiabat with the Rayleigh line determine the final state.
Aside: Taub-adiabat and Rayleigh line
Taub-adiabat for final states:
E.g. Bag Model EoS:
[P + (4B + P
o)/3] (X – X
o/3) = [w
o– 4(B + P
0)/3] X
o/3 Eg. Ideal gas EoS:
(P + 2P
o/3) (X – 2X
o/3) = (e
o- 2P
o/3) 2X
o/3 P
“1”
time-like space-like
space-like
“2” Taub-adiabat
Rayleigh-line
Space-like hypersurface - Problem II
Space-like hypersurface II
Deflagration front of fusion core
• Idealized transition layer of <10-20 cm thickness
• Conservation laws must be satisfied, EM-fields included !!!
• Results simple and strong connections
between the core, crust
and transfer
Freeze Out
Rapid and simultaneous FO and
“hadronization”
• Improved Cooper-Frye FO:
• - Conservation Laws:
• - Post FO distribution:
• Hadronization ~ CQ-s
• - Pre FO: Current and , QGP
• - Post FO: Constituent and
• - are conserved in FO!!!
0,
0 N
T
0 ) ( )
(
p
f p
q q
q
q
N
N and
q q
Rapid and simultaneous FO and “hadronization” can and must be assumed based on experiments as well as studies of phase transition dynamics.
Experiments indicate small source size and large strangeness abundance, as well as CNQ scaling. This means flow and strangeness develop in QGP phase and no time is left for reestablishing chemical balance among light and heavy strange hadrons, or to change the flow via interactions among hadrons.
Observed
Observed n n
qq– – scaling scaling
Flow develops in quark phase, Flow develops in quark phase, there is no further flowthere is no further flow
development after hadronization development after hadronization
R. A. Lacey (2006), nucl-ex/0608046.
Matching Conditions for core/crust boundary
Conservation laws Conservation laws
Nondecreasing entropy Nondecreasing entropy
If the final state is out of Eq., the energy-momentum
Let us consider sudden freeze out and hadronization from QGP:
• Start with 2 flavours (u,d) end with 3 flavours (u,d,s)
• Start with massless quarks and Bbag end with massive constituent quarks (CQs)
• Start with and in QGP end with either
(a) keeping all quarks post FO, i.e. both (very fast FO) (b) keeping only , & re-equilibrating CQs (fast)
Although, these processes happen gradually, during the reaction, the rate of quark equilibration increases exponentially due to increasing quark degeneracy, so we simplify our treatment assuming that these processes happen in the FO layer.
For a time-like FO surface, in RFF, with v0 = v = 0 nB = nB0 & e = e0 and T:
q q
B
n n
n n ~ n
q n
qn
Bn n
B& ~
C q C
q
C
n n
n ~
For small, finite incoming velocities the velocity change (due to pressure change), can be obtained from the momentum conservation:
Fig. The ratio of post and pre FO velocity as function of ε and n for B = 397GeV/ fm3.
Fig. The ratio of post and pre freeze out velocity , δ = (vx – v0)/v0 [%]. Contour lines of δ
In general the FO hyper-surface is not orthogonal to the flow velocities, so this acceleration (deceleration) is an essential consequence of the correct FO description!
In early simplified approach [see mentioned in L.P. Csernai: Introduction to Relativistic Heavy Ion Collisions] it was argued that in a flow one can
choose a ragged FO hyper-surface like this to the right:
t t
x x
Measurable, v2, calculated at FO from pre- & post- FO flow pattern
Fusion device by implosion
• Sufficient
implosion did not work (1 st ) because of Rayleigh – Taylor
instabilities
Radiation dominated implosion Timelike Detonation
Existing nuclear fusion Existing nuclear fusion device
device – – simple simple analytic model analytic model /correcting
/correcting
A. Taub A. Taub (1948) (1948) [L.P. Csernai,
[L.P. Csernai, Zh Zh. . Eksp. Eksp . Teor Teor. . Fiz Fiz . 92 . 92 (l987) 379, (in
(l987) 379, (in Russian);
Russian); Sov Sov. JETP . JETP 65 (l987) 216 (in 65 (l987) 216 (in English).
English).
Post F.O. - Cut-Jüttner distribution
[Bugaev, Nucl.Phys.A606(96)559]
[Anderlik et al., Phys.Rev.C59(99)3309]
Proposed by:
Solved:
p
p
x
y
Post F.O.
distribution:
V-parameter
V-flow
Matching conditions determine 5
parameters only . Ansatz in needed for
) ( )
( ) (
p f
d p p f
J FO
Cut – Jüttner distribution:
Θ(p.dσ) f(x,p)
Problem
II is
partly
solved
Kinetic freeze
Kinetic freeze - - out models out models
Kinetic approach Kinetic approach
f (x,p) out of equilibrium f (x,p) out of equilibrium
Asymmetry Asymmetry
Freeze out model with rescattering
[Anderlik et al., Phys.Rev.C59 (1999) 388-394]
Freeze out distribution with rescattering
V=0 V=0
Freeze out model with rescattering
V = 0.5
V = 0.5
Change of the rest temperature in FO
Change of the rest velocity during FO
[V. Magas, et al., Heavy Ion Phys.9:193-216,1999]
P-t distribution (T=130 MeV)
[V. Magas et al., Phys.Lett.B459(99)33]
Croonin
effect ?
Phase-Space FO probability
Phase-Space FO probability
A B C
D E F
Uniform
=1
Time-like F.O.
Space- like F.O.
d
3s
=u
[A. Anderlik, E. Molnar, et al.]
Freeze out in the layer
Cos L-x q
Freeze out distribution with rescattering from kinetic model across a layer
V=0 V=0
Analytic fit to Kinetic Model Solution :
.
.
[ K. Tamosiunas and L.P. Csernai,
Eur. Phys. J. A20 (04) 269]
Cancelling Juttner Distribution
[Karolis Tamosiunas et al.]
Conclusions
• Hydro works amazingly well! Stronger and stronger hydro effects are observed!
• Equilibrium and EoS exists ( in part of the reaction )
• We have a good possibility to learn more
and more about the EoS, with improved
experimental and theoretical accuracy!
Multi Module Modeling
• Initial state - pre-equilibrium: Parton Cascade; Coherent Yang-Mills [Magas]
• Local Equilibrium Hydro, EoS
• Final Freeze-out: Kinetic models, measurables
• If QGP Sudden and simultaneous
hadronization and freeze out (indicated
by HBT, Strangeness, Entropy puzzle)
Freeze out distribution with rescattering from kinetic model across a layer
V=0 V=0
Analytic fit to Kinetic Model Solution :
.
[Karolis Tamosiunas et al.] .
Cancelling Juttner Distribution
[Karolis Tamosiunas et al.]
Continuation
• Hydro works amazingly well! Stronger and stronger hydro effects are observed!
• Equilibrium and EoS exists ( in part of the reaction )
• We have a good possibility to learn more, beyond the EoS, with improved
experimental and theoretical accuracy!
• Transport properties & scaling
Sun-surface - plasma
The picture was made The picture was made using the Swedish Solar using the Swedish Solar Telescope on the Canary Telescope on the Canary Island of La Palma. The Island of La Palma. The filaments' newly revealed filaments' newly revealed dark cores are seen to be dark cores are seen to be thousands of kilometers thousands of kilometers long but only about 100 long but only about 100 kilometers wide.
kilometers wide.
Resolving features 100 Resolving features 100 kilometers wide or less At kilometers wide or less At optical wavelengths,
optical wavelengths,
these images are sharper these images are sharper than even current space than even current space- - based solar observatories based solar observatories can produce. Recorded can produce. Recorded on 15 July 2002
on 15 July 2002
Stability of the core/crust HS.
• Landau-Lifsitz: mechanical stability is limited V2,
rocket engine- gas-turbine- accidents,
• Fusion device instabilities
• Solved by Bethe /Los
Alamos publ. - Zeldovich,
Raiser: High Temp.
RT – instabilities in Tokamak
• The figure
above shows three-
dimensional isosurfaces of the pressure as the instability develops along ridges
dominantly
aligned along
the ambient
Preventing turbulence
The instability of deflagration- (flame-) front is not desirable at supersonic fronts.
With increasing temperature the radiation becomes
dominant and stabilizes the
flame front.
The radiative transfer also modifies the dissipative transport. This is of vital importance, because radiative transport propagates with the speed of light, and able to stabilize processes which cannot be stabilized by mechanical pressure. This is actually the reason of the failure of different rocket engines, and the success of the space shuttle rockets as well as of the implosion devices in the nuclear bombs. One should just look at the extremely stable, blue-ultraviolet flame fronts (15 000 oK) of the Space-Shuttle's liquid fuel rockets stabilized by radiative energy-momentum transfer, in contrast to the hardly stable, turbulent red flames at ignitions with
Viscosity vs. T has a
Viscosity vs. T has a minimum at the 1minimum at the 1stst order phase transition. order phase transition. This might signal the phase transition if viscosity is measured.
This might signal the phase transition if viscosity is measured.
At lower energies this was done.
At lower energies this was done.
Stability, Reynolds number
- kinematic viscosity
- viscosity - density
- length - velocity
In an ideal fluid any small perturbation increases and leads to turbulent flow. For stability
sufficiently large viscosity and/or heat conductivity are needed!
Re < 1000 - 2000
(Calculations are also stabilized by numerical
viscosity!)
Re – studies in HICs
Theoretical [D. Molnar, U. Heinz, et al., ] Theoretical [D. Molnar, U. Heinz, et al., ] η η = 50 = 50 – – 500 MeV/fm 500 MeV/fm
22c Re c Re º º 10 10 – – 100 100 Exp.: 50
Exp.: 50 – – 800 Mev 800 Mev/nucleon energies 80 /nucleon energies 80’ ’s s [Bonasera [ Bonasera , Schurmann , Schurmann , Csernai] , Csernai]
scaling analysis of flow parameters.
scaling analysis of flow parameters. Re Re º º 7 7 – – 8 ! 8 ! (more dilute, more viscous matter)
(more dilute, more viscous matter)
In both cases
In both cases η η/s /s ª ª 1 (0.5 – 1 (0.5 – 5) , 5) ,
This is a value large enough to keep the This is a value large enough to keep the flow laminar in Heavy Ion Collisions !!!
flow laminar in Heavy Ion Collisions !!!
Stability, Reynolds number
- kinematic viscosity
- viscosity - density
- length - velocity
In an ideal fluid any small perturbation increases and leads to turbulent flow. For stability
sufficiently large viscosity and/or heat conductivity are needed!
Re < 1000 - 2000
(Calculations are also stabilized by numerical viscosity.)
Interesting and important: in RFD detonation
Re – studies in HICs
Theoretical [D. Molnar, U. Heinz, et al., ] Theoretical [D. Molnar, U. Heinz, et al., ] η η = 50 = 50 – – 500 MeV/fm 500 MeV/fm
22c Re c Re º º 10 10 – – 100 100 Exp.: 50
Exp.: 50 – – 800 Mev 800 Mev/nucleon energies 80 /nucleon energies 80’ ’s s [Bonasera [ Bonasera , Schurmann , Schurmann , Csernai] , Csernai]
scaling analysis of flow parameters.
scaling analysis of flow parameters. Re Re º º 7 7 – – 8 ! 8 ! (more dilute, more viscous matter)
(more dilute, more viscous matter)
In both cases
In both cases η η/s /s ª ª 1 (0.5 – 1 (0.5 – 5) , 5) ,
This is a value large enough to keep the This is a value large enough to keep the flow laminar in Heavy Ion Collisions !!!
flow laminar in Heavy Ion Collisions !!!
Flow patterns
• Strong, correlated and dominant “Elliptic”, V 2 , flow observed (CERN/BNL).
• The flow is laminar (η is sufficiently large),
& not dissipated (η is sufficiently small) !?
• V 1 , „directed flow” measurements are not as detailed yet.
• The strong and dominant flow
measurements raised large, international
attention!
Origin of the news:
In superstring theory, „based on analogy between black hole physics and equilibrium thermodynamics, ... there exist solutions called black branes, which are black holes with translationally invariant horizons. ... these
solutions can be extended to hydrodynamics, ... and black branes possess hydrodynamic characteristics of ... fluids: viscosity, diffusion constants, etc.”
In this model the authors concluded that η / s = 1 / 4π
And then they „speculate” that in general η / s > 1 / 4π vagy η / s > 1.
They argue that this is a lower limit especially for such strongly interacting
systems where up to now there is no reliable estimate for viscosity, like the
(Kovtun, et al., PRL 2005)
With With Kapusta Kapusta and and McLerran McLerran we we have studied these results and have studied these results and assumptions and found that : assumptions and found that :
-η vs. T has a typical decreasing and then increasing behaviour, due to classical reasons (Enskog’21) - η/s has a minimum exactly at the critical point in systems, which
have a liquid-gas type of transition - η vs. T shows a characteristic shows a characteristic behaviour
behaviour in all in all systems near the systems near the
critical point (not only in the case of
critical point (not only in the case of
He). He).
Viscosity – Momentum transfer
Via VOIDS
Via VOIDS Via PARTICLESVia PARTICLES
Helium (NIST)
QGP (Arnold, Moore, Yaffe)
This phenomenon can help us This phenomenon can help us to detect experimentally the to detect experimentally the critical point:
critical point:
η can be determined from (i)
[Prakash, Venugopalan, .]
