How perfect is the RHIC liquid?. Jean-Yves Ollitrault, IPhT Saclay BNL colloquium, Feb. 24, 2009 based on Drescher Dumitru Gombeaud & JYO, arXiv:0704.3553 Gombeaud, Lappi, JYO, arXiv:0901.4908 . A nucleus-nucleus collision at RHIC. Two Lorentz-contracted nuclei. collide.
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Jean-Yves Ollitrault, IPhT Saclay
BNL colloquium, Feb. 24, 2009
Drescher Dumitru Gombeaud & JYO, arXiv:0704.3553
Gombeaud, Lappi, JYO, arXiv:0901.4908
Two Lorentz-contracted nuclei
« hard » processes, accessible to perturbative calculations
High-density, strongly-interacting hadronic matter (quark-gluon plasma?) is created and expands, and eventually reaches the detectors as hadrons.
If interactions are strong enough, the matter produced in a
nucleus-nucleus collision at RHIC reaches local thermal equilibrium
However, some quantities are very hard to compute on the lattice, such as the viscosity
Using the AdS/CFT correspondence, one can compute exactly the viscosity to entropy ratio in strongly coupled supersymmetric N=4 gauge theories, and it has been postulated that the result is a universal lower bound.
Kovtun Son Starinets hep-th/0405231
Martinez, Strickland arXiv:0902.3834
Luzum Romatschke arXiv:0901.4588
Bouras Molnar Niemi Xu El Fochler Greiner Rischke arXiv:0902.1927
Song Heinz arXiv:0812.4274
Denicol Kodama Koide Mota arXiv:0807.3120
Molnar Huovinen arXiv:0806.1367
Dusling Teaney arXiv:0710.5932
Csernai et al nucl-th/0604032
Non-central collision seen in the transverse plane: the overlap area, where particles are produced, is not a circle.
A particle moving at φ=π/2 from the x-axis is more likely to be deflected
than a particle moving at φ=0, which escapes more easily.
Initially, particle momenta are distributed isotropically in φ.
Collisions results in positive v2.
v2 scales like the eccentricityε of the initial density profile, defined as :
This eccentricity depends on the collision centrality, which is well known experimentally.
Ideal (nonviscous) hydrodynamics (∂μTμν= 0) is scale invariant:
v2= αε, where α constant for all colliding systems
(Au-Au and Cu-Cu) at all impact parameters at a given energy
The initial eccentricity ε is model dependent
The color-glass condensate prediction is 30% larger than the Glauber-model prediction
Drescher Dumitru Hayashigaki Nara, nucl-th/0605012
Eccentricity fluctuations are important
PHOBOS collaboration, nucl-ex/0510031
The ratio v2/ε depends on the equation of state
A harder equation of state gives more elliptic flow for a given eccentricity.
v2/ε is not constant at a given energy: non-viscous hydro fails !
But viscosity breaks scale invariance
η= viscosity, usually scaled by the mass/energy density:
η/ε ~ λvthermal, where λmean free path of a particle
R = typical (transverse) size of the system
vfluid= fluid velocity ~vthermal because expansion into the vacuum
The Reynolds number characterizes viscous effects :
Re≡ R vfluid/(η/ε) ~ R/λ
Viscous corrections scale like the viscosity: ~ Re-1 ~ λ/R.
If viscous effects are not « 1, the hydrodynamic picture breaks down!
In this talk, I use instead the Knudsen number K= λ/R
1/K ~ number of collisions per particle
Convergence to ideal hydro clearly seen!
Elliptic flow increases with number of collisions (~1/K)
Smooth convergence to ideal hydro as K→0
The mean free path of a particle in medium is λ=1/σn
1/K= σnR ~ (σ/S)(dN/dy), where
K can be tuned by varying the system size and centrality
K~0.3 for central Au-Au collisions
v2 : 30% below ideal hydro!
(remember, more collisions means lower viscosity)
depending on which initial conditions we use.
For particles with a given momentum pt along x
Romeasures the dispersion of xlast-v tlast
Rs measures the dispersion of ylast
Where (tlast,xlast,ylast) are the space-time coordinates of the particle at the last scattering
HBT puzzle : Ro/Rs~1.5 in hydro, 1 in RHIC data
A simulation with R/λ=3 (inferred from elliptic flow) gives a value
compatible with data, and significantly lower than hydro.
RegionEstimating the initial eccentricity
Until 2005, this was thought to be the easy part.
But puzzling results came:
1. v2 was larger than predicted by hydro in central Au-Au collisions.
2. v2 was much larger than expected in Cu-Cu collisions.
This was interpreted by the PHOBOS collaboration as an effect of fluctuations in initial conditions
[Miller & Snellings nucl-ex/0312008]
In 2005, it was also shown that the eccentricity depends significantly on the model chosen for initial particle production.
We compare two such models, Glauber and Color Glass Condensate.
They involve intrinsic properties of the fluid (mean free path λ, thermal/sound velocity cs, shear viscosity η, mass density ρ)as well as quantities specific to the flow pattern under study (characteristic size R, flow velocity v)
Knudsen number K=λ/R
K «1 : local equilibrium (fluid dynamics applies)
Mach number Ma= v/cs
Ma«1 : incompressible flow
Reynolds number R= Rv/(η/ρ)
R»1 : non-viscous flow (ideal fluid)
They are related ! Transport theory: η/ρ~λcsimpliesR * K ~ Ma
Remember: compressible+viscous = departures from local eq.
The hydrodynamic regime requires both D«1 and K«1.
a huge number of particles must be simulated.
(even worse in 3d)
The Boltzmann equation requires D«1
This is achieved by increasing N (parton subdivision)
Initial conditions: monoenergetic particles.
Number of particles with energy <E
in the system
Number of particles with energy <E in thermal equilibrium
Convergence to ideal hydro clearly seen!
The density is estimated at the time t=R/cs
(i.e., when v2 appears), assuming 1/t dependence.
The effective density that we see through elliptic flow depends little on colliding system & centrality !
(Bai Yuting, STAR)
Recall : RHIC data above ideal hydro
Boltzmann also above ideal hydro but still below data
Boltzmann again converges to ideal hydro for small Kn
However, Ro is not very sensitive to thermalization
pt dependence is already present in (CGC-inspired) initial conditions
due to the hard equation of state and 2 dimensional geometry