Elliptic flow and Thermalization at RHIC

J-Y Ollitrault1st International Workshop on Soft Physics in

ultrarelativistic Heavy Ion Collisions

(SPHIC’06), Catania, Sept. 28, 2006

Outline

• Which are the robust observables for thermalisation? (Bhalerao Blaizot Borghini & JYO, nucl-th/0508009)

• The centrality dependence of elliptic flow and the magnitude of v4 show deviations from ideal hydro

• Can we model deviations from ideal hydro? Preliminary results from a new transport calculation (C. Gombeaud & JYO, work in progress)

Good probe of thermalisation:Elliptic flow v2

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Interactions among the produced particles: Pressure gradients generate positive elliptic flow v2

(v4 smaller, but also measured)

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In hydro, at a time of order R/cs where R = transverse size cs= sound velocity

When does elliptic flow build up?

For a given equation of state, v2 scales roughly like the initial eccentricity ε

What is the density then?Assuming particle number conservation, the density at t=R/cs is(this is particle density, not energy density)

It varies little with centrality and system size !!

How can we probe hydro behaviour?(= thermalisation)

• We want to measure the equation of state so that we should not assume any value of cs a priori, but rather obtain it from the data The robust method is to compare systems with the same density, hence the same cs , and check that they have the same v2/ε

• Au-Au collisions and Cu-Cu collisions at midrapidity, and moderate centralities do a good job

• The rapidity dependence of v2 is interesting, but interpretation is more difficult since the density varies significantly with rapidityv

• v4 /(v2)2 is another robust observable (=1/2 in ideal hydro)

Bhalerao Blaizot Borghini & JYO, nucl-th/0508009Borghini & JYO, nucl-th/0506045

Why does this really probe thermalisation?

Varying centrality and system size, one does not change the density,

but one varies Kn-1~σ/S (dN/dy)~Number of collisions per particle

Notation: mean free path/system size =Kn: Knudsen number. The hydro limit is Kn«1. If not satisfied, one expects smaller v2 than in hydro.

v2/ε: Data from SPS and RHIC

Continuous increase with Kn-1, no saturation seen in dataSPS and RHIC fall on the same curve although ≠ densities:Suggests similar values of cs at both densities (?)

Modelling deviations from ideal hydro

• Need a theory that goes to ideal hydro in some limit.

• First method: viscous hydrodynamics (papers by Teaney, Muronga, Baier Romatschke & Wiedemann, Heinz & Chaudhuri, Pratt) : this is a general approach to small deviations from ideal hydro, but quantitative results are not yet available

• Second method: Boltzmann equation. Limitation : applies only to a dilute system (not to the liquid produced at RHIC). Advantage: directly involves microscopic physics through collisional cross-sections

Previous transport calculations

Molnar, Huovinen, nucl-th/0404065, Phys. Rev. Lett.

Boltzmann ≠hydro although Kn«1??

A new transport calculation

(C. Gombeaud & JYO, in preparation)

• Two-dimensions (three later)• Massless particles (mass later)• Billiard-ball calculation, but with Lorentz contraction taken into account: this ensures Lorentz invariance of the number of collisions (≠Molnar-Gyulassy)• N particles of size r in a box of size R: dilute system if r«R/√N

pT dependence of elliptic flow

The transport calculation coincides with the hydro calculation in the limit of small Knudsen number, as it should!

v2/ε

pT

Time dependence of elliptic flow

The transport calculation again coincides with the hydro calculation in the limit of small Kn, as it should!

Variation of v2 with Kn-1

~Nb collisions/particle

Best fit: v2=v2hydro/(1+1.76 Kn): goes to hydro for Kn→0

Hexadecupole flow : v4

Ideal hydro : universal prediction v4=0.5 (v2)2 at large pt . Confirmed by the transport calculation.

Data ~1.2 suggest Kn~1:No thermalisation at RHIC!

v4/(v2)2

pT

Revisiting the perfect liquid scenario

Model inputs Exp. constraints

What is wrong with this scenario?

• Initial density profiles: participant scaling

• Global observables: multiplicity

• Equation of state • Pt-spectra

• Thermalization assumed: Kn«1

• Elliptic flow « saturates the hydro limit »

•The color glass condensate gives much larger values of ε!

Drescher Dumitru Hayashigaki Nara nucl-th/0605012

• Elliptic flow no longer saturates the hydro limit!

• Thermalization not seen!!

Qualitative predictions for LHC

• Higher multiplicity : smaller Kn: closer to hydro.

• There is room for significant increase of v2

• v4/(v2)2 somewhat smaller than at RHIC

Work in progress

• Obtain the value of Kn by comparing the shape of the curve with exp. data

• Generalize to massive particles

• Generalize to three dimensions with longitudinal expansion

Test of the algorithm: thermalisation in a static system

Initial conditions: monoenergetic particles.Relaxation time = mean free path= tau=S/(Nr)

# particles with energy <E in the gas versus# particles with energy <E in thermal equilibrium