What do RHIC data tell us about thermalisation?

J-Y Ollitrault

Journée thématique

IPN Orsay, 2 juin 2006

Outline

• Which data can be interpreted in a fairly model-independent way, and why? (based on Bhalerao Blaizot Borghini & JYO, nucl-th/0508045)

• The centrality dependence of elliptic flow shows deviations from ideal hydro (taking into account fluctuations in initial conditions, Bhalerao & JYO, in preparation)

• Can we model deviations from ideal hydro? A preliminary transport calculation (Clément 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)

...)2cos2cos21(2

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dN

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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

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

• 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 is also interesting (not covered in this talk)

Bhalerao Blaizot Borghini & JYO, nucl-th/0508045

Why does this really probe thermalisation?

Varying centrality and system size, the density does not change, but the number of collisions per particle ~ σ/S (dN/dy) does !

Notation: # of collisions=1/K where K=Knudsen number. The hydro limit is K<<1. If not satisfied, one expects smaller v2 than in hydro.

Problems with RHIC data

Au +Au 200 GeV

STAR

prelimin

ary

Gang Wang, Quark Matter 2005, nucl-ex/0510034

Results depend on the method used for the analysis

Eccentricity fluctuations

A nice idea by the PHOBOS collaboration, nucl-ex/0510031

Positions of participant nucleons at the time when the collision occurs are randomly distributed throughout the overlap area.

The « participant eccentricity » εp differs from the « standard eccentricity» εs due to statistical fluctuations

v2{ZDC-SMD} should scale like εs

V2{2} should scale like <εp>

Bhalerao & JYO, in preparation

STAR data revisited

With the proper scaling by ε, the discrepancy between methods disappears:Little room for « nonflow effects »v2/ε increases with # of collisions per particle : clearly NOT HYDRO

Modelling deviations from ideal hydro

• Need a theory that goes to ideal hydro in some limit.• First method: viscous hydrodynamics (Teaney, Muronga

et: al, Romatschke et al, Heinz et al, Pal) : this is a general approach to small deviations from ideal hydro, but quantitative results are not yet available

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

What is the literature on the subject?

Molnar, Huovinen, nucl-th/0404065

Our approach to the Boltzmann equation

(C. Gombeaud, stage M1)

• Two-dimensions (three later)• Massless particles (mass later)• Billiard-ball-type calculation, but with Lorentz contraction taken into account: this ensures Lorentz invariance of the number of collisions.• N particles of size r in a box of surface S:Dilute system if r<<sqrt(S/N)

Test of the algorithm: thermalisation in a static system

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

Elliptic flow: preliminary results

Initial conditions: homogeneous density inside a rectangular box. Particle then escape freely from the box.Two dimensionless parametersD=r sqrt(N/S)1/K=R/λ

Time evolution of elliptic flow

Variation with number of collisions

Perspectives

• Study the pt-dependence (saturation of v2)

• Hexadecupole flow v4

• Generalize to three dimensions with longitudinal expansion

• Obtain the value of K by comparing the shape of the curve with data?