1. What do RHIC data tell us about thermalisation? J-Y Ollitrault Journée thématique IPN Orsay, 2 juin 2006

2. 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)

3. Good probe of thermalisation: Elliptic flow v 2 Interactions among the produced particles: Pressure gradients generate positive elliptic flow v 2 (v 4 smaller, but also measured)

4. In hydro, at a time of order R/c s where R = transverse size c s = sound velocity When does elliptic flow build up? For a given equation of state, v2 scales roughly like the initial eccentricity ε

5. What is the density then? Assuming particle number conservation, the density at t=R/c s is It varies little with centrality and system size

6. How can we probe hydro behaviour? (= thermalisation) We want to measure the equation of state so that we should not assume any value of c s a priori The robust method is to compare systems with the same density, hence the same c s, and check that they have the same v 2 /ε Au-Au collisions and Cu-Cu collisions at midrapidity, and moderate centralities do a good job The rapidity dependence of v 2 is interesting, but interpretation is more difficult since the density varies significantly with rapidityv v 4 is also interesting (not covered in this talk) Bhalerao Blaizot Borghini & JYO, nucl-th/0508045

7. 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 v 2 than in hydro.

8. Problems with RHIC data Au +Au 200 GeV STAR preliminary Gang Wang, Quark Matter 2005, nucl-ex/0510034 Results depend on the method used for the analysis

9. 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 » ε p differs from the « standard eccentricity» ε s due to statistical fluctuations v 2 {ZDC-SMD} should scale like ε s V 2 {2} should scale like Bhalerao & JYO, in preparation

10. STAR data revisited With the proper scaling by ε, the discrepancy between methods disappears: Little room for « nonflow effects » v 2 /ε increases with # of collisions per particle : clearly NOT HYDRO

11. 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

12. What is the literature on the subject? Molnar, Huovinen, nucl-th/0404065

13. 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)

14. Test of the algorithm: thermalisation in a static system Initial conditions: monoenergetic particles. Relaxation time = mean free path= tau=S/(Nr)

15. Elliptic flow: preliminary results Initial conditions: homogeneous density inside a rectangular box. Particle then escape freely from the box. Two dimensionless parameters D=r sqrt(N/S) 1/K=R/λ

16. Time evolution of elliptic flow

17. Variation with number of collisions

18. Perspectives Study the pt-dependence (saturation of v 2 ) Hexadecupole flow v 4 Generalize to three dimensions with longitudinal expansion Obtain the value of K by comparing the shape of the curve with data?