2009-4-291 Elliptic flow and incomplete equilibration in AMPT Jian-Li Liu Harbin Institute of Technology.

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Elliptic flow and incomplete equilibration in AMPT Jian-Li Liu Harbin Institute of Technology

Outline  Eccentricity scaled v 2 and incomplete equilibration  Variation of v 2 /εwith cross section  Variation of v 2 /εwith centrality  Summary

Eccentricity scaled v 2 and incomplete equilibration Heiselberg & Levy, PRC59, 2716(1999) Collisionless limit: Ideal hydrodynamic limit: From collisionless limit to hydrodynamic limit: Bhalerao et al., PLB627, 49(2005) work well in 2-D transport model. Gombeaud & Ollitrault, PRC77, (2008)

Knudsen coefficient Time scale of formation of elliptic flow: Only longitudinal expansion: (dN/dy is constant) Bhalerao et al., PLB627, 49(2005)

AMPT model data Differences: 1.c s is not constant from parton to hadron, BUT c s is approximately constant in parton stage 2. dN/dy is not constant D expansion 4. Non-isotropic differential cross section. Molnar & Gyulassy, NPA697, 459(2002) Lin et al., PRC72, (2005)

Relation between isotropic cross section and anisotropic cross section For isotropic cross section: AMPT model: μis turned to fixed total cross section. Transport cross section is related to μ and s.

Variation of v 2 /εwith cross section for quark Fitting parameters ： Gombeaud et al. 2-D transport model (2008): Ideal hydrodynamics ： Initial dN/dy ： final s ： initial dN/dy ， final s ： K 0 is sensitive to parameters used 27% 19% σ=3,6,10,14mb Issah et al., arXiv:nucl-ex/

HIJING hadron initial parton final parton hadron(c) hadron(f) Fitting parameters ： 15% 11% ε ， S is calculated for hadron from HIJING K 0 （ c ） is much larger than K 0 for quark: 1.Multiplicity difference 2. Variation of v 2 (c)/v 2 (quark) from 1.27 to 1.1 for cross section from 3mb to 14mb. 27% 19% Variation of v 2 /εwith cross section for hadron

% 14% Fitting parameters ： HIJING hadron initial parton final parton hadron(c) hadron(f) 15% 11% Keeping dN/dy unchanged and replace V 2 (f) with V 2 (c): Deviation of elliptic flow from its hydrodynamic limit is almost the same as hadron(c). Keeping elliptic flow unchanged and Replace dN(f)/dy with dN(c)/dy: Deviation of elliptic flow from its hydrodynamic limit is almost unchanged. ε ， S is calculated for hadron from HIJING Variation of v 2 /εwith cross section for hadron

Variation of v 2 /εwith centrality for quark

Changing the calculation of Knudsen coefficient

Changing the calculation of eccentricity Calculate eccentricity for quarks in all rapidity range. consistent with ideal hydrodynamic result

Only changing the calculation of eccentricity

Knudsen coefficient at initial stage ： Possible reasons for changed calculation of Knudsen coefficient 2. “Effective” calculation of Knudsen coefficient defined by Bhalerao et al.: The transverse expansion of system maybe important and is related to the size of system.

Original calculation of Knudsen coefficient New calculation of Knudsen coefficient Dependence on cross section ： 1.Relative distance between quark is related to cross section. 2.Quarks are coalesced according to their relative cross section Variation of v 2 /εwith centrality for hadron

Original calculation of Knudsen coefficient New calculation of Knudsen coefficient Dependence on cross section. Variation of v 2 /εwith centrality for hadron

Summary v 2 /εvariation with cross section for fixed parameter in AMPT model could be described well by the formula suggested Bhalerao et al. The deviation of v 2 /εof quark from its hydrodynamics limit is 19% ～ 27% for cross section from 6mb to 10mb. v 2 /εvariation with centrality for different cross section and collision energy in AMPT model could not be described by the formula suggested by Bhalerao et al, except the calculation of knudsen coefficient is changed.