Shear Viscosity and Viscous Entropy Production in Hot QGP at Finite Density 报告人： 刘 绘 华中师范大学 粒子所

Perfect fluid ? Well fitted by the ideal hydrodynamic model at PT<2GeV. How to understand? Dissipative structures! This figure, which has been demonstrated a lot of times, is about the anisotropic parameter v2 in terms of transverse momentum pT. Everyone notices that before the mid-rapidity region, the ideal hydrodynamics can describe the experimental data very well, as many people pointed out. I will say no more about this figure but one thing that is how we could understand the perfect fluid behavior of the matter produced at RHIC? The answers might be various in which the dissipative structure should be highlighted. The reason for that is as following: PRL89(2002)132301 Elliptic flow v2 as a function of pt for the strange particles and from minimum-bias in Au+Au 130GeV collisions. 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Irreversible thermodynamics & dissipative structure Entropy production ( -- driving force) Thermal flux ( -- transport coefficients) Energy-momentum tensor Driving force Xij environment Transport coefficient Intrinsic property The ratio of transport coefficient to the entropy production reflects the driving force Evolution of entropy density Superstring theory in equilibrium state 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Kinetics theory I Energy-momentum tensor fermion anti-fermion boson Correspondingly, Fluctuation of distribution (s: species) 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Kinetics theory II Boltzmann Equation collision term two-body scattering amplitude Recast the Boltzmann equation P.Arnold, G.D.Moore and G.Yaffe, JHEP 0011(00)001 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Shear viscosity With a definition of inner product and expanded distribution functions, where 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Collision terms Performing the integral over dk’ with the help of Scattering amplitude Distribution function term \chi term 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Matrix elements In Fig. 1(a) and (e), tu/s^2, the constant 3 and u/s are not singular, i.e., no contribution to leading-log. In the approx. 2 s≈-t in t-channel s≈-u in u-channel 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Distribution functions N_f is the quark flavor. The factors scaling the distribution functions are the freedom of degeneration, relevant to the distinguished reaction channels. For example, fermion-aitifermion< -- > fermion-antifermion appears 4N_f times in the sum over species 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

-functions Fig.1(b) has two sets of \chi functions because it involves different channels which bring on different momentum dependence of \chi^q and \chi^g. 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Variational approach Two-component fucntion Expand by the same basis Shear viscosity in this basis-set 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Shear viscosity (Nf=2) Right hand side: Left hand side: One function ansatz 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Non-equilibrium entropy density: viscous process (scheme I) Entropy density in kinetics theory With expanded distribution function Entropy in equilibrium state 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Viscous Entropy production: Scheme I (continued) Inserting the results from variational approach, the entropy produced in viscous process becomes = Depends on the dynamic parameter fine structure constant , the thermodynamic parameters T μ and the driving force. How to understand these dependences? 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Viscous entropy production: Scheme II Entropy production is Entropy in non-equilibrium state in local rest frame Notice and replace the proper time with the relaxation time which is solved from the Boltzmann equation in the relaxation time approximation the entropy density is: 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Longitudinal evolution maximum estimation of the velocity gradient STAR@RHIC, PRL_91(2003)052303 u z x Pseudo-rapidity plateau: Notice With the maximal velocity gradient Chemical potential enhances the ratio! 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Discussion on the ratio Conditions make ratio meaningful: viscosity/entropy(eq.) >0 Factor > 0 T=126.0MeV Temperature bound Minimum value T=181.15MeV assume: When T=181.15MeV, the ratio has a minimum value of 0.438, with μ=46MeV 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Summary & Outlook Shear viscosity of hot QCD at finite temperature has been calculated in the kinetics theory. The ratio of viscosity to viscous non-equilibrium entropy density demonstrates a minimum value and presents a temperature bound by some physical conditions. Chemical potential enhances the ratio. Besides the entropy sources we discussed here, others like increase of degree of freedom excited by phase transition…might be also contribute to the entropy production. The calculation in weakly coupled limit shows that it might be not sufficient to reproduce the recent experiment data. Strong coupling or correlation mechanism should be introduced to explain the experiment. 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Relaxation time Boltzmann equation in the relaxation time approximation With the help of the viscosity which is already obtained in the kinetics theory The relaxation time is 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Non-equilibrium entropy density: parton energy loss Non-equilibrium entropy can be obtained by reversing the evolution, i.e., All lost energy are converted into thermal energy Total energy dissipation of a jet (10GeV) over 2RA ~ 11.05GeV Uncertainty I: VOLUME (RHIC: Au+Au 200GeV) Uncertainty II: NUMBER OF ‘JETS’ and soft parton energy loss One, two or many? Not only jets, but also soft parton energy loss! Entropy production from jet quenching 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Outline Introduction Shear viscosity at finite chemical potential Non-equilibrium entropy density Entropy production from viscous process Entropy production from parton energy loss Ratio of viscosity to entropy density Summary The outline of my speech is listed on this slide. First I will give a brief introduction and motivations of our work. Then the calculation of shear viscosity in the framework of kinetics theory will be presented. Consequently, the entropy density will be evaluated based on the idea that the non-equilibrium state entropy density at some space-time point can be obtained by abstracting the entropy production from the maximum value of entropy production in the equilibrium state. Therefore the entropy production is the crucial factor in our calculation. Entropy production produced both in the jet quenching and viscous processes have been involved in this work. Finally the ratio of viscosity and entropy density will be discussed. This speech will be end by a summary. 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所

Non-equilibrium entropy density: jet energy loss Non-equilibrium entropy can be obtained by reversing the evolution, i.e., All lost energy are converted into thermal energy Even if all the initial energy of jet are converted into thermal energy, a typical jet contributes 10-3GeV3 to the entropy density in a volume of 1000fm3 Uncertainty I: VOLUME (RHIC: Au+Au 200GeV) Uncertainty II: NUMBER OF ‘JETS’ and soft parton energy loss One, two or many? Not only jets, but also soft parton energy loss! 高能物理年会 桂林（2006） 刘绘 华中师范大学粒子所