1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. -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

11. 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

12. 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

13. 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

14. 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

15. 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

16. Longitudinal evolution maximum estimation of the velocity gradient
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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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