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

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 2

Perfect fluid ?

Elliptic flow v2 as a function of pt for the strange particles and from minimum-bias in Au+Au 130GeV collisions.

• Well fitted by the ideal hydrodynamic model at PT<2GeV.

• How to understand?

• Dissipative structures!

PRL89(2002)132301PRL89(2002)132301

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 3

Evolution of entropy

density

Irreversible thermodynamics & dissipative structure

Entropy production

Thermal flux ( -- transport coefficients)

( -- driving force)

Energy-momentum tensor

The ratio of transport coefficient to the entropy production reflects the driving force

Superstring theory in equilibrium state

Driving force Xij

environment

Transport coefficient

Intrinsic property

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 4

Kinetics theory IEnergy-momentum tensor

fermion anti-fermion bosonCorrespondingly,

Fluctuation of distribution (s: species)

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 5

Kinetics theory IIBoltzmann Equation

collision term two-body scattering amplitude

Recast the Boltzmann equation

P.Arnold, G.D.Moore and G.Yaffe,

JHEP 0011(00)001

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 6

Shear viscosityWith a definition of inner product and expanded distribution functions,

where

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 7

Collision termsPerforming the integral over dk’ with the help of

Scattering amplitude

Distribution function term \chi term

2006高能物理年会 桂林（） 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

2006高能物理年会 桂林（） 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-aitifermi

on< -- > fermion-antifermion appears 4N_f times in the sum over species

2006高能物理年会 桂林（） 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.

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 11

Variational approachTwo-component fucntion

Expand by the same basis

Shear viscosity in this basis-set

2006高能物理年会 桂林（） 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

2006高能物理年会 桂林（） 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

2006高能物理年会 桂林（） 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?

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 15

Viscous entropy production: Scheme II

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:

Entropy in non-equilibrium state in local rest frame

Entropy production is

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 16

Longitudinal evolutionmaximum estimation of the velocity gradient

STAR@RHIC, PRL_91(2003)052303

u

z

x

Pseudo-rapidity plateau: Notice

Chemical potential enhances the ratio!

With the maximal velocity gradient

2006高能物理年会 桂林（） 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

80 100 120 140 160 180 200T

-3

-2

-1

0

1

2

3

s 0

80 100 120 140 160T

-5

-2.5

0

2.5

5

7.5

rotcaf

100 125 150 175 200 225 250T

-4

-2

0

2

4

s

T=126.0MeVTemperature bound

Minimum valueT=181.15MeV

When T=181.15MeV, the ratio has a minimum value of 0.438,

with μ=46MeV

assume:

2006高能物理年会 桂林（） 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.

2006高能物理年会 桂林（） 刘绘 华中师范大学粒子所 22

Non-equilibrium entropy can be obtained by reversing the evolution, i.e.,

Non-equilibrium entropy density:jet energy loss 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!