Generalized Entropy and Transport Coefficients of Hadronic Matter

Azwinndini Muronga1,2

1Centre for Theoretical Physics & AstrophysicsDepartment of Physics, University of Cape Town

2UCT-CERN Research CentreDepartment of Physics, University of Cape Town

Zimanyi 75 Memorial Workshop

02-04 July 2007, Budapest, Hungary

Transport properties of relativistic nuclear matter

Viscosities, diffusivities, conductivities.

Determine relaxation to equilibrium in heavy ion collisions – chemical equilibration (by flavor, spin and color diffusion)

In astrophysical situations such as in neutron stars – cooling and burning of neutron star into a strange quark star

In cosmological applications such as the early universe – electroweak baryogenesis

QED and QCD plasmas

Complete fluid dynamics solution requires

- initial conditions - equation of state - transport coefficientsExtract the transport coefficients

and associated time/length scales for a given model of interacting hadrons and/or partons.

Study the sensitivity of the space-time evolution of the system and the calculated distribution of the hadrons to dissipative, non-equilibrium processes

Compare the predicted distribution with those observed in experiments

Baym et. al.; Gavin, Prakash et. al.; Davesne; Heiselberg, Muroya et. al.; Arnold et. al.; AM; Z. Xu and C. Greiner

The interest in shear viscosity to entropy ratio

Energy equation

EoS and Transport coefficients

Temperature evolution

e wher1

Or

1 i.e., 0

11

1

TspR

sR

s

pR

p

3

2

01

03

1

0

0

12

1

R

T

T

11

3

21

2

1

2

2

2

222

Td

dT

d

d ndndnd

0 ideal

1

3

41st

9

815

1

2

1

3

2

1234

3

aTT

d

d

Tb

aT

d

d

a

TTT

d

d

,

4

3 ,

ln6

1

342.0)7.11( ,

902

2116 ,

3

1

23

12

3

24

pT

NNbbT

NaaTp

ssf

f

f

AM, 2002; 2004

Time evolution of thermodynamic quantities

Generalized entropy 4-current

Entropy 4-current:Muller-Israel-Stewart

Entropy density and entropy flux

Entropy production

qquqqqsuS 10212

02

1

02 11211

qqS

See AM, nuc-th/0611090 for details

212

0eq 2

1),( qqnsSus

qqqS 10

Relaxation equations for dissipative fluxes

Relaxation equations for the dissipative fluxes

Relaxation times/lengths

q

aT

TTqq

q

q

qqq

q

2

1100

210

2 , , ,

2 , ,

qqqq

q

TT

T

Fluctuations and Transport Coefficients

Green-Kubo

From generalized entropy

dttqqT

V

dttT

V

jidttT

V

ii

ijij

02

0

0

)().0(

)().0(

)().0(

/1

/12

/1

)()().0(

)()().0(

)()().0(

t

tijqji

tijklklij

eVTt

eVTtqq

eVTt

q

Transport Coefficients and Equation of State

From Maxwell-Cattaneo-type equations

Thermodynamics from transport models

Thermodynamics from hadronic gas model (e.g. mesons)

)().0(

)().0(

)().0(

tT

V

tqqT

V

tT

V

ijij

iiq

particles all

1

2

particles all

1

3

1

1

i i

i

ii

E

p

Vp

EV

1

1

3)2(),(

1

1

)2(),(

1)2(),(

)(

2

3

3

)(3

3

)(3

3

k

k

k

Ekk

k

Ek

k

Ek

kk

eE

ppdgTp

e

pdgTn

e

EpdgT

Relaxation Coefficients

Shear viscosity and shear relaxation time

AM, 2004;

See also , A. El, C. Greiner and Z. Xu, hep-ph 0706412, using parton model

• Transport coefficients are as important as the equation of state.

• Transport coefficients and relaxation times/lengths probe different time/length scales in fluid dynamics (physics of many scales)

• They should be calculated/extracted self consistently together with the equation of state.

• The relaxation times/lengths should be compared with the characteristic time/length scales of the system under consideration.

• Knowledge of transport coefficients and associated length/time scales provides good ground for comparison of theoretical prediction with experiments

Looking beyond the perfect picture