Viscous hydrodynamics and Transport Models

Azwinndini Muronga 1,2

1 Centre for Theoretical Physics and AstrophysicsDepartment of Physics, University of Cape

Town, South Africa2 UCT-CERN Research Centre

Department of Physics, University of Cape Town, South Africa

Workshop on “Viscous Hydrodynamics and Transport Models in heavy Ion Collisions”

May 2, 2008 : BNL, Long Island, NY, USA

Dissipative Relativistic Fluid Dynamics

Summary and Conclusions It concerns conservation of net

charges, energy-momentum and balance of fluxes. The primary state quantities such as number current, energy-momentum-tensor and entropy current differ from the ideal fluid by additional dissipative fluxes

It concerns non-linear, coupled partial differential equations

Formulation is relativistic and this add another complexity.

The system of equations is still closed by the equation of state.

In addition the balance of fluxes is controlled by the transport coefficients. Together with the equation of state they determine the relaxation times/lengths

Analytic solutions are rare. Numerical solution poses a

challenge. Initial conditions are more

interesting. DFD open s a window that

one can use to connect the macroscopic and microscopic dynamics of a system under consideration: in our case the system is the hot and dense matter produced in relativistic nuclear collisions.

The statement: transport coefficients are as important as the equation of state can no longer be overemphasized.

Refers to: A. Muronga (2007) I&II

Non-equilibrium fluid dynamics from kinetic theory

The equations for the first three moments of distribution function

where

Non-equilibrium fluid dynamics from kinetic theory

Thermodynamic integrals for relaxation/coupling coefficients

See A. Muronga (2007) II

14-Fields theory of non-equilibrium fluid dynamics

The conservation of net charge and of energy-momentum and the balance of fluxes

2nd order entropy 4-current

2nd order relaxation/coupling coefficients

Entropy production and transport coefficients

Relaxation equations for dissipative fluxes

Relaxation equations for the dissipative fluxes

Transport and relaxation times/lengths

Make the equations tractable

Macroscopic dynamics

where

See A. Muronga (2007) I

Make equations attractable

Microscopic dynamics

where

Physical problems: People’s ideas

Simple scaling solution: A. Muronga (2001/2002/2004)

zt

ztyzt

yzytt

zv

ln2

1 ,

sinh ,cosh ,

22

Ideal fluid vs non-ideal fluid

Energy equation

EoS and Transport coefficients

Temperature evolution

;

1 or 0

1

1

pR

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

Time evolution of thermodynamic quantities

A. Muronga (2002/2004)

Transport coefficients and relaxation times

A. Muronga (2004)

Physical problems

Boost invariance + symmetric transverse: A. Muronga + D. H. Rischke (2004)

Physical problems: People’s realizations

(2+1) viscous hydro:

Formulations U. Heinz, H. Song and A.K. Chaudhuri (2006); A. Muronga (2007)

ApplicationsA.K. Chaudhuri (2007)P. Romatchke and U. Romatschke (2007);H. Song and U. Heinz (2008);K. Dusling and D. Teaney (2008);P. Huovinen and D. Molnar (2008);

See the talks by P. Huovinen, P. Romatschke, H. Song and K. Dusling.

Physical problems: People’s realizations

(2+1) viscous hydro:

Formulations U. Heinz, H. Song and A.K. Chaudhuri (2006); A. Muronga (2007)

ApplicationsA.K. Chaudhuri (2007)P. Romatchke and U. Romatschke (2007);H. Song and U. Heinz (2008);K. Dusling and D. Teaney (2008);P. Huovinen and D. Molnar (2008);

See the talks by P. Huovinen, P. Romatschke, H. Song and K. Dusling.

Viscous hydro vs transport models

Bin Zhang et. al.

Viscous hydro vs transport models

Viscous hydro vs transport models

P. Huovinen and D. Molnar

Viscous hydro vs transport models

Slide from J.Y. Ollitrault’s talk

Viscous hydro vs transport models

Extracting transport coefficients from transport

Viscous hydro vs transport models

Extracting transport coefficients from transport A. El, A. Muronga (2007)