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)
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