Matching Pre-Equilibrium Models
to Viscous Hydrodynamics
Daniel White
Advisor: Ulrich Heinz
Outline
• Basic idea – motivation
• The matching process
and relevant equations
• Models for comparison
• Results, conclusions,
and future plans
1
Overview
• Quark-Gluon Plasma (QGP) described by viscous hydro
• What happens before thermal equilibrium?
– Detailed microscopic dynamics presently unclear
– Characterization by thermal parameters no longer meaningful
– Ultimate goal: complete description of pre-equilibrium dynamics
• Ultimate goal: complete description of pre-equilibrium system
• More immediate goal: explore sensitivity of the system on
pre-equilibrium dynamics
– Want to propagate pre-equilibrium effects through viscous hydro
– Needed to establish uncertainties in late-time observables due to
incomplete knowledge of early dynamics
– To do so, need some way of converting pre-equilibrium to viscous
hydrodynamics
– Procedure: Matching the stress-energy tensor to viscous hydro
parameters
2
The Matching Process
• To start, we use the standard expression:
T    eu  u  ( p  )     
Stress-energy tensor
Fluid velocity
Energy density
Bulk viscous
pressure
Pressure
Shear viscous
pressure tensor
Spatial projector
in local rest frame
• Need to invert this equation to find hydrodynamic variables in
terms of the elements of T  
• To aid in this process, we make a couple assumptions
– Longitudinal boost invariance – perform matching at
– Massless equation of state - p  e / 3
z  0,   0
3
The Matching Process – Projections
T

 

 eu u  ( p  )  

• Next, we find projections of T   to isolate the hydrodynamic
variables:
1



   T    3 p  e
3
1    
 1   

  
            T
3
2

with     g    u u
e  uT u
• Finding the velocity allows us to compute all the other variables
• Evaluating at   0 makes this somewhat simpler -
u    1, v cos  , v sin  ,0
Magnitude of transverse velocity
Angle in transverse plane
4
The Matching Process - Landau
• Beginning with the Landau matching condition eu   T  u , we
have three equations:
e  T 00  T 01v cos   T 02v sin 
e v cos   T 01  T 11v cos   T 12v sin 
e v sin   T 02  T 12v cos   T 22v sin 
• Eliminating

v sin  T
e
from the last two equations yields:

v sin    T
v cos  T 00  T 01v cos   T 02v sin   T 01  T 11v cos   T 12v sin 
00
 T 01v cos   T 02
02
 T 12v cos   T 22v sin 
• These equations can then be solved numerically, allowing us to
calculate all the relevant hydrodynamic variables
5
Models to Match
• To-date, we have used two models
– Coherent electromagnetic discussed by Vredevoogd and Pratt*:
– Free-streaming from Gaussian initialization
• Will add more in the coming months
–
–
–
–
Ideal* (and possibly viscous) hydrodynamics
Incoherent electromagnetic*
Free-streaming from Color Glass Condensate initialization
More?
*J. Vredevoogd and S. Pratt, Phys. Rev. C 79, 044915 (2009)
6
Coherent Electromagnetic Model
• Assume dynamics are
governed by charge densities
  ( x, y)
on nuclei receding at the
speed of light
• Treat as oppositely charged
capacitor plates with density
 ( x, y) in the transverse plane
• Compute electric and
magnetic fields

• Find T as shown
 ( x, y)


1
1  
2
2
0i
T 
E B
T 
EB i
8
4
1 
1
ij
2
2 
T 
Ei E j  Bi B j   ij E  B 

4 
2

00




7
Free-Streaming Model
• Initialize system with Gaussian phase-space distribution
2
2
2
2


p

p


x
y
x
y
f x , 0 ; pT , y   exp  2  2 

2
 2 Rx 2 Ry 2Q ( y) 
• Evolve without interactions – momenta never change, future
positions determined by momenta



 
pT 
f x , ; pT , y   f  x  (   0 ) ; pT , y 
pT


• Calculate T

T
from kinetic theory

1 d 2 pT    
 
p p f x , ; pT , y 

pT
y  s
8
Comparison
• To aid in comparison, we assume that each model has the
same initial T 00 :
2
2 

x
y
T 00  exp  2  2 
 2 Rx 2 Ry 
• This assumption specifies charge densities and phase-space
distributions, allowing numerical calculation of each model
• Models are then compared at different matching times (time
corresponding to the transition between early model and
viscous hydrodynamics)
• For the following comparisons, we used Rx  Ry  3.0 fm
– Similar conclusions can be drawn from Rx  Ry
– This condition isolates effects of matching process, making its
importance more obvious
9
Comparison - T
fg

Coherent Electromagnetic
Free-Streaming
T 01
T 00
x-position (fm)
(Similar comparison in J. Vredevoogd and S. Pratt, Phys. Rev. C 79, 044915 (2009))
10
Comparison – Fluid Velocity
Coherent Electromagnetic
Free-Streaming
Velocity
Radial Position (fm)
11
Comparison - Viscosity
• Use
     as a measure of the viscosity
e p
Coherent Electromagnetic
Free-Streaming
Viscous
Effects
Radial Position (fm)
12
Conclusion and Future Work
• Difference seems to be the result of viscosity
• However, effects on late-time observables still unclear
– Need to propagate the models through viscous hydrodynamic
evolution to compare hadron spectra, elliptic flow coefficients
to estimate error bars
– Include other pre-equilibrium models (hydro, free-streaming
from CGC)
13