Nonequilibrium Dynamics of Quark
Gluon Plasma: Boltzmann-Langevin
(Boedeker) Equation
Tanja Horn
University of Maryland
Outline

Introduction



Nonperturbative Physics




Quark Gluon Plasma
Relevant Particles and Scales
Kinetic Theory
Hydrodynamics
Effective theories: HTL, LLO, NLLO
Langevin Dynamics

Boedeker equation (LLO)


Classical Transport Theory




Fluctuation-Dissipation Relation
Master Equation


Origin of noise term in Boedeker equation
Linearized Boltzmann equation with forcing term
Fokker-Planck equation
Particle Dynamics
Summary
Quark Gluon Plasma (QGP)




State of matter in which quarks and gluons are unconfined.
Manifest of asymptotic freedom in Quantum Chromodynamics
(QCD)
Characteristic phase transition at 10  5  10  6 s in the cosmological
standard model.
Different models of heavy ion collisions are studied at
RHIC/Brookhaven and LHV/CERN.
Lifetime of a plasma state is short, so a large fraction of QGP
lifetime spent in nonequilibrium states.
 Need for an understanding of nonequilibrium in QGP
Relevant Particles
Length

Degrees of Freedom in QCD




Gluons  spin 1 gauge fields,
color
1
Quarks  spin 2 matter
particles
Particles in QGP are characterized
by the temperature T and the gauge
coupling constant g.
Characteristic energy scales



Description/
Classification
QCD: T ~ 200 MeV
QGP: T ~ 150 MeV
Typical scales for QGP are shown in
Table 1.
Particle
Wavelength
Particle
Separation
T
T
Hard Modes
1
Thermal
Wavelength
1
Plasma
Frequency
( gT )
1
Semihard Modes
Debye Length
( gT )
1
Screening of
static fields at
large distances
Nonperturbative
magnetic scale
(g T )
Soft modes
(g T )
2
1
2
1
Unscreened
magnetic fields
Transport Coefficients

 
 

[  v    Fext   ] f ( x , p , t )   C [ f ]
t
x
p

•View Boltzmann equation as an effective theory with Quantum Fluctuations
integrated out.
•On the scale O(T), the dispersion relation of the hard particles is
suppressed by O(g2) and collisions can be treated as local collisions.
•Hard modes are weakly interacting and can be treated perturbatively.
•Method: Linearized Boltzmann equation with Driving Term
f ( x , p )  f 0 ( x , p )  f 0 ( x , p )[ 1  f 0 ( x , p )] f 1 ( x , p )
C [ f ]  C [ f 0 ]  Cf i
Transport Coefficients
[

t


 
 p   x  Fext   p ] f 0 ( x , p , t )   ( Cf 1 )( x , p , t )
Driving Fields:
(i) Conductivity:
Ei
(ii) Diffusion:
i
(iii) Shear Viscosity:
1
6
[ i u j   j u i 
2
3
 ij   u ]
Nonperturbative Physics
•Perturbation Theory:
•At high temperature (T >>mq,
T>>  QCD ), g << 1.
•Long distance modes of hot nonAbelian gauge fields are strongly
coupled resulting in breakdown of
perturbation theory.
•Nonperturbative Physics:
•Lattice calculations for static
situations.
•Kinetic Theory: for large times and
distances compared to particle
energy and momentum.
•Hydrodynamics: particles propagate
classically between collisions, which
are independent, uncorrelated
events.
Mean free time
Perturbation
Theory
Kinetic Theory
Hydrodynamics
T
Quantum Field
Theory
1
4
( g T ln( g
4
(g T )
1
1
))
1
Soft, nonperturbative
dynamics
Transport
Coefficients
(Arnold & Yaffe)
Effective Theory


Effective Theories depend on scale separation.
Expansion parameter:
 
LowEnergyS cale
HighEnergy Scale
•Eliminates the high energy scale from the equations of motion,
which implies that the high energy scale is not dynamical anymore
•Obtain an effective Lagrangian in which the information about the
high energy scale is only contained implicitly.
Classical Field Approximation



Soft field modes have a large occupation number
 Classical Field Approximation
Hard field modes (p~T) have occupation number unity, so to use
classical field approximation
 Integrate out hard field modes
Can think of this as hard field modes propagating in a slowly
varying classical background field.

Effective Hard Thermal Loop Theory (HTL)
HTL Effective Theory





Infrared divergences due to massless modes in standard thermal
theories.
Found that physical observables are gauge dependent.
Resummation of all 1-loop diagrams with hard internal momenta and
soft external momenta (Braaten & Pisarski).
HTL corrections to the propagator and vertices lead to gauge
invariant results for physical observables.
Resulting effective theory for the soft modes contains non-local
interactions.
NLLO : Next To Leading Logarithmic
Order
•Dynamics of low frequency gauge fields.
D  B      .
  color conductivi
ty
•Effective Leading Logarithmic Order (LLO) obtained by Boedeker after
integrating out
•Quantum Fluctuations: T
•Thermal Fluctuations: gT
•Goal: Effective Theory valid beyond LLO
•All higher order corrections are suppressed by one or more powers of (ln( g  1 ))  1
(Yaffe)
 Form of Effective Theory unchanged to NLLO
 Need only to calculate the single parameter to NLLO.
Langevin Dynamics: Boedeker

Soft field modes affect gauge field dynamics on the magnetic scale
 Interactions between higher and lower
frequency modes
Soft modes
(P~ g 2T )

Semihard
modes
(P~gT)
•Goal: Effective Theory for soft modes in QGP
 Treat soft modes as the system, semihard modes as the
environment
•Coarse-grained effective action (coarse-grain parameter are loop
expansions)
g T    gT
2
Classical Effective Theory


Open system introduced through coarse-graining condition.
Procedure:
(i) Integrate out hard field modes
 HTL effective theory
(ii)Integrate out degrees of freedom with p~gT.
 Resulting Effective Theory
D F
a
2
 mD W

A ( x )  A
 
( v  D )W  v  E  
BoltzmannLangevin
Equation
 ( x , v )  gf

D      igA  ( x )
 b
c
(v  at ( x ) w ( x, v )

Tr (T T ) 
a
b
( x )T
1

a
ab
2
W ( x , v )  W ( x , v )T
a
Fa
abc
a





a
  A a   A a  gf abc A


v  (1, v )

b
A

c
Origin of noise term

To leading logarithmic order equations of motion can be
approximated by a Langevin equation.
ijk
E F
0i
i
D  B      .
White, Gaussian
  ( x )
ia
m
 ( x )  4
i
jb
 
m
1
D
2
2
D
2
i
ij
Ng T log( g
4
2
( x )  2T  
2
0 (x)
ab
 ( x  x )
4
i
1
)
Stochastic forcing term
1
3 Ng T log( g
1
)
1

B 
Color conductivity
F
jk
Origin of the noise term


Coarse-graining in closed system reduces one of the subsystems to environment
(semihard modes)
Introduce Open System
System
Environment
(P~ g 2T )
(P~gT)
Averaged Effect of environment backreacts on the system.
 Random Force  due to thermal fluctuations of semihard field modes
 Dissipative behaviour of the system.
i
Dissipation
of system

Fluctuations in
environment
Classical Transport Theory



Boltzmann-Langevin equation can be obtained from full QFT as the kinetic
limit if the full stochastic Dyson equations (Calzetta/Hu)
But, in the semiclassical limit the soft modes have a large occupation
number, so can arrive at the Boltzmann-Langevin equation even in classical
transport theory.
 Classical colored point particles
 Phase Space trajectories determined by Lorentzian type equations
Litim and Manuel introduce a Boltzmann-Langevin equation including QCD
effects

p [

x

 gf
abc
A
b

m
m
Q

c
dp
Q
a
 gQ a F

 gQ Fa
a
d
dQ
d
a



p

] f ( x, p, Q )  C [ f ]   ( x, p, Q )
p
a
  gf
abc

p A
b

Q
c
non-Abelian: Color charge
evolves in time.
Classical Transport Theory: Origin of
Noise Term




Phenomenological Langevin Approach.
Basically stipulating that equation of motion is incomplete. To include the
effect of the environment need to include the noise term.
Fluctuation-Dissipation Theorem for classical linear dissipative systems
(Landau&Lifshitz).
Assume: Dissipative process is known:
S
(i) Central idea: kinetic entropy and thermodynamical forces: F  
x
F


x
(ii) Near equilibrium, linear response:
(iii) Using fluctuation-dissipation-relation: dx i    ij F  
i
i
j
i
ij
dt
j
i
 The fluctuations near equilibrium can be identified without detailed
knowledge of underlying microscopic dynamics responsible for dissipation.
Mean Field Fluctuations

Fluctuations at equlibrium, equilibrium distribution function.
  f ( x , p , Q )  f ( x , p , Q  )  f eq ( p ) ( x  x  ) ( p  p  ) ( Q  Q  )
3


3
  f  0
soft and semihard modes can be treated classically due to the large
occupation number
 Quantum Statistics: Bose-Einstein, Fermi-Dirac
 Fluctuation-Dissipation-Relation reduced to classical relation.
Nonequilibrium fluctuations
f  gf
(1 )
 f
  f  g f
(1 )
  f  0
  f ( x , p , Q ) f ( x , p , Q  )  f eq ( p ) ( x  x  ) ( p  p  ) ( Q  Q  )  g f
3
3
 Fluctuations still present in nonequilibrium
2
(1 )
( x, p, Q ) f
(1 )
( x , p , Q  )
Fluctuation-Dissipation-Relation

Dissipative Linear System (Landau&Lifshitz)
 
Dissipation
Fluctuation
Key idea: Kinetic entropy S(x), thermodynamic force
  i ( t ) j ( t )    ( t  t  )
ij
Dynamics :
dx i
dt
  F j   i
ij
dissipative

White,
Random,microscopic
fluctuations, induced by
environment
Gaussian
Fluctuation-Dissipation

Near equilibrium: Phenomenological linear response
F i   ij x

j
Equal time statistics of fluctuations  Einstein’s Law
 x ( t ) F j ( t )  
i

i
j
Assuming: White, Gaussian noise

Noise-noise
Autocorrelation
function
ik
 [
ik
 ]
ki
Symmetrized
dissipative function
Master Equation

Consider :

Homogeneous gas

Set of velocity states available to N particles divided into K cells

State of the system defined by complete
set of occupation numbers
K

n ( t )  ( n1 ( t ) n 2 ( t )... n K ( t ))


 n (t )  N
i
i 1
Goal: Treat evolution of the system as a stochastic process
Assumptions:
N
n
Thermodynamic Limit
N 
K  
K






P ( n ; t   | m ; t )  P ( n ;  | m ;0 )
stationary
P ( n K ; t   | n1t1 ... n K 1t K 1 )  P ( n K t K | n K 1t K 1 )
Markovian
Master Equation


Process entirely determined by

 
P (m | 0 )
P ( m | n ; )
Transition Probability
Master Equation

P ( n ; ) 

 

P ( m | n ;  ) P ( m ;0 )
m
Assume n changes only be a small amount in  t :    t  t





 P ( n ;   t | m )
P ( n ;   t )  P ( n ; | m ; )   t[
|t  0  C ]
 (t )
 
Q (n | m )
Transition Matrix
Master Equation

 
Q (n | m )  0
C can be calculated using:
 
 P (m | n; t )  1

n
 
 
Q ( m | m )    Q (m | n )
i   j  k  l
 
mn
 
 
 
P (m | n; t )   (m | n )  Q (m | n )t
 ij  0
kl
Only for
allowed
transitions
 ij  
kl
No transition
Transition m->n in  t
kl
ji
  ij
 ii   ij  0
kl
kk
 ij   kl
kl
As  t  0 obtain a Master Equation


 
dP ( n ; t )
  P ( m ; t )Q ( m | n )

dt
m
ij
N-particle stochastic process,
position coordinates suppressed
 
1
kl
ij
Q ( m | n )  (   ij m i m j  n i m i 1  n j m j 1  n k m k 1  n l m l 1   n r m r    ij m i m j   n r m r )
4 ij  kl
ij
r  ijk
r
lk
Master Equation

Probability of collision between point particles
P ( ij  kl ) dt  d  ( ij  kl ) m i m j dt
d  ( ij  kl )   ij
kl
 ij  
kl
kl
ji
Occupation numbers before
collision
  ij
lk
 ii   ij  0
kl
kk
 ij   kl
kl
ij
microreversibility
Boltzmann Equation

Introduce a characteristic moment generating function (Fourier transform of
the probability density)

 ( x; t) 
K


n

n
xi i P ( x ; t )

i 1
 n s 

xs
| x1... x
K 1
Summing over all but 1,2…of the occupation numbers equations obtain
the BBGKY hierarchy
d  ns 
dt


1

2
ks
ij
 ni n j 
ijk
d  ns 
dt

1

2
ks
ij
 ni n j  
ij  ks
Molecular Chaos assumption: Neglect
to
 n i  n j 
1

2
ij
ks
 nk ns 
microreversibilty
k
 ( n i   n i  )( n j   n j  ) 
compared
Boltzmann Equation
d  ns 

dt
1
2
  ij  n i  n j    n k  n s 
ks
ij  ks
For small deviations from equilibrium:
d  ns 
dt


1

2
ks
ij
 n i  n i (1  hi )
n i n j ( hi  h j  h k  h s )
ys 
ij  ks
d  ys 

dt
1
ns

ns
ns
g s       j n n j 
ks
g s   y s  0
jk
1

2
s
ij
ni n j 
jk
1
2
 s

ks
ij
ni n j
jk

Langevin idea, Linear response, then a linearized Boltzmann equation
with a white, Gaussian noise term is obtained:
d  ys 
dt

1
s
g
s

 y s  c~s ( t )
s 

ks
ij
ns
s 
N
 N
ks
ij
ns
N
Fokker-Planck Equation

dP ( x ; t )
dt
 
s
A



1
[ a s ( x ) P ( x )] 
xs
2
Drift Term

 x x
s
s
B
2


[ b s  ( x ) P ( x )]

Diffusion Term
Comparing terms A and B to the stochastic Boltzmann equation
d  ys 
dt

1
s
A
g
s
 y s  c~s ( t )

B
Then the Fokker-Planck equation is equivalent to the linearized
Boltzmann equation with forcing term.
Particle Dynamics


Fermions? Quark Dynamics?
Consider: Worldline Influence Functional Method (Johnson/Hu)





Quantum/Vacuum fluctuations – Thermal Fluctuations
Decoherence – Dissipation
Key Concept: Quantum Open System


Langevin Equation for QED
Relativistic particles moving in an electromagnetic field.
Closed vs. open system
System
Environment
Collection of Worldlines
Gauge Field
Method = Influence Functional Formalism
Summary

Degrees of Freedom and scales in Quark Gluon Plasma
Gluons (p~g2T, p~gT)




Semihard field modes backreact on the soft field modes
The Boltzmann equation needs to be “upgraded” with a noise term
Boltzmann-Langevin (Boedeker) equation
Three ways to derive the noise term in kinetic theory:





Quarks (p~T)
Boedeker: HTL, coarse-grained effective action, open system
Litim/Manuel: Phenomenological Langevin idea, Fluctuation-Dissipation relation
Kac/Logan: Master equation, stationary Markovian transition process
White, Gaussian noise in all three cases
Nonlocal noise? Fermions?
Effective Theories
ys 
ns
 ij  0

kl
ns
g s      n n j 
ks
j
jk
1
2

s
ij
ni n j 
1
jk
 s
2

ks
ij
d  ns 
ni n j
jk

dt
1

2
ks
ij
 ni n j  
ij  ks
1
 n s 

2
ij
ks
 nk ns 
xs
| x1... x
K 1
 n i  n j 
k




P ( n ; t   | m ; t )  P ( n ;  | m ;0 )





 P ( n ;   t | m )
P ( n ;   t )  P ( n ; | m ; )   t[
|t  0  C ]


 (t )
 n (t   ) n (t ) 
 
 P ( m | n ;  t )  1
 
 
n
Q
(
m
|
m
)


Q
(
m
| n)

 n  n (1  h )
i
i

d  ys 
 
 
 
P (m | n; t )   (m | n )  Q (m | n )t

dP ( n ; t )


1

dt
g
s
ns
N
t
 y s  0
 
d  ns 

1
2

dt

n
i 1
kl
K


ks
ij
kl
d  ys 
ni (t )  N
i 1
dt
 
s
lk
kk
kl
ij
K  

P (m | 0 )
n i n j ( hi  h j  h k  h s )

dt
1

2
 n i  n j    n k  n s  P ( n K ; t   | n1t1 ... n K 1t K 1 )  P ( n K t K | n K 1t K 1 )
ks
ij
ij  ks

1
s
g
s
 
P ( m | n ; )
 y s  c~s ( t )


P ( n ; ) 
 

 P ( m | n ;  ) P ( m ;0 )
s 
m

  ij
 ij   kl
ij  ks
 
1
kl
ij
Q ( m | n )  (   ij m i m j  n i m i 1  n j m j 1  n k m k 1  n l m l 1   n r m r    ij m i m j   n r m r )
4 ij  kl
ij
r  ijk
r

dP ( x ; t )
kl
ji
 ii   ij  0

d  ns 

 ij  

N 
P ( ij  kl ) dt  d  ( ij  kl ) m i m j dt

n
xi i P ( x ; t )
kl
 
Q (n | m )
  t  t
dt
K
d  ( ij  kl )   ij

 P ( m ; t )Q ( m | n )
i   j  k  l

 ( x; t) 
t2

n ( t )  ( n1 ( t ) n 2 ( t )... n K ( t ))

m
dt
n
K


m  n
 
mn
i
s 
t1


1
[ a s ( x ) P ( x )] 
xs
2

2
 x x
s
s


[ b s  ( x ) P ( x )]

 ( n i   n i  )( n j   n j  ) 
d  ns 
dt

1

2
ijk
ks
ij
 ni n j 

ks
ij
ns
N
 N
ks
ij
ns
N
  f  0
f  gf
(1 )
 f
  f ( x , p , Q )  f ( x , p , Q  )  f eq ( p ) ( x  x  ) ( p  p  ) ( Q  Q  )
3
  f ( x , p , Q ) f ( x , p , Q  )  f eq ( p ) ( x  x  ) ( p  p  ) ( Q  Q  )  g f
3
2
m
 ( x )  4
i
2
  ( x )
ia
1
D
3
0 (x)
i
1
2
Ng T log( g
)
E F
i
jb
Fi  
dp

 gQ Fa
a
d

S
x

d
ij

]  mD
2
d
 4 v
ab
( x , p , Q  )
 ( x  x )

  f  0
4
i
M
D      igA  ( x )
Tr (T T ) 
a
b
1

a

g T    gT
a
 A ( t ) A ( t  )  D A  ( t  t  )
 
( v  D )W  v  E  
ab
2
 Fext  Fenv
v   v  Aext  A fluct
W ( x, v)
( x )T
dv
dt
F i   ij x
2
A ( x )  A
 
[ v  D , W ( x , v )]  v  E   ( x , v )
m
( x )  2T  
(1 )
2
g T
p
i
dQ
( x, p, Q ) f
0i
[D , F
m
(1 )
3
 ( x , v )  gf
a
abc
j
 b
c
(v  at ( x ) w ( x, v )
 x ( t ) F j ( t )  
i
W ( x , v )  W ( x , v )T
a
  f  g f
a
(1 )
B 
i
1

ijk
F
jk
i
j
2
a
  gf
abc

p A
b

Q
c
Fa



 
4
m
2


3 Ng T log( g
b
A


v  (1, v )

c
D F

2
 mD W

  i ( t ) j ( t )    ( t  t  )
ij

ik
D  B      .
1
)

p [
dt

1
D
2
dx i

  A a   A a  gf abc A
  F j   i
ij

x

 gf
abc
A
b

Q
c

Q
a
 gQ a F
a


p

] f ( x, p, Q )  C [ f ]   ( x, p, Q )
 [
ik
 ]
ki
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NonequilibriumDynamicsofQuarkGluonPlasma