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A new implementation of the
Boltzmann – Langevin Theory in 3-D
• Beyond mean field: the Boltzmann – Langevin equation
• Two existing applications:
An idealized 2-D model by Chomaz et al.
Toward nuclear reactions: 3-D implementation by Bauer et al.
• A new procedure: checking the Pauli blocking
• Preliminary results: tests of the code by looking at fluctuations
• Conclusions and perspectives
Philippe Chomaz, GANIL Caen
Maria Colonna and Joseph Rizzo, INFN – LNS Catania
Boltzmann-Langevin equation
How to introduce fluctuations in mean field approaches?
Semi-classical approximation: BUU equation + fluctuations

f (r , p )  h( f ), f (r , p )  K ( f )  K (r , p, t )
t
Random
force
Collision
term
Ensemble averaging
K (r , p, t )K (r , p, t )   ( p, p, r ) (r  r )
Brownian phase-space motion
 ( p, p)   f 2 ( p) ( p  p)   cov ( p, p)
Correlation function
= Gain+Loss terms
   f ( p)  f (1  f )
2
Instantaneus
equilibrium
value
W 
d
 (W   W  ) 
dt
df
 fW   fW 
dt
1
dp dp dp
h  32  31  32 f ( p2 ) f ( p1 ) f ( p2 ) w(12  1 2)
2
h
h
h
W+,W- gain and loss rates
w(12  1 2)   ( p f  pi ) ( E f  Ei )
d NN
d
2-D application: an idealized model
Fixed grid of phase-space points
Number of expected
Actual number of transitions:
2
random walk    
1, 2;1, 2 t transitions in an interval Δt


Configuration: uniform r space; two touching Fermi spheres in p space
 ( p, p)   f 2 ( p) ( p  p)   cov ( p, p)
Equil.
value
Observable: variance in
2
momentum space  f ( p)
 f ( p)
2
Successful, but:
• only 2-Dimensional
• time consuming
Occupancy at
a single point
3-D Application: Correlated pseudo-particle collisions
Cross section     NN
NN
reduction
NTEST
•Clouds translated (no rotation) to final states
•Pauli blocking checked only for i and j t.p.
Phase-space distance

dik  ( pi  pk )2  ( pF / R)2 (ri  rk )2
2

1 nucleon = NTEST nearest neighbours
<pi> and <pj> ; Δp assigned to each “cloud”
ff
 f ( p)
2
Equil.
value
-Δp
Δp
Shape of  f ( p) more similar
to classical than to quantum case
2
d ij
Difficult to extrapolate the
strength of fluctuations
Problems:
•Definition of dij
•Wrong f and  f 2 ( p)
But:
• 3-Dim
• suitable for HIC
simulations
A new procedure: no Pauli-blocking violation
Grid in p-space
ntrans  min( nI , nJ , nI  , nJ  )
“clouds” rotated to final states
Nucleon with arbitrary shape in p-space
(suitable for any configuration)
Pauli blocking checked in each cell
Objectives:
I
J
 Procedure easily applicable to nuclear reactions
 Preserve localization of the nucleon in p-space
 Take into account possible nucleon deformations
 Correct fluctuations preserving averages ( f , K(f ) )
Still arbitrary:
•r-space distance dik 2  (ri  rk )2
•p-cell size
Results (1): cloud sizes
Initial configuration: Fermi-Dirac (equilibrium)
Is it possible to build expected fluctuations?
Periodic boundaries 3-D box l = 26 fm; kT = 5 MeV; ρ = 0.16 fm-3 (2820 nucleons); 500 t.p.
Collision = correlation in p-space
best result is to correlate a volume h3/4
In a single collision we move particles
within a volume Vpnuc  (35MeV / c)3
Vpnuc  (30MeV / c)3
Volume larger
than V pnuc
•1 nucleon in bigger volume
•f (p)<1, f (p) smoothed
•Clouds are not sharp, surface effects
Cloud sizes
t  50 fm / c
p
x
p
yz
p
Is it possible to observe the expected
fluctuations in a fixed grid?
Centroids of clouds can have any position:
fluctuations will be smoothed for volumes
containing a few nucleons
“Centroid effect”
Results (2): fixed grid of cubic cells
h3 / 4 f (1  f )
 f ( p)  f (1  f ) 3 
, N nuc  1
L
N nuc
2
L
(MeV/c)
σ2f
Larger cubes = more nucleons
Reduced “centroid effect”
Centroid positions
Better geometry:
Spherical coordinates
• allow to go to larger volumes
• focus on max. correlation region
p  pF
 p  30MeV / c
t ~ 50 fm/c
f (1  f )
1
N nuc
 2f N nuc
30
1
9.0
45
0.3
4.6
65
0.1
3.0
80
0.05
2.7
90
0.04
2.6
Cloud sizes
Results (3): spherical coordinates
Δθ = 30°
f (1  f )
 2f N nuc
L
(MeV/c)
1
N nuc
130
0.044
4.7
150
0.028
3.6
170
0.020
190
210
Δθ°
1
N nuc
f (1  f )
 2f N nuc
10
0.124
3.1
20
0.031
2.2
30
0.015
2.5
2.8
45
0.006
3.2
0.015
2.5
60
0.004
3.4
0.010
2.7
L = 190 MeV/c
L3
V  2  cos( )
3
Set of coordinates dp, pd , p sin(  )d
t = 0 fm/c
t = 100 fm/c
L=190 Mev/c
Δθ=20°
f (1  f )
 2.2
 2f N nuc
p = 260 MeV/c
Δp = 10 MeV/c
  sin(  )
f (1  f )
2
 f 2  ,
Conclusion and perspectives
Improved correlated pseudo-particle collision procedure
• easy to insert into existing BUU codes for nuclear reactions
• Pauli-blocking carefully checked
Results of preliminary tests: variance in momentum space
• In each collision cloud volumes are larger than h3/4: f smoothed
• Additional smearing due to the “centroid effect”
Looking for the best way to see fluctuations
• Cartesian vs spherical coordinates
• Maximum correlation region
At present:
• Expected fluctuations reproduced within a factor of 2
• Right shape of  2f ( p)
Next steps: going to nuclear reactions
• dependence on free parameters (r-space distance, p-cell size)
•A “better” configuration: 2 touching Fermi spheres; comparison with other models
• Covariance: correlations between different phase-space point
• Effects on intermediate energy HIC