1. 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

2. Boltzmann-Langevin equation Instantaneus equilibrium value W +,W - gain and loss rates Ensemble averaging Semi-classical approximation: BUU equation + fluctuations Collision term Random force Brownian phase-space motion How to introduce fluctuations in mean field approaches? Correlation function = Gain+Loss terms

3. 2-D application: an idealized model Occupancy at a single point Equil. value Observable: variance in momentum space Configuration: uniform space; two touching Fermi spheres in space Fixed grid of phase-space points Number of expected transitions in an interval Δt Actual number of transitions: random walk Successful, but: only 2-Dimensional time consuming

4. 3-D Application: Correlated pseudo-particle collisions Equil. value Shape of more similar to classical than to quantum case 1 nucleon = N TEST nearest neighbours and ; Δp assigned to each “cloud” Clouds translated (no rotation) to final states Pauli blocking checked only for i and j t.p. Cross section reduction Phase-space distance Difficult to extrapolate the strength of fluctuations -Δp-Δp ΔpΔp Problems: Definition of d ij Wrong f and But: 3-Dim suitable for HIC simulations

5. A new procedure: no Pauli-blocking violation Grid in p-space Pauli blocking checked in each cell Nucleon with arbitrary shape in p-space (suitable for any configuration) I J J’ I’ Still arbitrary: r-space distance p-cell size “clouds” rotated to final states Objectives: 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 ) )

6. Results (1): cloud sizes Centroids of clouds can have any position: fluctuations will be smoothed for volumes containing a few nucleons Initial configuration: Fermi-Dirac (equilibrium) Is it possible to build expected fluctuations? Periodic boundaries 3-D boxl = 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 h 3 /4 In a single collision we move particles within a volume pypy pxpx pzpz Is it possible to observe the expected fluctuations in a fixed grid? Cloud sizes Volume larger than 1 nucleon in bigger volume f (p)<1, f (p) smoothed Clouds are not sharp, surface effects “Centroid effect”

7. L (MeV/c) 3019.0 450.34.6 650.13.0 800.052.7 900.042.6 σ2fσ2f Results (2): fixed grid of cubic cells Larger cubes = more nucleons Reduced “centroid effect” Better geometry: Spherical coordinates allow to go to larger volumes focus on max. correlation region t ~ 50 fm/c Centroid positionsCloud sizes

8. Δθ = 30° Results (3): spherical coordinates L (MeV/c) 1300.0444.7 1500.0283.6 1700.0202.8 1900.0152.5 2100.0102.7 Set of coordinates p = 260 MeV/c Δp = 10 MeV/c t = 0 fm/c t = 100 fm/c L=190 Mev/c Δθ=20° L = 190 MeV/c Δθ° 100.1243.1 200.0312.2 300.0152.5 450.0063.2 600.0043.4

9. 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 h 3 /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 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 Next steps: going to nuclear reactions