TRANSPORT 2017 March 27-30, 2017 FRIB-MSU, East Lansing, Michigan, - - PowerPoint PPT Presentation

Stochastic Mean Field (SMF) description

Maria Colonna

INFN - Laboratori Nazionali del Sud (Catania)

March 27-30, 2017 FRIB-MSU, East Lansing, Michigan, USA

TRANSPORT 2017

) (12,1'2' ) (2,2' )ρ (1,1' ρ ) (12,1'2' ρ

1 1 2

  

1,2

v H H  

] , δK[ρ ] K[ρ 1' | (t)] ρ , [H | 1 t) , (1,1' ρ t i

1 1 1 1

        

Dynamics of many-body system I

Mean-field Residual interaction

) | | , (

2 1 v

K  F 

) , (   v K F 

  K K 

Average effect of the residual interaction

• ne-body

Fluctuations

TDHF

  K 

• ne-body

density matrix two-body density matrix

• Mean-field (one-body) dynamics
• Two-body correlations
• Fluctuations

Dynamics of many-body systems II

• - If statistical fluctuations larger than quantum ones

Main ingredients:

Residual interaction (2-body correlations and fluctuations) In-medium nucleon cross section

Effective interaction (self consistent mean-field) Skyrme, Gogny forces

) ' ( ) ' , ' ( ) , ( t t C t p K t p K      

f f  1

Transition rate W interpreted in terms of NN cross section

K

Collision Integral

 

     ˆ H E E

eff

H ˆ

 

 ˆ

Effective interactions

Energy Density Functional theories: The exact density functional is approximated with powers and gradients of one-body nucleon densities and currents.

The nuclear Equation of State (T = 0)

• analogy with Weizsacker

mass formula for nuclei (symmetry term) !

β = asymmetry parameter = (ρn - ρp)/ρ

... 3 ) (         L S Esym

• r J

expansion around normal density Energy per nucleon E/A (MeV) Symmetry energy Esym (MeV)

) ( ) ( ) , ( ) , (

4 2

       O E A E A E

sym

   

• symm. matter
• symm. energy

stiff soft poorly known … predictions of several effective interactions

25 ≤ J ≤ 35 MeV 20 ≤ L ≤ 120 MeV

• 1. Semi-classical approximation to Nuclear Dynamics

Transport equation for the one-body distribution function f

Chomaz,Colonna, Randrup

• Phys. Rep. 389 (2004)

Baran,Colonna,Greco, Di Toro

• Phys. Rep. 410, 335 (2005)

     

, , , , ,

0 

    H f t t p r f dt t p r df

Semi-classical analog of the Wigner transform of the one-body density matrix

f = f (r,p,t) Phase space (r,p)

Vlasov Equation,

like Liouville equation: The phase-space density is constant in time Density

• H 0 = T + U

Semi-classical approaches …

       

coll coll

I f I h f t t p r f dt t p r df        , , , , ,

Correlations, Fluctuations

k δk

Vlasov Semi-classical approximation transport theories

Boltzmann-Langevin

The mean-fiels potential U is self-consistent: U = U(ρ) Nucleons move in the field created by all other nucleons

From BOB to SMF ……

Fluctuations from external stochastic force (tuning of the most unstable modes)

Chomaz,Colonna,Guarnera,Randrup PRL73,3512(1994)

Brownian One Body (BOB) dynamics

λ = 2π/k multifragmentation event

From BOB to SMF ……

Fluctuations from external stochastic force (tuning of the most unstable modes) Stochastic Mean-Field (SMF) model : Thermal fluctuations (at local equilibrium) are projected on the coordinate space by agitating the spacial density profile

M.Colonna et al., NPA642(1998)449 Chomaz,Colonna,Guarnera,Randrup PRL73,3512(1994)

Brownian One Body (BOB) dynamics

λ = 2π/k multifragmentation event

Details of the model

• triangular function

l = 1 fm

[ ]

Total number of test particles: Ntot = Ntest * A

• System total energy (lattice Hamiltonian):

lattice size

• Potential energy
• Symmetry energy parametrizations :

New Skyrme interactions (SAMi-J family) recently introduced

Hua Zheng et al.

Negative surface term to correct surface effects induced by the the use of finite width t.p. packets β = asymmetry parameter

• Test particle positions and momenta are propagated

according to the Hamilton equations (non relativistic) Ground state initialization with Thomas-Fermi

• Initialization and dynamical evolution

Details of the model: Collision Integral

• Mean free path method :

each test particle has just one collision partner

• Δt = time step

Free n-p and p-p cross sections, with a maximum cutoff of 50 mb

Fluctuation tuning

When local equilibrium is achieved:

• F

T V     2 3

2 

Stochastic Mean-Field (SMF) model : Fluctuations are projected on the coordinate space by agitating the spacial density profile Fluctuation variance for a fermionic system at equilibrium

Some applications ……

 Fragmentation studies in central and semi-peripheral collisions

 Small amplitude dynamics (collective modes) and low-energy reaction dynamics

 Isospin effects at Fermi energies

stiff soft

• J. Rizzo et al., NPA(2008)
• E. De Filippo et al., PRC(2012)

Data

• Calculations

Charge distribution

J.Frankland et al., NPA 2001

Hua Zheng et al.