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

Stochastic Mean Field (SMF) description TRANSPORT 2017 March 27-30, 2017 FRIB-MSU, East Lansing, Michigan, USA Maria Colonna INFN - Laboratori Nazionali del Sud (Catania )

Dynamics of many-body system I o Mean-field (one-body) dynamics o Two-body correlations o Fluctuations one-body density matrix two-body density matrix      ρ ρ ) ρ H H v (12,1'2' ) (1,1' (2,2' ) (12,1'2' ) 0 1,2 2 1 1 Mean-field Residual interaction one-body        ρ ρ K[ ρ δK[ρ  i (1,1' , t) 1 | [H , (t)] | 1' ] , ]  1 0 1 1 1 t TDHF   F 2 ( , | | ) K 1 v Average effect of the residual interaction   (  F   K    K   , ) K v 0 K Fluctuations

Dynamics of many-body systems II Transition rate W interpreted in terms of Collision Integral K NN cross section  1  f f -- If statistical fluctuations larger than quantum ones       ( , ) ( ' , ' ) ( ' ) K p t K p t C t t Main ingredients: Residual interaction (2-body correlations and fluctuations) In-medium nucleon cross section Effective interaction (self consistent mean-field) Skyrme, Gogny forces ˆ    Effective interactions E H Energy Density Functional theories: The exact    ˆ ˆ     E density functional is approximated with powers and H eff gradients of one-body nucleon densities and currents.

The nuclear Equation of State (T = 0) Energy per nucleon E/A (MeV) Symmetry energy E sym (MeV) soft poorly known … stiff predictions of several effective interactions E E            2 4 ( , ) ( , 0 ) ( ) ( ) E O sym A A symm. energy symm. matter expansion around normal density    β = asymmetry parameter = ( ρ n - ρ p )/ ρ     0 ( ) ... E sym S L  0  analogy with Weizsacker 3 0 or J mass formula for nuclei (symmetry term) ! 25 ≤ J ≤ 35 MeV 20 ≤ L ≤ 120 MeV

 1. Semi-classical approximation to Nuclear Dynamics Chomaz,Colonna, Randrup Phys. Rep. 389 (2004) Baran,Colonna,Greco, Di Toro Transport equation for the one-body distribution function f Phys. Rep. 410, 335 (2005) Semi-classical analog of the Wigner transform of the one-body density matrix f = f ( r , p ,t) Phase space ( r , p ) Density      Vlasov Equation,  , , , ,  df r p t f r p t   0   , 0 like Liouville equation: f H  dt t The phase-space density is constant in time H 0 = T + U The mean-fiels potential U is self-consistent: U = U( ρ ) Nucleons move in the field created by all other nucleons Semi-classical approximation transport theories       Boltzmann-Langevin , , , ,   df r p t f r p t  δ k      k , f h I f I  coll coll Correlations, dt t Semi-classical approaches … Fluctuations Vlasov

From BOB to SMF …… Fluctuations from external stochastic force (tuning of the most unstable modes) Brownian One Body ( BOB ) dynamics Chomaz,Colonna,Guarnera,Randrup PRL73,3512(1994) λ = 2 π /k multifragmentation event

From BOB to SMF …… Fluctuations from external stochastic force (tuning of the most unstable modes) Brownian One Body ( BOB ) dynamics Chomaz,Colonna,Guarnera,Randrup PRL73,3512(1994) λ = 2 π /k multifragmentation event 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

Details of the model  l = 1 fm triangular function lattice size Total number of test particles : N tot = N test * A  System total energy (lattice Hamiltonian): [ ]

 Potential energy β = asymmetry parameter  Negative surface term to correct surface effects induced by the the use of finite width t.p. packets  Symmetry energy parametrizations : New Skyrme interactions (SAMi-J family) recently introduced Hua Zheng et al.

 Initialization and dynamical evolution o Ground state initialization with Thomas-Fermi Test particle positions and momenta are propagated o according to the Hamilton equations (non relativistic)

Details of the model: Collision Integral  Mean free path method : each test particle has just one collision partner Δ t = time step   Free n-p and p-p cross sections, with a maximum cutoff of 50 mb

Fluctuation tuning When local equilibrium is achieved: Fluctuation variance  for a fermionic system at equilibrium  2  3 T    2 V F Stochastic Mean-Field ( SMF ) model :  Fluctuations are projected on the coordinate space by agitating the spacial density profile

Some applications …… Charge distribution E. De Filippo et al., PRC(2012)  Fragmentation studies in central and semi-peripheral collisions Data - Calculations  Isospin effects at Fermi energies J.Frankland et al., stiff NPA 2001 soft  Small amplitude dynamics (collective modes) and low-energy reaction dynamics Hua Zheng et al. J. Rizzo et al., NPA(2008)