Stochastic Mean Field (SMF) description TRANSPORT Maria Colonna INFN - Laboratori Nazionali del Sud (Catania)

Transcription

1 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)

2 Dynamics of many-body system I two-body density matrix ρ 1 2(12,1'2') ρ1(1,1')ρ (2,2') (12,1'2') H H 0 v 1,2 Mean-field ρ1(1,1', t) 1 [H0,ρ1(t)] 1' K[ρ1] δk[ρ, ] t i 1 K F( 1, v K 2 ) F( v, ) one-body density matrix one-body Average effect of the residual interaction K 0 o Mean-field (one-body) dynamics TDHF o Two-body correlations KK o Fluctuations Residual interaction Fluctuations

3 Dynamics of many-body systems II Collision Integral K f 1 f Transition rate W interpreted in terms of NN cross section -- 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 Effective interactions Energy Density Functional theories: The exact density functional is approximated with powers and gradients of one-body nucleon densities and currents. E Ĥ E ˆ Ĥ eff

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

5 1. Semi-classical approximation to Nuclear Dynamics Transport equation for the one-body distribution function f Semi-classical analog of the Wigner transform of the one-body density matrix Chomaz,Colonna, Randrup Phys. Rep. 389 (2004) Baran,Colonna,Greco, Di Toro Phys. Rep. 410, 335 (2005) Density f = f (r,p,t) Phase space (r,p) df r, p, t f r, p, t dt t f, H 0 0 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 Vlasov Equation, like Liouville equation: The phase-space density is constant in time Semi-classical approximation df r, p, t f r, p, t dt t, Vlasov Semi-classical approaches f h Icollf I coll k transport theories δk Boltzmann-Langevin Correlations, Fluctuations

6 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 = 2π/k multifragmentation event

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

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

9 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.

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

11 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

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

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