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 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
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
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 0 3 0 L 0... 25 J 35 MeV 20 L 120 MeV
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
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
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
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): [ ]
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 o Test particle positions and momenta are propagated 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 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: 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
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.