stochastic mean field (smf) description

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 112 1,20 vHH

],δK[ρ]K[ρ1'|(t)]ρ,[H|1t),(1,1'ρt

i 11101

Dynamics of many-body system I

Mean-field Residual interaction

)||,( 2

1 vK F

),( vK F KK

Average effect of the residual interaction

one-body

Fluctuations

TDHF

0 K

one-body

density matrix two-body density matrix

o Mean-field (one-body) dynamics

o Two-body correlations

o 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

)'()','(),( ttCtpKtpK

ff 1

Transition rate W

interpreted in terms of

NN cross section KCollision Integral

ˆ H

E

EeffH

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)(

0

00

LSEsym

or J

expansion around normal density

Energy per nucleon E/A (MeV) Symmetry energy Esym (MeV)

)()()0,(),( 42 OEA

E

A

Esym

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,

,,,,0

Hf

t

tprf

dt

tprdf

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 …

collcoll IfIhf

t

tprf

dt

tprdf

,

,,,,

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 o

o

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

32

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.