Stochastic Mean Field (SMF) description
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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
Transcript of 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.
