Transcript time

Extensions of mean-field with stochastic methods
Denis Lacroix
Laboratoire de Physique Corpusculaire - Caen,
FRANCE
Mapping the nuclear N-body dynamics
into a open system problem.
Stochastic one-body mechanics applied
to nuclear physics
Quantum jump approach to the
many-body problem
TDHF and beyond … -Saclay 2006
Mapping the nuclear dyn. to a system-environment problem
Assuming an initial uncorrelated state :
Evolution
in time
Deg3
Mean-field approximation:
One can improve the mean-field
Deg1
approximation by considering one-body
degrees of freedom as a system coupled
to an environment of other degrees of freedom.
Environment
One-body
subspace
Deg2
Illustration:
D. Lacroix, S. Ayik and Ph. Chomaz, Progress in Part. and Nucl. Phys. (2004)
{
Starting from
The correlation propagates as :
where
{
Propagated initial
correlation
Two-body effect
projected on the
one-body space
Molecular chaos assumption
The initial correlations could be treated as a stochastic operator :
where
{
Link with semiclassical approaches in Heavy-Ion collisions
t
t
t
Vlasov
BUU, BNV
BoltzmannLangevin
Adapted from J. Randrup et al, NPA538 (92).
t
time
Application to small amplitude motion
Standard RPA states
Coupling
to 2p2h states
Coupling
to ph-phonon
More insight in the fragmentation of the GQR of 40Ca
EWSR repartition
Intermezzo: wavelet methods for fine structure
Observation
Basic idea of the wavelet method
D. Lacroix and Ph. Chomaz, PRC60 (1999) 064307.
+1
-1
Recent extensions :
D. Lacroix et al, PLB 479, 15 (2000).
A. Shevchenko et al, PRL93, 122501 (2004).
E
Discussion on one-body evolution from projection technique
Success
Results on small amplitude motions looks fine
The semiclassical version (BOB) gives a good reproduction of
Heavy-Ion collisions
Critical aspects
Numerical Implementation of Stochastic methods for
large amplitude motion are still an open problem
(No guide to the random walk)
Instantaneous reorganization of internal degrees of freedom?
Theoretical justification of the introduction of noise ?
Quantum jump method -introduction
Environment
System
Exact dynamics
Dissipative dynamics
At t=0
If waves follow stochastic eq.
{
At t=0
Weak coupling approx.
Projection technique
Markovian approx.
with
Then, the average dyn. identifies with
the exact one
1
For total wave
2
For total density
In fermionic self-interacting systems
Breuer, Phys. Rev. A69, 022115 (2004)
1 Stochastic
mean-field
Lacroix,
Phys. Rev.
A72, 013805 (2005)
Juillet and Chomaz, PRL 88 (2002)
2
Stochastic BBGKY
Lacroix, PRC 71 (2005)
Lindblad master equation:
Can be simulated by stochastic eq. on |F>,
The Master equation being recovered using :
Gardiner and Zoller, Quantum noise (2000)
Breuer and Petruccione, The Theory of Open Quant. Syst.
Quantum jump in the weak coupling regime
GOAL: Restarting from an uncorrelated state
we should:
1-have an estimate of
2-interpret it as an average over jumps between “simple” states
Weak coupling approximation : perturbative treatment
R.-G. Reinhard and E. Suraud, Ann. of Phys. 216, 98 (1992)
Residual interaction in the mean-field
interaction picture
Statistical assumption in the Markovian limit :
We assume that the residual interaction
can be treated as an ensemble of
two-body interaction:
Time-scale and Markovian dynamics
Mean-field time-scale
t+t
t
Collision time
Replicas
{
Hypothesis :
Two strategies have been considered:
Considering waves directly
(philosophy of exact treatment)
Considering densities directly
(philosophy of dissipative treatment)
Average time between two collisions
Simplified scenario for introducing fluctuations beyond MF
Interpretation of the equation on waves as an average over jumps:
Let us simply assume that
with
Matching with a quantum jump process between “simple states” ?
We consider densities
and focus on one-body density:
Additional hypothesis:
We end with:
Mean-field like term
D. Lacroix, arXiv:quant-ph/ 0509038
Nature of the Stochastic one-body dynamics
Important properties
r remains a projector
At all time
with
Average evolution
t
One-body
Correlations beyond mean-field, denoting by
similar to Ayik and Abe,PRC 64,024609 (2001).
Numerical implementation : flexible and rather simple.
time
Application
Stoch. Schrödinger Equation (SSE) on single-particle states:
Assuming
t<0
and
All the information on the system is contained
in the one-body density
r
2
Associated quantum jumps on single particle states:
Monopole vibration in nuclei
rms (fm)
Mean-field part :
Residual part :
Application :
Root mean-square radius evolution:
TDHF
Average evol.
40Ca
nucleus
 = 0.25 MeV.fm-2
time (fm/c)
Diffusion of the rms around the mean value
Standard deviation
Compression
Similar to Nelson quantization theory
Dilatation
Nelson, Phys. Rev. 150, 1079 (1966).
Ruggiero and Zannetti, PRL 48, 963 (1982).
Summary and Critical discussion on the simplified scenario
The stochastic method is directly applicable to nuclei
It provide an easy way to introduce fluctuations beyond mean-field
It does not account for dissipation.
In nuclear physics the two particle-two-hole components dominates
the residual interaction, but
!!!
Generalization: quantum jump with dissipation
Second Philosophy
Master equation for the one-body evolution
Starting from
Contains an
additional term
and its one-body density
with
Matching with the nuclear many-body problem
The residual interaction is dominated by 2p-2h components
Equivalent to the
collision term of
extended TDHF
Existence and nature of the associated quantum jump ?
All interaction of 2p-2h nature can be decomposed into a sum of separable
interaction, i.e.
with
Koonin, Dean, Langanke, Ann.Rev.Nucl.Part.Sci. 47 (1997).
Juillet and Chomaz, PRL 88 (2002).
We can use standard quantum jump methods to
simulate this equation
Again
The equation can be interpreted as the feedback
action of the On operators on the one-body density
time
Application to Bose condensate
1D bose condensate with gaussian two-body interaction
N-body density:
SSE on single-particle state :
with
The numerical effort is fixed by the number of Ak
t>0
mean-field
average evolution
r
width of the
condensate
r(r) (arb. units)
t=0
average evolution
mean-field
time (arb. units)
Summary
Quantum Jump (QJ) methods to extend mean-field
Mean-field
Simplified QJ
Fluctuation
Dissipation
Fluctuation 
Dissipation
Generalized QJ
Fluctuation 
Dissipation 
Variational QJ
Exact QJ
Everything

Partially
everything
Numerical issues
Flexible
Fixed
Fixed
O. Juillet (2005)
Flexible

Giant resonances
Introduction to stochastic theories in nuclear physics
n
Bohr picture of the nucleus
n
Mean-field
N-N collisions
Historic of quantum stochastic one-body transport theories :
Ff
Fi
Statistical treatment of the
residual interaction
(Grange, Weidenmuller… 1981)
-Statistical treatment of one-body
configurations (Ayik, 1980)
-Random phases in final wave-packets
(Balian, Veneroni, 1981)
-Quantum Jump (Fermi-Golden rules)
(Reinhard, Suraud 1995)
Average ensemble evolutions
Fluctuations around the mean density :
Evolution of the average density :
{
Incoherent nucleon-nucleon
collision term.
Coherent collision
term
Linear response
Extended mean-field
Mean-field
Notations for RPA equations
Response to harmonic vibrations
Using
Mean-field
Extended mean-field
+
Fourier transform and coupling to decay channels
Incoherent damping
Coherent damping
Ph. Chomaz, D. Lacroix, S. Ayik, and M. Colonna
PRC 62, 024307 (2000)
S. Ayik and Y. Abe, PRC 64, 024609 (2001).
Coupling to 2p-2h states
Coupling to ph-phonon states
2 p 2 h  
2 p 2h V Coll
  E2 p 2 h
2
  ph  
  ph V Coll
  E  ph
2
Average GR evolution in stochastic mean-field theory
D. Lacroix, S. Ayik and Ph. Chomaz, Progress in Part. and Nucl. Phys. (2004)
RPA response
Full calculation
with fluctuation and
dissipations
Mean energy variation
Full
fluctuation
RPA
dissipation
Effect of correlation on the GMR and incompressibility
Evolution of the
main peak energy :
Incompressibility in finite system
in 208Pb E0  1MeV
{
K ARPA  156MeV
K AERPA  135MeV
Systematic improvement of the GQR energy
Calculated strength
Main peaks energies ,
comparison with experiment
Experiments
N-body exact
Functional integral and stochastic quantum mechanics
General strategy
S. Levit, PRCC21 (1980) 1594.
S.E.Koonin, D.J.Dean, K.Langanke,
Ann.Rev.Nucl.Part.Sci. 47, 463 (1997).
Given a Hamiltonian
and an initial State
Write H into a
quadratic form
Use the Hubbard
Stratonovich
transformation
Interpretation of the
integral in terms of
quantum jumps
and stochastic
Schrödinger equation
t
Example of application:
-Quantum Monte-Carlo Methods
-Shell Model Monte-Carlo ...
time
Recent developments based on mean-field
Carusotto, Y. Castin and J. Dalibard, PRA63 (2001).
O. Juillet and Ph. Chomaz, PRL 88 (2002)
Nuclear Hamiltonian applied to Slater determinant
Self-consistent
one-body part
Residual part
reformulated
stochastically
Quantum jumps between Slater determinant
Thouless theorem
Stochastic schrödinger equation
in one-body space
Stochastic schrödinger equation
in many-body space
Fluctuation-dissipation theorem
Generalization to stochastic motion of density matrix
D. Lacroix , Phys. Rev. C71, 064322 (2005).
The state of a correlated system could be described by
a superposition of Slater-Determinant dyadic
Stochastic evolution of non-orthogonal Slater determinant dyadics :
t
Quantum jump in
one-body density space
with
Quantum jump
in many-body density space
time
Discussion of exact quantum jump approaches
Many-Body
Stochastic Schrödinger equation
One-Body
Stochastic Schrödinger equation
Generalization :
Stochastic evolution
of many-body density
Stochastic evolution
of one-body density
Each time the two-body density evolves as :
with
Then, the evolution of the two-body density can be replaced by an average (
stochastic one-body evolution with :
Actual applications :
-Bose-condensate
-Two and three-level systems
-Spin systems
(Carusotto et al, PRA (2001))
(Juillet et al, PRL (2002))
(Lacroix, PRA (2005))
) of
Perturbative/Exact stochastic evolution
Exact
Perturbative
Properties
Many-body density
Many-body density
Projector
Projector
Number of particles
Number of particles
Entropy
Entropy
Average evolution
One-body
One-body
Correlations beyond mean-field
Correlations beyond mean-field
Numerical implementation :
Flexible: one stoch. Number or more…
Fixed :
“s” determines the number of stoch. variables
Summary
Stochastic mean-field from
statistical assumption
(approximate)
Stochastic mean-field from
functional integral
(exact)
Dab
Dac
Dde
t
Applications:
Vibration :
time
Stochastic mean-field
in the perturbative
regime
Sub-barrier fusion :
Violent collisions :