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Extensions of mean-field with stochastic methods
Denis LacroixLaboratoire de Physique Corpusculaire - Caen,
FRANCE
Mapping the nuclear N-body dynamicsinto a open system problem.
Quantum jump approach to the many-body problem
One Body space
TDHF and beyond … -Saclay 2006
Stochastic one-body mechanics applied to nuclear physics
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Mapping the nuclear dyn. to a system-environment problem
Assuming an initial uncorrelated state :
Evolution in time
One can improve the mean-field approximation by considering one-body degrees of freedom as a system coupled to an environment of other degrees of freedom.
Mean-field approximation:
Deg1
Deg2
Deg3
One-body subspace
Environment
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Illustration:
The correlation propagates as :
where
{ Propagated initial correlation Two-body effect projected on the one-body space
Starting from
D. Lacroix, S. Ayik and Ph. Chomaz, Progress in Part. and Nucl. Phys. (2004)
{
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The initial correlations could be treated as a stochastic operator :
where
{Link with semiclassical approaches in Heavy-Ion collisions
t t t t time
Vlasov
BUU, BNV
Boltzmann- Langevin
Adapted from J. Randrup et al, NPA538 (92).
Molecular chaos assumption
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Application to small amplitude motion
Standard RPA states Coupling
to ph-phononCoupling
to 2p2h states
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More insight in the fragmentation of the GQR of 40Ca
EWSR repartition
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Intermezzo: wavelet methods for fine structure
Observation
E
-1
+1
D. Lacroix and Ph. Chomaz, PRC60 (1999) 064307.
Basic idea of the wavelet method
Recent extensions : D. Lacroix et al, PLB 479, 15 (2000). A. Shevchenko et al, PRL93, 122501 (2004).
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Discussion on one-body evolution from projection technique
Results on small amplitude motions looks fine
The semiclassical version (BOB) gives a good reproduction of Heavy-Ion collisions
Success
Critical aspects
Numerical Implementation of Stochastic methods for large amplitude motion are still an open problem
(No guide to the random walk)
Theoretical justification of the introduction of noise ?
Instantaneous reorganization of internal degrees of freedom?
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Quantum jump method -introduction
Environment
System
{If waves follow stochastic eq.
with
Exact dynamics
At t=0
Breuer, Phys. Rev. A69, 022115 (2004)Lacroix, Phys. Rev. A72, 013805 (2005)
Then, the average dyn. identifies with the exact one
1 For total wave
For total density2
Projection technique
Weak coupling approx.
Markovian approx.
At t=0
Dissipative dynamics
Lindblad master equation:
Gardiner and Zoller, Quantum noise (2000) Breuer and Petruccione, The Theory of Open Quant. Syst.
Can be simulated by stochastic eq. on |>, The Master equation being recovered using :
1
In fermionic self-interacting systems
2
Stochastic mean-field Juillet and Chomaz, PRL 88 (2002)
Stochastic BBGKY Lacroix, PRC 71 (2005)
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Quantum jump in the weak coupling regime
We assume that the residual interaction can be treated as an ensemble of two-body interaction:
Statistical assumption in the Markovian limit :
Weak coupling approximation : perturbative treatment
Residual interaction in the mean-field interaction picture
R.-G. Reinhard and E. Suraud, Ann. of Phys. 216, 98 (1992)
GOAL: Restarting from an uncorrelated state we should:
2-interpret it as an average over jumps between “simple” states
1-have an estimate of
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Time-scale and Markovian dynamics
{t t+t
Rep
licas
Collision time
Average time between two collisions
Mean-field time-scale
Hypothesis :
Two strategies have been considered:
Considering densities directly (philosophy of dissipative treatment)
Considering waves directly(philosophy of exact treatment)
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Simplified scenario for introducing fluctuations beyond MF
Additional hypothesis:
We end with:
Mean-field like term
D. Lacroix, arXiv:quant-ph/ 0509038
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” ?
and focus on one-body density:We consider densities
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Nature of the Stochastic one-body dynamics
Important properties remains a projector
Numerical implementation : flexible and rather simple.
t timeAverage evolution
One-body
Correlations beyond mean-field, denoting by
similar to Ayik and Abe,PRC 64,024609 (2001).
At all time
with
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Application
Root mean-square radius evolution:
rms (
fm)
time (fm/c)
TDHFAverage evol.
Stoch. Schrödinger Equation (SSE) on single-particle states:
Assuming and
All the information on the system is contained in the one-body density
2rt<0
Residual part :
Mean-field part :
Application : 40Ca nucleus = 0.25 MeV.fm-2
Monopole vibration in nuclei
Associated quantum jumps on single particle states:
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Diffusion of the rms around the mean value
Standard deviation
No const
rain
t
Compression Dilatation
= 0.25 MeV.fm-2
Similar to Nelson quantization theoryNelson, 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 !!!
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Generalization: quantum jump with dissipation
Second Philosophy
Contains an additional term
Master equation for the one-body evolution
Starting from and its one-body density
Matching with the nuclear many-body problem
The residual interaction is dominated by 2p-2h components
with
Equivalent to the collision term of extended TDHF
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Existence and nature of the associated quantum jump ?
with
All interaction of 2p-2h nature can be decomposed into a sum of separable interaction, i.e.
Koonin, Dean, Langanke, Ann.Rev.Nucl.Part.Sci. 47 (1997). Juillet and Chomaz, PRL 88 (2002).
time
Again
We can use standard quantum jump methods tosimulate this equation
The equation can be interpreted as the feedback action of the On operators on the one-body density
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SSE on single-particle state :
with
time (arb. units)
wid
th o
f th
e
con
den
sate
mean-field
average evolution
Condensate size
Application to Bose condensate
N-body density:
1D bose condensate with gaussian two-body interaction
The numerical effort is fixed by the number of Ak
r
(r)
(arb
. u
nit
s)
t=0t>0
mean-field
average evolution
Density evolutio
n
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SummaryQuantum Jump (QJ) methods to extend mean-field
Simplified QJ
FluctuationDissipation
Generalized QJ
FluctuationDissipation
Exact QJ
Everything
Mean-field
FluctuationDissipation
Variational QJ
Partially everything
Numerical issues
FlexibleFlexible Fixed Fixed
O. Juillet (2005)
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Giant resonances
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Introduction to stochastic theories in nuclear physics
Mean-field
Bohr picture of the nucleus
n
N-N collisions
n
Statistical treatment of the residual interaction(Grange, Weidenmuller… 1981)
-Random phases in final wave-packets (Balian, Veneroni, 1981)
-Statistical treatment of one-body configurations (Ayik, 1980)
-Quantum Jump (Fermi-Golden rules) (Reinhard, Suraud 1995)
Historic of quantum stochastic one-body transport theories :
if
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{Incoherent nucleon-nucleon collision term.
Coherent collision term
Evolution of the average density :
One Body space Fluctuations around the mean density :
Average ensemble evolutions
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Linear response
Mean-field Mean-field Extended mean-fieldExtended mean-field
Response to harmonic vibrations
Notations for RPA equations
Using
+
Mean-field Mean-field Extended mean-fieldExtended mean-field
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Fourier transform and coupling to decay channels
Incoherent damping
Ph. Chomaz, D. Lacroix, S. Ayik, and M. Colonna PRC 62, 024307 (2000)
Coherent damping
S. Ayik and Y. Abe, PRC 64, 024609 (2001).
hphp E
CollVhp
22
2
22
22
phph E
CollVph
2
Coupling to 2p-2h states Coupling to ph-phonon states
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Average GR evolution in stochastic mean-field theory
Full calculation with fluctuation and dissipations
RPA response
D. Lacroix, S. Ayik and Ph. Chomaz, Progress in Part. and Nucl. Phys. (2004)
Mean energy variation
fluctuation
dissipationRPA
Full
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Effect of correlation on the GMR and incompressibility
Incompressibility in finite system
in 208Pb MeVE 10 MeVK RPA
A 156
MeVK ERPAA 135{
Evolution of the main peak energy :
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Systematic improvement of the GQR energy
Calculated strength Main peaks energies , comparison with experiment
Experiments
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N-body exact
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Functional integral and stochastic quantum mechanics
Given a Hamiltonianand an initial State
Write H into a quadratic form
Use the HubbardStratonovich transformation
Interpretation of the integral in terms of quantum jumpsand stochastic Schrödinger equation
t time
Example of application: -Quantum Monte-Carlo Methods -Shell Model Monte-Carlo ...
General strategy S. Levit, PRCC21 (1980) 1594.S.E.Koonin, D.J.Dean, K.Langanke, Ann.Rev.Nucl.Part.Sci. 47, 463 (1997).
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Carusotto, Y. Castin and J. Dalibard, PRA63 (2001).O. Juillet and Ph. Chomaz, PRL 88 (2002)
Recent developments based on mean-field
Nuclear Hamiltonian applied to Slater determinant
Self-consistent one-body part
Residual partreformulated 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
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Stochastic evolution of non-orthogonal Slater determinant dyadics :
Quantum jump in one-body density space
Quantum jump in many-body density space
with
Generalization to stochastic motion of density matrix D. Lacroix , Phys. Rev. C71, 064322 (2005).
The state of a correlated system could be described bya superposition of Slater-Determinant dyadic
t time
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Discussion of exact quantum jump approaches
Many-Body Stochastic Schrödinger equation
Stochastic evolutionof many-body density
One-Body Stochastic Schrödinger equation
Stochastic evolutionof one-body density
Generalization : Each time the two-body density evolves as :
with
Then, the evolution of the two-body density can be replaced by an average ( ) of stochastic one-body evolution with :
Actual applications : -Bose-condensate (Carusotto et al, PRA (2001)) -Two and three-level systems (Juillet et al, PRL (2002)) -Spin systems (Lacroix, PRA (2005))
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Perturbative/Exact stochastic evolution
Perturbative Exact
Many-body density
Properties
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
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Summary
One Body space
Stochastic mean-field from statistical assumption
(approximate)
t time
DabDac
Dde
Stochastic mean-field from functional integral
(exact)
Stochastic mean-field in the perturbative
regime
Sub-barrier fusion : Violent collisions :Vibration :
Applications: