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Stochastic quantum dynamics beyond mean-field.
Denis Lacroix
Laboratoire de Physique Corpusculaire - Caen, FRANCE
Introduction to stochastic TDHF
Application to collective motions
Alternative exact stochastic mechanics
Functional integrals for dynamical
Many-body problems
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 :
f
i
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)
Introduction to stochastic mean-field theories :
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
Vlasov
BUU, BNV
BoltzmannLangevin
Adapted from J. Randrup et al, NPA538 (92).
t
t
time
Average ensemble evolutions
Fluctuations around the mean density :
Evolution of the average density :
{
Incoherent nucleon-nucleon
collision term.
Coherent collision
term
Application to small amplitude motion
Standard RPA states
Coupling
to 2p2h states
Coupling
to ph-phonon
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
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 :
A. Chevchenko et al, PRL93 (2004) 122501.
E
Discussion on approximate quantum stochastic theories
based on statistical assumptions
Success
Results on small amplitude motions looks fine
The semiclassical version (BOB) gives a good reproduction of Heavy-Ion collisions
Critical aspects
Stochastic methods for large amplitude motion are still an open problem
(No guide to the random walk)
Instantaneous reorganization of internal degrees of freedom?
Which interaction for the collision term
Theoretical justification of the introduction of noise
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. C (2005) in press.
The state of a correlated system could be described by
a superposition of Slater-Determinant dyadics
Dab
Dac
Dde
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
Exact stochastic dynamics guiding approximate quantum stochastic mechanics
Weak coupling approximation : perturbative treatment
Residual interaction in the mean-field
interaction picture
Statistical assumption in the quasi-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
{
Average time between two collisions
Hypothesis :
Interpretation in terms of average evolution of quantum jumps :
with
Stochastic
term
From stochastic many-body to stochastic one-body evolution
We need additional simplification
Following approximate dynamics
We focus on one-body degrees of freedom
Following the exact stochastic dynamics
We introduce the density
Gaussian approximation for quantal fluctuations
We obtain a new stochastic one-body evolution in the perturbative regime:
D. Lacroix, in preparation (2005)
Mean-field like term
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
Application to spherical nuclei
t<0
Mean-field part :
r 2
Residual part :
Application :
rms (fm)
Root mean-square radius evolution:
TDHF
Average evol.
40Ca
nucleus
 = 0.25 MeV.fm-2
Lifetime of the determinant:
s0=500
s0=300
s0=200
s0=100
tD=179
tD=260
tD=340
tD=1040
time (fm/c)
time (fm/c)
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 :