Stochastic methods beyond the independent particle picture

Denis Lacroix IPN-Orsay

Collaboration: S. Ayik, D. Gambacurta, B. Yilmaz, K. Washiyama,G. Scamps

Outline:

Mean-field dynamics: advantages, limitations and recent progress

Stochastic mean-field with initial fluctuations

Two-body effects through quantum jump

Quantum Monte-Carlo approach to the N-body problem

Deterministic approach beyond the single-particle picture

Nuclear Time-dependent Density Functional Theory

two-body

three-body

one-body

Mean-field: (DFT/EDF)

“Simple” Trial state:

Self-consistentMean-field

-Kim, Otsuka, Bonche, J. Phys.G23, (1997).-Nakatsukasa and Yabana, PRC71, (2005).-Maruhn, Reinhard, Stevenson, Stone, Strayer, PRC71 (2005).-Umar and Oberacker, PRC71, (2005).-Simenel, Avez, Int. J. Mod. Phys. E 17, (2008).-Washiyama, Lacroix, PRC78 (2008)

Current status

Typical examples of application

Courtesy C. Simenel

Ion-Ion collisionsCollective motion

System size particles

Interaction (hard-core, spin orbit, tensor, 2, 3-body… )

Recent progress

Relevant space

irrel

evan

t sp

ace

Mean-field

exact

Selection of few degrees of freedom

Find closed equation for relevant DOF

Mean-Field strategy

andActual trend: Inclusion of pairing

Y. Hashimoto and K. Nodeki, arXiv:0707.3083.B. Avez et al Phys. Rev. C 78, 044318 (2008).S. Ebata et al, Phys. Rev. C 82, 034306 (2010).I. Stetcu et al, Phys. Rev. C 84, 051309 (2011).G. Scamps et al, Phys. Rev. C 85, 034328 (2012).G. Scamps et al Phys. Rev. C 87, 014605 (2013).

(see G. Scamps talk)

Missing quantum effects

No tunneling

Pote

ntial

Ene

rgy

Surf

ace

Collective variable

Absence of quantum tunneling in collective spaceMissing quantum fluctuationIn collective space

Wrong dynamics close to a symmetry breaking point

Mean-field stays there

Mean-Field

Underestimation of dissipation

DL, Ayik, Chomaz, Prog. Part and Nucl. Phys. (2004)

Collective motionMean-field

Assié and Lacroix, PRL102 (2009)

Some deterministic Beyond Mean-Field Approach

Relevant space

irrel

evan

t sp

ace

Mean-field

exact

Enlarging the space of relevant DOF

Time Dependent Density MatrixTime Dependent Coupled ClusterIllustration: BBGKY

Difficulty

Some deterministic Beyond Mean-Field Approach

Relevant space

irrel

evan

t sp

ace

Mean-field

exact

Enlarging the space of relevant DOF

Time Dependent Density MatrixTime Dependent Coupled Cluster

Reproject the effect of irrelevant DOF

Extended TDHF

Theories beyond the mean-field

One Body space

<A1>

<A2>

<B>Exact evolution

Mean-field

Miss

ing

info

rmati

on

Y. Abe et al, Phys. Rep. 275 (1996)D. Lacroix et al, Progress in Part. and Nucl. Phys. 52 (2004)

Short time evolution

Approximate long time evolution+Projection

Correlation

with

Propagated initial correlation

Quantum zero point motion

Dissipation

projected two-body effect

MF Fluctuations

Dynamics beyond mean-fieldNon-Markovian effects

with

Non-Markovian master equation

Ave

rag

e p

osi

tio

n

Occupation number evolution

Occ

up

atio

n n

um

ber

s

DL, Chomaz, Ayik, Nucl. Phys. A (1999).

1D

Example: two interacting fermions in 1dimension

Difficulty: memory effect!

Starts to look like TDDMFT with memory

Strategy of stochastic methods tackling the N-body problem

Question: Is it possible to recover some of the quantum mechanics aspects by considering an ensemble of independent mean-field trajectories?

Quantum Monte-Carlo

Stochastic TDHF

Stochastic Mean-Field

Correlations that built up in time Direct NN collisions

Initial fluctuations

All Correlations

D. Lacroix and S. Ayik EPJA Review (in preparation)

Correlations that built-up in time: in medium collisions

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

Reinhard and Suraud, Ann. of Phys. 216 (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

Markovian limit, quantum-diffusion and stochastic Schrödinger Equation

{t t+Dt

Rep

licas

Collision time

Average time between two collisions

Mean-field time-scale

Hypothesis :

Average Density Evolution:

Time-scale and Markovian dynamic

with

Initial simple state

One-body densityMaster equation

step by step

2p-2h nature of the interaction

with

Separability of the interaction

Dissipation contained in Extended TDHF is included

The master equation is a Lindblad equation

Associated SSE Lacroix, PRC73 (2006)

Dissipation: link between Extended TDHF and Lindblad Eq.

SSE on single-particle state :

with

time (arb. units)

wid

th o

f the

co

nden

sate

mean-field

average evolution

Condensate size

N-body density:

1D bose condensate with gaussian two-body interaction

The numerical effort is fixed by the number of Ak

r

r(r)

(a

rb. u

nits

)

t=0t>0

mean-field

average evolution

Density e

volutio

n

Application to Bose-Einstein condensates

Correlations that are here initially and propagates

Collective space

Strategy to construct a stochastic mean-field theory

Ayik, Phys. Lett. B 658, (2008).

MF

Collective phase-space Quantum fluctuations

The dynamics is described by a set of mean-field

evolutions with random initial conditions

Mean-Field theory at all time

Stochastic Mean-Field

at all time

Constraint:

The stochastic mean-field (SMF) concept applied to many-body problem

Ayik, Phys. Lett. B 658, (2008).

MF

Collective phase-space Quantum fluctuations

The dynamics is described by a set of mean-field

evolutions with random initial conditions

The average properties of initial sampling should identify with properties of the mean-field.

SMF in density matrix space

SMF in collective space

Description of large amplitude collective motion with SMFThe case of spontaneous symmetry breaking

Lipkin Model

e

See for instance : Ring and Schuck book Severyukhin, Bender, Heenen, PRC74 (2006)

p=1 p=2 … p=N

N=40 particlesExact dynamics

Mean-fieldis stationary

Description of large amplitude collective motion with SMFThe stochastic mean-field solution

Initial condition

One-body observables

ExactSMF

Lacroix, Ayik, Yilmaz, PRC 85 (2012)

Formulation in quasi-spin space

Description of large amplitude collective motion with SMFThe stochastic mean-field solution

Formulation in quasi-spin space

Initial condition

Fluctuations

Lacroix, Ayik, Yilmaz, PRC 85 (2012)

The stochastic mean-field (SMF) with pairing

Following the general strategy:

TDHFB with initial fluctuations

Setting the initial fluctuations

Quasi-particle occupation

Lacroix, Gambacurta, Ayik, PRC 87 (2013)

Requires to fix:

and

Illustration with the pairing Hamiltonian

Quasi-spin operators

: degeneracy

Occupation probabilities

Pair creation/annihiliationoperators

TDHFB equations

Illustration with the pairing Hamiltonian

=2

Occupation number evolution and dissipation

Exact

SMF

Illustration with the pairing Hamiltonian

Departure from the independent Particle picture

Quantum Fluctuations

weak coupling

strong coupling

Comparison between deterministic and stochastic methods

Strong coupling

Exact

Stochastic MF

Evolution of both One- and two-body densities

SMF might even be better than BBGKY hierarchy truncation

Open systems

One Body space

<A1>

<A2>

<B>

Exact evolution

Mean-field

Mis

sing

in

form

atio

n

Brownian motion

N-body

Towards Exact stochastic methods for N-body and Open systems

Environment

System(one-body)

(others)Environment

System

Exact stochastic formulation

More insight in mean-field dynamics:

Exact state Trial states

{The approximate evolution is obtained by minimizing the action:

Included part: average evolution

exact Ehrenfestevolution

Missing part: correlations

Environment

System

Complexself-interacting

System

Hamiltonian splitting

SystemEnvironment

One Body space

<A1>

<A2>

<B>

Exact evolution

Mean-field

Mis

sing

in

form

atio

n

Relevant degrees of freedom

The idea is now to treat the missing informationas the Environment for the Relevant part (System)

Mean-field from variationnal principle

<A1>

Exact evolution

<A2>

<A1A 2

>- <

A 1A 2

> MF

with

D. Lacroix, Ann. of Phys. 322 (2007).

…Mean-field

Mean-field level

Mean-field + noise

Theorem :One can always find a stochastic process for trial states such thatevolves exactly over a short time scale.

Valid for

orIn practice

Existence theorem : Optimal stochastic path from observable evolution

t>0Mean-field evolution:

x

t>0

Reduction of the information:I want to simulate the expansion with Gaussian wave-function having fixed widths.

t=0

with

Relevant/Missing information:

Relevant degrees of freedom

Missing information

Trial states

Coherent states

illustration: simulation of the free wave spreading with “quasi-classical states”

Stochastic c-number evolution from Ehrenfest theorem

Densities

with

Nature of the stochastic mechanics

with

the quantum wave spreading can be simulated by a classical brownian motion in the complex plane

x

x

tim

e

x

fluctuationsmean values

Guess of the SSE from the existence theorem

The method is general.the SSE are deduced easily

Ehrenfest theorem BBGKY hierarchy

D. Lacroix, Ann. Phys. 322 (2007)

Starting point:

with

Observables

Fluctuations

with

Stochastic one-body evolution

The mean-field appears naturally and the interpretation is easier

extension to Stochastic TDHFB DL, arXiv nucl-th 0605033

Occ

upati

on p

robabili

ty

time

two-level systemBosons

but…

the numerical effort can be reduced by reducing the number of observables

unstable trajectories

SSE for Many-Body Fermions and bosons

Stochastic MF

FluctuationDissipation

Stochastic TDHF

FluctuationDissipation

Exact QD

Everything

Mean-field

FluctuationDissipation

variational QD

Partially everything

Numerical issues

FlexibleFlexible Fixed Fixed

Approximate evolution

Summary, stochastic methods for Many-Body Fermionic and bosonic systems

Numericalinstabilities