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Extensions of mean-field with stochastic methods Denis Lacroix Laboratoire de Physique Corpusculaire - Caen, FRANCE Mapping the nuclear N-body dynamics.

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Presentation on theme: "Extensions of mean-field with stochastic methods Denis Lacroix Laboratoire de Physique Corpusculaire - Caen, FRANCE Mapping the nuclear N-body dynamics."— Presentation transcript:

1 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. Quantum jump approach to the many-body problem One Body space TDHF and beyond … -Saclay 2006 Stochastic one-body mechanics applied to nuclear physics

2 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

3 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) {

4 The initial correlations could be treated as a stochastic operator : where { Link with semiclassical approaches in Heavy-Ion collisions time Vlasov BUU, BNV Boltzmann- Langevin Adapted from J. Randrup et al, NPA538 (92). Molecular chaos assumption

5 Application to small amplitude motion Standard RPA states Coupling to ph-phonon Coupling to 2p2h states

6 More insight in the fragmentation of the GQR of 40 Ca EWSR repartition

7 Intermezzo: wavelet methods for fine structure Observation +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).

8 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?

9 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 density 2 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)

10 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

11 Time-scale and Markovian dynamics { t t+  t Replicas 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)

12 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

13 Nature of the Stochastic one-body dynamics Important properties  remains a projector Numerical implementation : flexible and rather simple. time Average evolution One-body Correlations beyond mean-field, denoting by similar to Ayik and Abe,PRC 64,024609 (2001). At all time with

14 Application Root mean-square radius evolution: rms (fm) time (fm/c) TDHF Average evol. Stoch. Schrödinger Equation (SSE) on single-particle states: Assumingand All the information on the system is contained in the one-body density t<0 Residual part : Mean-field part : Application : 40 Ca nucleus = 0.25 MeV.fm -2 Monopole vibration in nuclei Associated quantum jumps on single particle states:

15 Diffusion of the rms around the mean value Standard deviation No constraint Compression Dilatation = 0.25 MeV.fm -2 Similar to Nelson quantization theory 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 !!!

16 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

17 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 to simulate this equation The equation can be interpreted as the feedback action of the O n operators on the one-body density

18 SSE on single-particle state : with time (arb. units) width of the condensate 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 A k r  (r) (arb. units) t=0 t>0 mean-field average evolution Density evolution

19 Summary Quantum Jump (QJ) methods to extend mean-field Simplified QJ Fluctuation Dissipation Generalized QJ Fluctuation Dissipation Exact QJ Everything Mean-field Fluctuation Dissipation Variational QJ Partially everything Numerical issues Flexible Fixed O. Juillet (2005)

20 Giant resonances

21 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 :

22 { Incoherent nucleon-nucleon collision term. Coherent collision term Evolution of the average density : One Body space Fluctuations around the mean density : Average ensemble evolutions

23 Linear response Mean-field Extended mean-field Response to harmonic vibrations Notations for RPA equations Using + Mean-field Extended mean-field

24 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). Coupling to 2p-2h states Coupling to ph-phonon states

25 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

26 Effect of correlation on the GMR and incompressibility Incompressibility in finite system in 208 Pb { Evolution of the main peak energy :

27 Systematic improvement of the GQR energy Calculated strengthMain peaks energies, comparison with experiment Experiments

28 N-body exact

29 Functional integral and stochastic quantum mechanics 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 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).

30 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 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

31 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 by a superposition of Slater-Determinant dyadic time

32 Discussion of exact quantum jump approaches Many-Body Stochastic Schrödinger equation Stochastic evolution of many-body density One-Body Stochastic Schrödinger equation Stochastic evolution of 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))

33 Perturbative/Exact stochastic evolution Perturbative Exact Many-body density Properties Many-body density Projector Number of particles Entropy Average evolution One-body Correlations beyond mean-field Numerical implementation : Flexible: one stoch. Number or more… Fixed : “s” determines the number of stoch. variables

34 Summary One Body space Stochastic mean-field from statistical assumption (approximate) time D ab D ac D de Stochastic mean-field from functional integral (exact) Stochastic mean-field in the perturbative regime Sub-barrier fusion : Violent collisions : Vibration : Applications:


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