1. FUSTIPEN-GANIL OCTOBER 13, 2014 Quantal Corrections to Mean-Field Dynamics Sakir Ayik Tennessee Tech University Stochastic Mean-Field Approach for Nuclear Dynamics Nucleon Exchange Mechanism in Deep-Inelastic Collisions Spinodal Instabilities in Nuclear Matter, Multi-fragmentation Recent colaborators: D. Lacroix, D. Gambacurta, K. Washiyama O. Yilmaz, B. Yilmaz, A. Gokalp, F. Acar Research supported by DOE since 1988

2. STOCHASTIC MEAN-FIELD APPROACH Standard Mean-Field approximation, many-body wave function is a single Slater Determinant constructed from single-particle wave functions determined by TDHF equations At low energies, E/A < 10 MeV, Pauli blocking is very important. With effective interactions mean-field provides good description for average dynamics including one-body dissipation. However  Collective motions is treated in deterministic manner nearly classical approximation! Fluctuations in collective motion are severely underestimated, and initial symmetries are preserved. Task: Improve transport approach beyond the mean-field by incorporating dynamics of fluctuation mechanism.

3. Probability distribution of an observable  P(Q) Mean-field  Dispersion  Spontaneous and induced fission of hot nucleus (Symmetry breaking)

4. Two different mechanisms for fluctuations: Fluctuations induced by two-body collisions : collisional mechanism can be incorporated into equation of motion in similar manner to Langevin description of Brownian particle  Semi-classical limit: Stochastic Boltzmann-Langevin Approach [Ayik and Gregoire, PL B212 (1988)174] Mean-Field Fluctuations: originating from quantal and thermal fluctuations in the initial state. Dominant mechanism for fluctuations in collective motion at low energies.  Stochastic Mean-Field Approach [Ayik, PL B658 (2008) 174]

5. A Classical Stochastic Model For Fusion [Esbensen et al. PRL 41 (1978) 296]  Relative motion is coupled to harmonic surface vibrations : Key Point  For harmonic modes quantal phase-space distribution evolves according to Vlasov Equation, Therefore, quantal effects enter through zero-point fluctuations in the initial state.

6. We can obtained solutions by generating ensemble of classical trajectories specified by quantal fluctuations in the initial state (double Gaussian in phase-space) Provides very good approximation to quantal couple-channel calculations.

7. Couple-channel calculations of Nobre et al. (NPA 786 (2007) 90) Ayik, Yilmaz, Lacroix, PRC 81(2010)

8. STOCHASTIC MEAN-FIELD  (Quantal Corrections) Generate an ensemble of s.p. density matrices by incorporating density fluctuations (quantal or thermal) in the initial state: Each matrix elements is a Gaussian random number with: Single-particle wave functions are determined by the self-consistent mean-field of each event Observables are calculated as averages over the ensemble.

9. MF BUU BL SMF

10. Demonstrations of SMF Approach Dispersion formula for one-body observables (Connection with Ballian-Veneroni’s approach) Transport description of DIC of Heavy-Ions (Connections with nucleon exchange model and Langevin dynamics of Mori formalism) Spinodal Instabilities in Nuclear Matter O. Yilmaz et al., EJP A 49 (2013) 13 Testing with Lipkin-Meshkov-Glick Model B. Yilmaz, Lacroix, Curebal, submitted

11. NUCLEON EXCHANGE IN DEEP-INELASTIC COLLISIONS Extract transport coefficients for macroscopic variables  mass and charge asymmetry, relative distance-relative momentum… Macro variables can be defined with help of window. Consider nucleon exchange: Nucleon number of target-like fragments 

12. Langevin description : fluctuating flux across the window act as random forces on macro variable Initial work: central collisions in semi-classical approximation. Gaussian random force determined by diffusion coefficient (Markovian, no memory effects) Similar structure with phenomenological nucleon-exchange model, but more refined description for transport coefficients, are not constraint by adiabatic or diabatic approximation. (3D TDHF Code with Sly4d by Bonch et al.) Washiyama, Ayik, Lacroix, Phys. Rev. C 80 (2009) 031602 (R)

14. 0ff- Central Collisions Window can be determined by diagonalizing the mass quadrupole tensor. Principle axis is along the symmetry and window is perpendicular passing trough the lowest density point. B.Yilmaz, S.Ayik, D. Lacroix, O.Yilmaz, PRC 90 (2014)

15. Diffusion coefficient and variance of fragment mass distribution

16. Heated and compressed matter expands, cools down and may enter into spinodal region. Uniform matter unstable, small density fluctuations grow rapidly leading the system to break-up into clusters  Dynamical mechanism for liquid-gas phase transformation. SPINODAL INSTABILITIES IN STOCHASTIC MEAN-FIELD

17. Baryon density correlation function Correlation length provides a measure for size of primary condensation regions Dynamics of liquid-gas phase transformation Yilmaz et al., preparation

18. Spontaneous symmetry breaking: It may be possible to microscopic description of induced fission ?

19. CONCLUSIONS Need tools for investigations of reactions mechanism with radioactive beams in upgraded heavy-ion facilities at FRIB -MSU, SPIRAL2 -GANIL, SPES-LNL-Italy, RIKEN-Japan. Development of Quantal transport models (such as SMF)  powerful tools for describing gross properties nuclear dynamics Dissipation and fluctuation mechanisms in DIC Dynamical description of phase transformations Fusion and induced fission dynamics Lacroix and Ayik, EPJ-A (review section) 50 (2014)