Theme de travail: DYNAMICAL CHAOS AND NUCLEAR FISSION

1. Yu.L.Bolotin, I.V.Krivoshei, Sov.J. of Nucl.Phys.(Yad. Fiz). 42, 53 (1985)DYNAMICAL CHAOS AND NUCLEAR FISSION Theme de travail: DYNAMICAL CHAOS AND NUCLEAR FISSION

2. REGULAR AND CHAOTIC CLASSIC AND QUANTUM DYNAMICS IN (2D) MULTI-WELL POTENTIALS YU.L.Bolotin, NSC KhFTI, Kharkov, Ukraine

3. Hamiltonian system with multi-well potential energy surface (PES) represents a realistic model, describing the dynamics of transition between different equilibrium states, including such important cases as chemical and nuclear reactions, nuclear fission, string landscape and phase transitions). One-well potential – rare exception Multi-well – common case Such system represents an important object, both for the study of classic chaos and quantum manifestations of classical stochasticity.

4. Research of any nonlinear system (in the context of chaos) includes the following steps • 1. Investigation of the classical phase space, detection of chaotic regimes • 2. Analytical estimation of the critical energy transition to chaos. • 3. Test for quantum manifestation of classical stochasticity • 4. Action of chaos on concrete physical effects. The basic subject of the current report is to realize the outlined program for two-dimensional multi-well Hamiltonian systems (of course, only in part)

5. I. Classical dynamics • SPECIFICS OF CLASSICAL DYNAMICS IN MULTI-WELL POTENTIALS —MIXED STATE

6. Quadrupole oscillations of nuclei

8. Why is such potential chosen? Authorities love order, but very don’t love chaos. My bosses this dislike briefly formulated as follows: Chaos could be studied only by those, who have nothing to do. We felt himself as partisan. In this distant time we lived in era Henon-Heiles potential. I remembered that this potential saw somewhere (W.Greiner book). We proudly went out from an undergroundand

9. The surfacesofpotentialenergyofKryptonisotopes.

10. Full list of «Our» potentials We worked also and with other potentials, but nothing substantially new (as compared to these) did not discover there

13. What is the mixed state? Yu.L.Bolotin, V.Yu.Gonchar, E.V.Inopin Chaos and catastrophes in quadrupole oscillations of nuclei, Yad.Fiz. 45, 350, 1987 (20 anniversary ) One-well case – Poincare section (nothing unusual!) As the energy increase the gradual transition from the regular motion to chaotic one is observed.

14. Change of the character of motion in left and right local minima is essentially different! It means that in this case so-called MIXED STATE may be observed: at one and the same energy in different local minima various dynamical regimes (regular or chaotic) are realized

15. Mixed state is common property of multi-well potentials

16. Why the dynamical behavior is so unlike in the different local minima: why in some local minima chaos begins below the saddle energy, but in others only above. If we want answer this question, we must use different criteria of chaos. It is a very complicated problem, separate question, and we do not have time for the detailed discussion. If there will be time at last, we will discuss some details.

17. We used: • Negative curvature criterion (Toda) • Geometrical approach (Pettini et al.) • 3. Overlap of nonlinear resonances (Chirikov) • 4. Destruction of stochastic layer (Delande et al). …. • and many others

18. Result: we can find critical energy of transition to, but, we can’t forecast specificity of behavior in arbitrary local minimum using only geometrical terms (for example, number of saddle, negative curvature etc.)

19. Regular-Chaos-Regular transition R-C-R transition is a possible only for the system with localized domain of instability (negative Gaussian curvature or overlap of nonlinear resonances) QO potential K<0 Thepartofphasespace S% withchaotic trajectories asafunctionoftheenergy

20. R-C-R TRANSITION IN MULTI-WELL POTENTIAL

21. Reason of the additional C-R transition: new intersection point Yu.L.Bolotin, V.Yu.Gonchar, M.Ya.Granovsky, Physica D 86 (1995) R-C-R transition in a periodically driven anharmonic oscillator

22. One comment Stochastization ofquadrupolenuclearoscillationsis confirmed by the direct observation of chaotic regimes at simulation of reaction with heavy ions. Umar et al. (1985) TDHF calculation head-on collisions: Poincare section for isoscalar quadrupole mode in

23. II. Quantum chaos Quantum manifestation of classical stochasticity in mixed state. (comparison of one-well and multi-well)

24. SPECTRAL METHOD M.D.Feit, J.A.Fleck, A.Steiger (1982) 1.Calculation of quasiclassical part of the spectrum for multi-well systems requires appropriate numerical methods. 2. Matrix diagonalization method (MDM) is attractive only for one-well potential. In particular, the diagonalization of the QO Hamiltonian with W > 16 in the harmonic oscillator basis requires so large number of the basis functions that go beyond the limits of the our computation power. The spectral method is an attractive alternative to MDM

25. The main instrument of spectral method is correlation function The solution can be accurately generated with the help of the split operator method

28. ANALITICAL METHODS For simplicity, we only will name analytical methods which we used (and plan to use) for description of the mixed state. • A.Auerbach and S.Kivelson (1985): The path decomposition expansion • Path integral technique which allows to break configuration space into • disjoint regions and express dynamics of full system in term of its parts • 2. KazuoTakatsuka ,HiroshiUshiyama,AtsukoInoue-Ushiyama (1998) • Tunnelingpathsinmulti-dimensionalsemiclassicaldynamics

29. Now we have methods of investigations both classical and quantum chaos, but ….. Do we have a research object? Chinese legend Once upon a time there lived Dzhu,Who learned to kill off dragonsAnd gave up all he hadTo master art like that.Three whole years it took,But, alas, never came up that chanceTo present skill and form. So he took on himself teachingothers the art of slaying dragons. Chaos vs. regularity Eternal battle The last two lines belong R. Thom We have a chaotic dragon and even can present some trophy

30. O.Bohigas, M.Giannoni, C.Shmit: (1983) Hypothesis of the universal fluctuations of energy spectra Fluctuation properties of QO spectra Rigid lines are Poisson’s prediction Dashed lines are GOA prediction Qualitative agreement with Bohigas hypothesis

31. Fluctuations of energy spectrum in mixed state Rigid lines are Poisson and Wigner prediction; dashed lines – fitting by Berry-Robnik- Bogomollny distribution (interpolation between Poisson and Wigner distribution) In that case we deal not with statistics of mixture of two spectral series with different NNSD, but with statistics of levels that none of them belongs to well-defined statistics. Statistical properties of such systems were not studied at all up to now, though namely such systems correspond to common situation.

32. Evolution of shell structure in the process R-C-R transition and in mixed state Very old problem (W.Swiateski, S.Bjornholm): how one could reconcile the liquid drop model of the nucleus (short means free path) with the gas-like shell model? To account for such contradiction investigation of shell effect destruction in the process R-C-R transition plays the key role More exact formulation: How do shell dissolve with deviation from regularity, or, conversely, How do incipient shell effects emerge as the system is approached to an integrable situation?

33. We used nonscale version of the Hamiltonian QO Classical prompting Interesting In the interval 0<W<4 for all energies the motion remains regular (in this interval K>0)

34. The destruction of shell structure can be traced, using analog of thermodynamic entropy Regular domain: change of entropy correlates with the transition from shell to shell Chaotic domain: 1. quasiperiodic dependence of entropy from energy is violated; 2. Monotone growth on average towards a plateau corresponding to entropy of random sequence.

35. We obtain this result for QO potential, but it is general result Regularity-chaos transition in any potential is always accompanied destruction of shell structure

36. Quantum chaos and noise Relano et al. 2002: theenergyspectrum fluctuationsofquantumsystemscanbe formallyconsideredasa discretetimeseries.Thepowerspectrumbehaviorofsuchasignalarecharacterizedby Spectral fluctuations described by Power spectrum of a discrete time series

37. Example of chaotic system is nucleus at high excitation energy The average power spectrum of the function for (sd shell) and (very exotic nucleus) using 25 sets from 256 levels for high level density region. Theplotsaredisplacedtoavoidoverlapping.

38. Power spectrum of the function for GDE (Poisson) energy levels compared to GOE,GUE, GSE (Relano et al. 2002)

40. Signature of quantum chaos in wave function structure In analysis of QMCS in the energy spectra the main role was given to statistical characteristic: quantum chaos was treated as property of a group of states In contrast, the choice of a stationary wave function as a basic object of investigation relates quantum chaos to an individual state! Evolution of wave function during R-C-R transition can be studied with help: 1. Distribution on basis. 2. Probability density. 3. Structure of nodal lines.

41. Degree of distribution of wave function Nordholm, Rice (1974) Degree of distribution of wave function arises in the average along with the degree of stochasticity. Yu.L.Bolotin, V.YU.Gonchar….Yud.Fiz. (1995)

42. Isolines of probability density

43. The topography of nodal lines of the stationary wave function. R.M.Stratt, C.N.Handy, W.N.Miller (1974): system of nodal lines of the regular wave function is a lattice of quasiorthogonal curves or is similar to such lattice. The wave function of chaotic states does not have such representation nonseparable, but integrable separable nonintegrable, avoided intersection of nodal lines A.G.Manastra et al. (2003)

44. Mixed state: QMCS in structure of wave function Usual procedure of search for QMCS in wave function implies investigation their structure below and above critical energy Problem: necessity to separate QMCS from modification of wave functions structure due to trivial changes in its quantum numbers The main advantage of our approach: In the mixed state we have possibility to detect QMCS not for different wave function, but for different parts of the one and the same wave function. V.P.Berezovoj, Yu.L.Bolotin, V.A.Cherkaskiy, Phys. Lett A (2004)

45. DecayoftheMixedStates Theescapeof trajectories(particles)fromlocalizedregionsofphaseorconfigurationspacehasbeenanimportanttopicindynamics,becauseitdescribesthedecayphenomenaofmetastablestatesinmany fieldsofphysics,asforexample chemicalandnuclearreactions,atomicionizationandinducednuclear fission.

46. acousto-optic deflectors laser beam horizontal vertical billiard plane 10 KHz 100 KHz Optic Billiard

47. 450 mm 55 mm 55 mm Cs How Do We Observe Chaos in the Wedge ? Stable trajectories do not “feel” the hole Chaotic trajectories leak through the hole

48. Experiment vs Numerical Simulations .4 .3 .2 .1 0 θ (deg) 20 25 30 35 40 45 50 55

49. (W.Bauer, G.F.Bertch, 1990) Exponential decay is a common property expected in strongly chaotic systems For the chaotic systems  exponentialdecay law For the nonchaotic systems  power decay law

50. Numerical experiment on the Sinai billiard