1. Connection between THE LARGEST LYAPUNOV EXPONENT,DENSITIY FLUCTUATION AND MULTIFRAGMENTATION in EXCITED NUCLEAR SYSTEMS Yingxun Zhang (CIAE) Xizhen Wu (CIAE), Zhuxia Li (CIAE) CCAST, Beijing, 2005.8.20

2. Outline 1. Motivations 2. Model 3. Results & Discussion 4. Summary

3. Phase transition in finite nuclear system MOTIVATIONS Anomalous increase of density fluctuation A rapid increase of chaoticity Multifragmentation the main goal of this work is to explore the relation between them ? ? ?

4. (the density fluctuation) (Macroscopic thermodynamical) (the largest lyapunov exponent ) Phase space distance between two trajectories at time n  Measurement of chaoticity a given trajectory in phase space come back close to the initial state of system Average along an infinite trajectory In general case:

5. finite size effects on critical temperature From calcium to superheavy nuclei Nuclear fragmentation Average on ensemble at local time Excited state Nucleons & Clusters A given trajectory in the phase space never come back close to the initial state

6. MODEL 1. QMD model The dynamic evolution of an excited nuclei 2. Create an initially excited nucleus a). RMF The nuclei in ground states b). Density distribution Each nucleon position & momnetum c). Resampled the momentum T T initial temperature Physics picture The latter stage of the heavy ion collisions

7. RESULTS & DISCUSSION a. LLE In our case rms is root mean square radius avp is average momentum Over an ensemble as a function of time Whose condition is consistent with a hot nucleus at a given temperature The relation between the chaoticity and density fluctuation Distance in phase space between two events At initial state:

8. evolution with time fragmentation take place  t~45fm/c 208 Pb (t) value at the plateau as the LLE PRC69, 044609(2004)

9. LLE as a function of temperature “Critical temperature” The raising branch Due to increase of fluctuation with temperature The descent branch System breaks up very soon and collective expansion

10. Time evolution of density fluctuation b. DENSITY FLUCTUATION In QMD modelMany-body correlation At T=11MeV Abnormal growth and jumps character time for abnormal growth~150fm/c Saturation values

11. The character time for abnormal density fluctuation growth ~150fm/c The inverse LLE ~ 40 fm/c > there is enough time to develop chaotic dynamics during the process of fragment formation the abnormal density fluctuation deterministic chaos Small uncertainty in the initial condition Produce a large dynamical fluctuation in final observances.

12. Saturation values of density fluctuation as a function of temperature

13. The heterogeneity of the phase density Density fluctuation Cross term What relation between the LLE &density fluctuation ???? Momentum distribution fluctuation J.P.Eckmann, Rev.Mod.Phys. 57,617(1985), Y.Gu,Phys.Lett.A 149,95(1990) LLE ~LLE

14. The LLE increase with the density fluctuation increasing. The relation between the LLE and the density fluctuation for finite nuclear system

15. c. MASS DISTRIBUTION OF MULTIFRAGMENTATION Nucleons,and heavy residues Nucleons, and light fragments Distributed over a wild range

16. At “critical temperature”, Power-law Fisher’s model of liquid-gas phase transition a drop with size A in the vapor For 208Pb, T=11MeV 124 Sn 144 Nd 197 Au 208 Pb 226 Ra 238 U Tc10MeV 11MeV mm 2.6792.6722.6962.6762.6422.700 zz 2.5142.4962.4772.4532.4062.453 Recently obtained experiment value PRL88(2002) 022701

17. “critical temperature” as a function of the size of systems From Ca to superheavy nuclei T c increase with the system size finite size effect on critical temperature PRC69, 044609(2004)

18. SUMMARY 3. The LLE peaks at the same temperature where the density fluctuation grows abnormally and the mass distribution of fragments is fitted well with the Fisher’s power law 4. The critical temperatures increase with system mass, after 197 Au it seems to reach a saturation value of about T=11MeV 1. At critical temperature, there appears a plateau in the time evolution of LLE and the density fluctuation show an abnormal growth 2. The time scale of the density fluctuation is much longer than the inverse largest Lyapunov exponent, which means that the chaotic motion can be well developed during the process of fragment formation.