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

OutlineOutline

1. Motivations

2. Model

3. Results & Discussion

4. Summary

Phase transition in finite nuclear system

MOTIVATIONSMOTIVATIONS

Anomalous increase of density fluctuation

A rapid increase of chaoticity

Multifragmentation

the main goal of this work is to the main goal of this work is to explore the relation between explore the relation between themthem

?

?

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(the density fluctuation)(Macroscopic thermodynamical)

2

222

)(

)()(

t

tt

(the largest lyapunov exponent )

0

ln1

limd

d

nn

n

Phase space distance between two trajectories at time n

Measurement of chaoticityMeasurement 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:

finite size effects on critical temperature

From calcium to superheavy nuclei

Nuclear fragmentation

Average on ensemble at local time

Excited stateNucleons & Clusters

A given trajectory in the phase space never come back close to the initial

state

MODELMODEL

1. QMD model 1. QMD model The dynamic evolution of an excited nuclei

2. Create an initially excited nucleus2. 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

RESULTS & DISCUSSIONRESULTS & DISCUSSION

a. LLE

In our case

rms is root mean square radius avp is average momentum

Over an ensembleas a function of time

Whose condition is consistent with a hot nucleus at a given temperature

0

ln1

)(Xd

Xd

nt

n

The relation between the chaoticity and density fluctuation

Distance in phase space between two events

At initial state:

0

ln1

)(Xd

Xd

nt

n

evolution with time

fragmentation take place

t~45fm/c208Pb

(t) value at the plateau as the LLE

PRC69, 044609(2004)

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

Time evolution of density fluctuation

b. DENSITY FLUCTUATIONb. DENSITY FLUCTUATION

In QMD model Many-body correlation

At T=11MeV

Abnormal growth and jumps

character time for abnormal

growth~150fm/c

Saturation values

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.

Saturation values of density fluctuation as a function of temperature

The heterogeneity of the phase density

Density fluctuation

Cross term

What relation between the LLE &density fluctuation ????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)

~LLELLE

The LLE increase with the density fluctuation increasing.

The relation between the LLE and the density fluctuation

for finite nuclear system

c. MASS DISTRIBUTION OF MULTIFRAGMENTATION c. MASS DISTRIBUTION OF MULTIFRAGMENTATION

Nucleons,and heavy residues

Nucleons, and light fragments

Distributed over a wild range

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

124Sn 144Nd 197Au 208Pb 226Ra 238U

Tc 10MeV 10MeV 11MeV 11MeV 11MeV 11MeV

m 2.679 2.672 2.696 2.676 2.642 2.700

z 2.514 2.496 2.477 2.453 2.406 2.453

Recently obtained experiment value

5.035.2 z

PRL88(2002) 022701

“critical temperature” as a function of the size of systems

From Ca to superheavy nuclei

Tc increase with the system size

finite size effect on critical temperature

PRC69, 044609(2004)

SUMMARY 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 197Au 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 growth2. 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.