connection between the largest lyapunov exponent,densitiy fluctuation and multifragmentation in...
TRANSCRIPT
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.