Time-dependent picture for trapping of an anomalous massive system
into a metastable well
Jing-Dong Bao ([email protected])
Department of Physics, Beijing Normal University
2005. 8. 19 – 21 Beijing
1. The scale theory
2. Barrier passage dynamics
3. Overshooting and backflow
4. Survival probability in a
metastable well
1. The model (anomalous diffusion)
ground state
saddle
exit0x
bx
scx
A metastable potential:
2
220 1
2
1)(
bl
xxmxU
What is an anomalous massive system?
(i) The generalized Langevin equation
Here we consider non-Ohmic model （ ）
(ii) the fractional Langevin equation
)()(')()(1
1
txUtxt
mtxm
).()()(,0)(
),()(')()()(0
stTkstt
txUdssxsttxm
B
t
)(J
Jing-Dong Bao, Yi-Zhong Zhuo: Phys. Rev. C 67, 064606 (2003).
J. D. Bao, Y. Z. Zhuo: Phys. Rev. Lett. 91, 138104 (2003).
memory effect, underdamped
),(),()('),(
2
21
0 txPx
txPxUx
Dt
txPt
(iii) Fractional Fokker-Planck equation
这里 是一个
x
a
xa axdyyfyxxfD )(,)()()(
1)( 1
分数导数，即黎曼积分10 tD
Jing-Dong Bao: Europhys. Lett. 67, 1050 (2004).Jing-Dong Bao: J. Stat. Phys. 114, 503 (2004).
Jing-Dong Bao, Yan Zhou: Phys. Rev. Lett. 94, 188901 (2005).
Normal Brownian motion
Fractional
Brownian motion
•Here we add an inverse and anomalous Kramers problem and report some analytical results, i.e., a particle with an initial velocity passing over a saddle point, trapping in the metastable well and then escape out the barrier.
•The potential applications:
(a) Fusion-fission of massive nuclei;
(b) Collision of molecular systems;
(c) Atomic clusters;
(d) Stability of metastable state, etc.
•The scale theory
(1) At beginning time: the potential is approximated to be an inverse harmonic potential, i.e., a linear GLE;
(2) In the scale region (descent from saddle point to ground state) , the noise is neglected, i.e., a deterministic equation;
(3) Finally, the escape region, the potential around the ground state and saddle point are considered to be two linking harmonic potentials, (also linear GLE).
2. Barrier passage process
)()()(2)(
)(')'(1)(
)(2
)()(exp
)(2
1),(
2
1)(
2121
0
2
0
12
00
0
2
2
2
22
1
ttttttdtdtTmkt
vtxdtttx
t
txtx
ttxW
xmxU
tt
Bx
t
b
xx
b
J. D. Bao, D. Boilley, Nucl. Phys. A 707, 47 (2002).D. Boilley, Y. Abe, J. D. Bao, Eur. Phys. J. A 18, 627 (2004).
The response function is given by
0
01
211
21
1
22
0222
0),(
1),(~2
)exp(
;1,)exp()exp(
21,~2
)exp(
)(
andwith
)(cos~2~
)exp(sin~)(
t
taa
ta
tataaa
aa
ta
tR
r
tRr
drtrrt
MM
M
MM
M
b
；
Where is the anomalous fractional constant ;
0~s
equation theofroot positivelargesttheis22 b
M
s
a
The effective friction constant is written as
)2(sin~ 11
r
The passing probability (fusion probability) over the saddle point is defined by
)(2
)(erfc
2
1
),(),,(000pass
t
tx
dxtxWtvxP
x
It is also called the characteristic function ),,( 00 tvx
0 1 2 3 4 5 6 7 8 9 100.0
0.2
0.4
0.6
0.8
1.0
1.2
(x 0,v
0,t)
t
(a) v0=6.0
0 2 4 6 8 100.0
0.1
0.2
0.3
0.4
0.5
(x 0,v
0,t)
t
(b) v0=3.0
subdiffusion
normal diffusion
Passing
Probability
3. Overshooting and backflow
J. D. Bao, P. Hanggi, to be appeared in Phys. Rev. Lett. (2005)
)(2
)()(where
)()()()(
)()(
)(2)()(
)(2
))(exp(
),,()(
2121
21
0
2
0
1002
2
0pass
1
t
txtz
tttttttt
ttdtdtt
tTzmkvtxt
t
tz
dt
tvxdPtj
x
tt
x
Bb
x
b
* For instance, quasi-fission mechanism
0.1 1-5
0
5
10
15
20
25
30
35
j(t)
t
(b)
0v
0 2 4 6 8 10-2
0
2
4
6
8
10(a)
x(t)
<x(t
)>
t
180 190 200 210 220 230 240 250 2600.00.10.20.30.40.50.60.70.80.91.01.1
Pfu
s
Ecm
/MeV
(a) 100Mo+100Mo
200 210 220 230 240 2500.00.10.20.30.40.50.60.70.80.91.01.1
Pfu
s
Ecm
/MeV
(b) 86Kr+123Sb
210 220 230 240 250 260 270 2800.00.10.20.30.40.50.60.70.80.91.01.1
Pfu
s
Ecm
/MeV
(c) 96Zr+124Sn
210 220 230 240 250 260 270 2800.00.10.20.30.40.50.60.70.80.91.01.1
(d) 100Mo+110Pd
1D
2D
Pfu
s
Ecm
/MeV
4. Survival probability in a metastable well
We use Langevin Monte Carlo method to simulate the complete process of trapping of a particle into a metastable well
.),exp()(
;),(),0()(
0sur
trktrpass
trpass
tttrtP
tttP
N
txNtP
0x
10-1 100 101 102 103
0.0
0.1
0.2
0.3
0.4
0.5
(a) v0=0.3
Ps
ur(t
)
t
10-1 100 101 102
0.0
0.2
0.4
0.6
0.8
1.0
(b) v0=2.0
Psu
r(t)
t
J.D. Bao et. al., to be appeared in PRE (2005).
Summary
1. The passage barrier is a slow process, which can be described by a subdiffusion;
2. When a system has passed the saddle point, anomalous diffusion makes a part of the distribution back out the barrier again, a negative current is formed;
3. Thermal fluctuation helps the system pass over the saddle point, but it is harmful to the survival of the system in the metastable well.
Thank you !
