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)
xb
saddle
x0
ground state
xsc
exit
2


1
x
2
A metastable potential: U ( x )  m0 x1  2 
2
lb 

What is an anomalous massive system?
memory effect, underdamped
(i) The generalized Langevin equation
t
mx(t )    (t  s ) x ( s )ds  U ' ( x )   (t ),
0
 (t )  0,  (t ) ( s )  k BT (t  s ).
J. D. Bao, Y. Z. Zhuo: Phys. Rev. Lett. 91, 138104 (2003).

J
(

)


Here we consider non-Ohmic model（
）
(ii) the fractional Langevin equation
mx(t )  m 
  1
 (t )  U ' ( x )   (t )
x
 1
t
Jing-Dong Bao, Yi-Zhong Zhuo: Phys. Rev. C 67, 064606 (2003).
(iii) Fractional Fokker-Planck equation
2


P( x, t )


1
U ' ( x) P( x, t )    2 P( x, t ) 
 0 Dt  
t
x
x


这里0 Dt1 是一个 分数导数，即黎曼积分
x
1

 1
D
f
(
x
)

(
x

y
)
f ( y )dy,
a
x

( ) a
( x  a)
Jing-Dong Bao: Europhys. Lett. 67, 1050 (2004).
Jing-Dong Bao: J. Stat. Phys. 114, 503 (2004).
Fractional
Brownian
motion
Normal Brownian
motion
Jing-Dong Bao, Yan Zhou: Phys. Rev. Lett. 94, 188901 (2005).
•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
1
U ( x )   mb2 x 2
2
W ( x, t ) 
 x (t )  x (t )
1
exp  
2


2
2  x (t )
x (t )


2




t


2
x (t )  1  b    (t ' )dt ' x0    (t )v0
0


t
t1
0
0
 x2 (t )  2mk B T  dt1  dt 2   (t  t1 )  (t  t 2 ) (t1  t 2 )
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
~ sin 
  (t ) 



0
r  exp( tr)dr
 R (t )
2
2 2

~
~
    r  2  cos 
with   r 2  b2 and
 exp( a M t )
 2a  ~ a  1 ,
 M
 M
a1  a 2 1 exp( a1t )  exp( a 2 t ),
R (t )  
 exp( a M t )
 (t ),

1
~
 2a M    a M

(t ),
1    2；
  1;
0    1
0    0
Where
 is the anomalous fractional constant ;
a M is the largest positive root of the equation
s 2  ~ s    2  0

b
The effective friction constant is written as
1
1
~
     r sin ( 2)
The passing probability (fusion probability)
over the saddle point is defined by

Ppass ( x0 , v0 , t )   W ( x, t )dx
0

1
 erfc  
2



2 x (t ) 
x (t )
It is also called the characteristic function  ( x0 , v0 , t )
1.2
normal diffusion
(a) v0=6.0
(x0,v0,t)
1.0
subdiffusion
0.8
0.6
Passing
0.4
Probability
0.2
0.0
0
1
2
3
4
5
6
7
8
9
10
t
0.5
(b) v0=3.0
(x0,v0,t)
0.4
0.3
0.2
0.1
0.0
0
2
4
6
8
10
3. Overshooting and backflow
j (t ) 
dPpass ( xb , v0 , t )
dt
exp(  z 2 (t ))

2  x (t )
 ( t  t )



where z (t ) 
1
t1
t
 2
2
mk
Tz
(
t
)
B
 (t )v 
dt1  dt 2 (t1  t 2 )
b   (t ) x0  

0


(
t
)
x
0
0


 ( t  t 2 )    ( t  t1 ) ( t  t 2 ) 

x (t )
2 x (t )
* For instance, quasi-fission mechanism
J. D. Bao, P. Hanggi, to be appeared in Phys. Rev. Lett. (2005)
35
(b)
30
25
j(t)
20
15
10
5
0
-5
0.1
t
1
v0
10
(a)
8
<x(t)>
6
x(t)
4
2
0
-2
0
2
4
t
6
8
10
Pfus
Pfus
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
100
100
0.2
(a) Mo+ Mo
0.1
0.0
180 190 200 210 220 230 240 250 260
Ecm/MeV
Pfus
Pfus
86
123
(b) Kr+ Sb
210
220
230
Ecm/MeV
Ecm/MeV
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
96
124
0.2
(c)
Zr+
Sn
0.1
0.0
210 220 230 240 250 260 270 280
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
200
240
250
1.1
1.0
1D
0.9
0.8
0.7
0.6
2D
0.5
0.4
0.3
100
110
0.2
(d) Mo+ Pd
0.1
0.0
210 220 230 240 250 260 270 280
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
t  ttr ;
N ( x  0, t )  Ppass (t ),
Psur (t ) 

N0
 Ppass (ttr ) exp(  rk t ), t  ttr .
0
x
0.5
(a) v0=0.3
Psur(t)
0.4
0.3
0.2
0.1
0.0
-1
10
0
1
10
2
10
3
10
t
10
1.0
(b) v0=2.0
Psur(t)
0.8
0.6
0.4
J.D. Bao et. al., to be
appeared in PRE
(2005).
0.2
0.0
-1
10
0
10
1
t
10
2
10
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 !