1. Time-dependent picture for trapping of an anomalous massive system
into a metastable well
Jing-Dong Bao
Department of Physics, Beijing Normal University
– 21 Beijing

2. The scale theory
Barrier passage dynamics
Overshooting and backflow
Survival probability in a
metastable well

3. 1. The model (anomalous diffusion)
saddle
exit
ground state
A metastable potential:

4. What is an anomalous massive system?
(i) The generalized Langevin equation
Here we consider non-Ohmic model（　　　　）
(ii) the fractional Langevin equation
memory effect, underdamped
J. D. Bao, Y. Z. Zhuo: Phys. Rev. Lett. 91, (2003).
Jing-Dong Bao, Yi-Zhong Zhuo: Phys. Rev. C 67, (2003).

5. (iii) Fractional Fokker-Planck equation
这里　　是一个
　　　　分数导数，即黎曼积分
Jing-Dong Bao: Europhys. Lett. 67, 1050 (2004).
Jing-Dong Bao: J. Stat. Phys. 114, 503 (2004).

6. Jing-Dong Bao, Yan Zhou: Phys. Rev. Lett. 94, 188901 (2005).
Fractional
Brownian motion
Normal Brownian motion
Jing-Dong Bao, Yan Zhou: Phys. Rev. Lett. 94, (2005).

7. The potential applications:
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.

8. 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).

9. 2. Barrier passage process
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).

10. The response function is given by

11. Where is the anomalous fractional constant ;
The effective friction constant is written as

12. The passing probability (fusion probability) over the saddle point is defined by
It is also called the characteristic function

13. normal diffusion
Passing
Probability
subdiffusion

14. J. D. Bao, P. Hanggi, to be appeared in Phys. Rev. Lett. (2005)
3. Overshooting and backflow
* For instance, quasi-fission mechanism
J. D. Bao, P. Hanggi, to be appeared in Phys. Rev. Lett. (2005)

19. 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 x

20. J.D. Bao et. al., to be appeared in PRE (2005).

21. 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.

22. Thank you !