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Time-dependent picture for trapping of an anomalous massive system into a metastable well Jing-Dong Bao (jdbao@bnu.edu.cn) Department of Physics, Beijing Normal University 2005. 8. 19 – 21 Beijing

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1. The scale theory 2. Barrier passage dynamics 3. Overshooting and backflow 4. Survival probability in a metastable well

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

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What is an anomalous massive system? (i) The generalized Langevin equation Here we consider non-Ohmic model （ ） (ii) the fractional Langevin equation 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

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(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).

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

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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.

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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).

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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).

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The response function is given by

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

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

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s ubdiffusion normal diffusion Passing Probability

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

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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 0 x

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

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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.

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Thank you !

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