Escape from the underdamped potential well

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Abstract
IntroductionIn this chapter we discuss a modest refinement of Kramers (1940) result for the noise activated escape out of an underdamped potential well. At very low damping, motion in a potential well (see Figure 3.1) is almost conservative, and the escape up in energy, out of the well, becomes difficult. This was appreciated by Kramers, who derived a result for extremely low damping. For a particle subject to thermal noise and damping proportional to the momentum p with a relaxation rate η, Kramers found an escape rate from the metastable state which is proportional to η. In the low friction limit it is diffusion up along the energy coordinate which limits the escape rate. On the other hand for large friction, the escape out of the potential well is limited by the diffusion along the reaction coordinate. For large friction the escape rate is proportional to η-1. Thus the escape rate as a function of the friction constant peaks at an intermediate value of the damping constant as shown in Figure 3.2. Kramers noted in his paper that it is the smaller of the two rates which applies. At very low damping, as the friction increases, the escape rate must eventually deviate from the linear behavior predicted by Kramers to merge at large η into the heavy damping behavior. It is this departure from the linear behavior for small damping which is the subject of this chapter.

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