Asymptotic Analytical Solutions Of First-Passage Rate

The asymptotic analytical methods of the first-passage problem about the stochastic vibration system have been studied. Using the common numerical method to solve the problem of single molecular bond rupture based on a existing model for the stochastic dynamics of a single molecular bond and the theory of first-passage problem on diffusion process, the backward Kolmogorov equation with respect to the conditional reliability function for rupture of a single molecule bond and the Pontryagin equation with respect to the mean first-passage time are derived. Solving these two equations by numerical method yields the bond survival reliability and the rate of bond rupture. In comparison with the existing method, the results of this method can be used to describe the probability of the single molecular bond rupture. Meanwhile, it is pointed out that the results obtained by the numerical method are dependent on the initial condition. In the present paper, a new technological method about acquiring the asymptotic analytical solution of the first-passage rate based on the averaged Ito stochastic differential equation for quasi non-integrable Hamiltonian system is also proposed. By using the probability current equation and the Laplace integral method, the asymptotic analytical expressions for the first-passage rate are obtained in the case of high passage threshold or long enough passage time. The conditional reliability, the conditional probability density of the first-passage time and the mean first-passage time can also be obtained analytically. As an illustrative example, the random bistable oscillator is considered. The obtained analytical solutions are consistent with the Kramers formula in high passage threshold, and they agree well with simulation results in a more wide range. A coupled two-degree-of-freedom nonlinear oscillator subjected to stochastic excitations is also studied to illustrate the procedure of acquiring the asymptotic analytical solution. Comparison between the results obtained from the analytical solution and those from numerical simulation of original system verifies the effectiveness of the proposed method.