Chen Yaming, Just Wolfram
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Feb;89(2):022103. doi: 10.1103/PhysRevE.89.022103. Epub 2014 Feb 6.
We provide an analytic solution to the first-passage time (FPT) problem of a piecewise-smooth stochastic model, namely Brownian motion with dry friction, using two different but closely related approaches which are based on eigenfunction decompositions on the one hand and on the backward Kolmogorov equation on the other. For the simple case containing only dry friction, a phase-transition phenomenon in the spectrum is found which relates to the position of the exit point, and which affects the tail of the FPT distribution. For the model containing as well a driving force and viscous friction the impact of the corresponding stick-slip transition and of the transition to ballistic exit is evaluated quantitatively. The proposed model is one of the very few cases where FPT properties are accessible by analytical means.
我们使用两种不同但密切相关的方法，为一个分段光滑随机模型，即具有干摩擦的布朗运动的首次通过时间（FPT）问题提供了一个解析解。一方面，这两种方法基于本征函数分解；另一方面，基于反向柯尔莫哥洛夫方程。对于仅包含干摩擦的简单情况，我们发现了频谱中的一个相变现象，它与出口点的位置有关，并影响FPT分布的尾部。对于同时包含驱动力和粘性摩擦的模型，定量评估了相应的粘滑转变和弹道出口转变的影响。所提出的模型是极少数可以通过解析方法获得FPT性质的情况之一。
