Shotorban B
Department of Mechanical and Aerospace Engineering, The University of Alabama in Huntsville, Huntsville, Alabama 35899, USA.
Phys Rev E. 2020 Jan;101(1-1):012113. doi: 10.1103/PhysRevE.101.012113.
An approach was developed to describe the first passage time (FPT) in multistep stochastic processes with discrete states governed by a master equation (ME). The approach is an extension of the totally absorbing boundary approach given for calculation of FPT in one-step processes [N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier Science Publishers, North Holland, Amsterdam, 2007)] to include multistep processes where jumps are not restricted to adjacent sites. In addition, a Fokker-Planck equation (FPE) was derived from the multistep ME, assuming the continuity of the state variable. The developed approach and an FPE based approach [C. W. Gardiner, Handbook of Stochastic Methods, 3rd ed. (Springer-Verlag, New York, 2004)] were used to find the mean first passage time (MFPT) of the transition between the negative and positive stable macrostates of dust grain charge when the charging process was bistable. The dust was in a plasma and charged by collecting ions and electrons, and emitting secondary electrons. The MFPTs for the transitioning of grain charge from one macrostate to the other were calculated by the two approaches for a range of grain sizes. Both approaches produced very similar results for the same grain except for when it was very small. The difference between MFPTs of two approaches for very small grains was attributed to the failure of the charge continuity assumption in the FPE description. For a given grain, the MFPT for a transition from the negative macrostate to the positive one was substantially larger than that for a transition in a reverse order. The normalized MFPT for a transition from the positive to the negative macrostate showed little sensitivity to the grain radius. For a reverse transition, with the increase of the grain radius, it dropped first and then increased. The probability density function of FPT was substantially wider for a transition from the positive to the negative macrostate, as compared to a reverse transition.
开发了一种方法来描述由主方程（ME）控制的具有离散状态的多步随机过程中的首次通过时间（FPT）。该方法是对用于计算一步过程中FPT的全吸收边界方法的扩展[N.G.范坎彭，《物理和化学中的随机过程》（爱思唯尔科学出版社，北荷兰，阿姆斯特丹，2007年）]，以包括跳跃不限于相邻位点的多步过程。此外，假设状态变量连续，从多步ME推导出了福克 - 普朗克方程（FPE）。当充电过程为双稳时，使用所开发的方法和基于FPE的方法[C.W.加德纳，《随机方法手册》，第3版（施普林格出版社，纽约，2004年）]来求尘埃颗粒电荷在负和正稳定宏观态之间转变的平均首次通过时间（MFPT）。尘埃处于等离子体中，通过收集离子和电子以及发射二次电子来充电。对于一系列颗粒尺寸，通过这两种方法计算了颗粒电荷从一个宏观态转变到另一个宏观态的MFPT。除了颗粒非常小时，两种方法对于相同颗粒产生的结果非常相似。两种方法对于非常小的颗粒的MFPT差异归因于FPE描述中电荷连续性假设的失效。对于给定的颗粒，从负宏观态到正宏观态转变的MFPT显著大于相反顺序转变的MFPT。从正到负宏观态转变的归一化MFPT对颗粒半径的敏感性很小。对于反向转变，随着颗粒半径的增加，它先下降然后上升。与反向转变相比，从正到负宏观态转变的FPT概率密度函数要宽得多。
